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5.76k
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13.6k
| state_after
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13.6k
|
|---|---|---|---|---|---|---|---|---|---|---|
Mathlib/SetTheory/ZFC/Ordinal.lean | [
[
"Mathlib.SetTheory.ZFC.Basic",
"Mathlib/SetTheory/ZFC/Basic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def IsTransitive (x : ZFSet) : Prop :=\n β y β x, y β x",
"end": [
41,
17
],
"full_name": "ZFSet.IsTransitive",
"kind": "commanddeclaration",
"start": [
39,
1
]
},
{
"code": "@[simp]\ntheorem empty_isTransitive : IsTransitive β
",
"end": [
45,
85
],
"full_name": "ZFSet.empty_isTransitive",
"kind": "commanddeclaration",
"start": [
44,
1
]
},
{
"code": "theorem IsTransitive.subset_of_mem (h : x.IsTransitive) : y β x β y β x",
"end": [
49,
6
],
"full_name": "ZFSet.IsTransitive.subset_of_mem",
"kind": "commanddeclaration",
"start": [
48,
1
]
},
{
"code": "theorem isTransitive_iff_mem_trans : z.IsTransitive β β {x y : ZFSet}, x β y β y β z β x β z",
"end": [
53,
73
],
"full_name": "ZFSet.isTransitive_iff_mem_trans",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "protected theorem IsTransitive.inter (hx : x.IsTransitive) (hy : y.IsTransitive) :\n (x β© y).IsTransitive",
"end": [
62,
53
],
"full_name": "ZFSet.IsTransitive.inter",
"kind": "commanddeclaration",
"start": [
59,
1
]
},
{
"code": "protected theorem IsTransitive.sUnion (h : x.IsTransitive) :\n (ββ x : ZFSet).IsTransitive",
"end": [
68,
50
],
"full_name": "ZFSet.IsTransitive.sUnion",
"kind": "commanddeclaration",
"start": [
65,
1
]
},
{
"code": "theorem IsTransitive.sUnion' (H : β y β x, IsTransitive y) :\n (ββ x : ZFSet).IsTransitive",
"end": [
74,
57
],
"full_name": "ZFSet.IsTransitive.sUnion'",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "protected theorem IsTransitive.union (hx : x.IsTransitive) (hy : y.IsTransitive) :\n (x βͺ y).IsTransitive",
"end": [
84,
14
],
"full_name": "ZFSet.IsTransitive.union",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "protected theorem IsTransitive.powerset (h : x.IsTransitive) : (powerset x).IsTransitive",
"end": [
90,
32
],
"full_name": "ZFSet.IsTransitive.powerset",
"kind": "commanddeclaration",
"start": [
87,
1
]
},
{
"code": "theorem isTransitive_iff_sUnion_subset : x.IsTransitive β (ββ x : ZFSet) β x",
"end": [
96,
79
],
"full_name": "ZFSet.isTransitive_iff_sUnion_subset",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem isTransitive_iff_subset_powerset : x.IsTransitive β x β powerset x",
"end": [
103,
100
],
"full_name": "ZFSet.isTransitive_iff_subset_powerset",
"kind": "commanddeclaration",
"start": [
102,
1
]
}
]
| 6 | ZFSet.IsTransitive.union | [
[
78,
29
],
[
84,
14
]
]
| 0 | 6 | rw [β sUnion_pair] | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
β’ (x βͺ y).IsTransitive | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
β’ (ββ {x, y}).IsTransitive |
Mathlib/SetTheory/ZFC/Ordinal.lean | [
[
"Mathlib.SetTheory.ZFC.Basic",
"Mathlib/SetTheory/ZFC/Basic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def IsTransitive (x : ZFSet) : Prop :=\n β y β x, y β x",
"end": [
41,
17
],
"full_name": "ZFSet.IsTransitive",
"kind": "commanddeclaration",
"start": [
39,
1
]
},
{
"code": "@[simp]\ntheorem empty_isTransitive : IsTransitive β
",
"end": [
45,
85
],
"full_name": "ZFSet.empty_isTransitive",
"kind": "commanddeclaration",
"start": [
44,
1
]
},
{
"code": "theorem IsTransitive.subset_of_mem (h : x.IsTransitive) : y β x β y β x",
"end": [
49,
6
],
"full_name": "ZFSet.IsTransitive.subset_of_mem",
"kind": "commanddeclaration",
"start": [
48,
1
]
},
{
"code": "theorem isTransitive_iff_mem_trans : z.IsTransitive β β {x y : ZFSet}, x β y β y β z β x β z",
"end": [
53,
73
],
"full_name": "ZFSet.isTransitive_iff_mem_trans",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "protected theorem IsTransitive.inter (hx : x.IsTransitive) (hy : y.IsTransitive) :\n (x β© y).IsTransitive",
"end": [
62,
53
],
"full_name": "ZFSet.IsTransitive.inter",
"kind": "commanddeclaration",
"start": [
59,
1
]
},
{
"code": "protected theorem IsTransitive.sUnion (h : x.IsTransitive) :\n (ββ x : ZFSet).IsTransitive",
"end": [
68,
50
],
"full_name": "ZFSet.IsTransitive.sUnion",
"kind": "commanddeclaration",
"start": [
65,
1
]
},
{
"code": "theorem IsTransitive.sUnion' (H : β y β x, IsTransitive y) :\n (ββ x : ZFSet).IsTransitive",
"end": [
74,
57
],
"full_name": "ZFSet.IsTransitive.sUnion'",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "protected theorem IsTransitive.union (hx : x.IsTransitive) (hy : y.IsTransitive) :\n (x βͺ y).IsTransitive",
"end": [
84,
14
],
"full_name": "ZFSet.IsTransitive.union",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "protected theorem IsTransitive.powerset (h : x.IsTransitive) : (powerset x).IsTransitive",
"end": [
90,
32
],
"full_name": "ZFSet.IsTransitive.powerset",
"kind": "commanddeclaration",
"start": [
87,
1
]
},
{
"code": "theorem isTransitive_iff_sUnion_subset : x.IsTransitive β (ββ x : ZFSet) β x",
"end": [
96,
79
],
"full_name": "ZFSet.isTransitive_iff_sUnion_subset",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem isTransitive_iff_subset_powerset : x.IsTransitive β x β powerset x",
"end": [
103,
100
],
"full_name": "ZFSet.isTransitive_iff_subset_powerset",
"kind": "commanddeclaration",
"start": [
102,
1
]
}
]
| 6 | ZFSet.IsTransitive.union | [
[
78,
29
],
[
84,
14
]
]
| 1 | 6 | apply IsTransitive.sUnion' fun z => _ | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
β’ (ββ {x, y}).IsTransitive | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
β’ β z β {x, y}, z.IsTransitive |
Mathlib/SetTheory/ZFC/Ordinal.lean | [
[
"Mathlib.SetTheory.ZFC.Basic",
"Mathlib/SetTheory/ZFC/Basic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def IsTransitive (x : ZFSet) : Prop :=\n β y β x, y β x",
"end": [
41,
17
],
"full_name": "ZFSet.IsTransitive",
"kind": "commanddeclaration",
"start": [
39,
1
]
},
{
"code": "@[simp]\ntheorem empty_isTransitive : IsTransitive β
",
"end": [
45,
85
],
"full_name": "ZFSet.empty_isTransitive",
"kind": "commanddeclaration",
"start": [
44,
1
]
},
{
"code": "theorem IsTransitive.subset_of_mem (h : x.IsTransitive) : y β x β y β x",
"end": [
49,
6
],
"full_name": "ZFSet.IsTransitive.subset_of_mem",
"kind": "commanddeclaration",
"start": [
48,
1
]
},
{
"code": "theorem isTransitive_iff_mem_trans : z.IsTransitive β β {x y : ZFSet}, x β y β y β z β x β z",
"end": [
53,
73
],
"full_name": "ZFSet.isTransitive_iff_mem_trans",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "protected theorem IsTransitive.inter (hx : x.IsTransitive) (hy : y.IsTransitive) :\n (x β© y).IsTransitive",
"end": [
62,
53
],
"full_name": "ZFSet.IsTransitive.inter",
"kind": "commanddeclaration",
"start": [
59,
1
]
},
{
"code": "protected theorem IsTransitive.sUnion (h : x.IsTransitive) :\n (ββ x : ZFSet).IsTransitive",
"end": [
68,
50
],
"full_name": "ZFSet.IsTransitive.sUnion",
"kind": "commanddeclaration",
"start": [
65,
1
]
},
{
"code": "theorem IsTransitive.sUnion' (H : β y β x, IsTransitive y) :\n (ββ x : ZFSet).IsTransitive",
"end": [
74,
57
],
"full_name": "ZFSet.IsTransitive.sUnion'",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "protected theorem IsTransitive.union (hx : x.IsTransitive) (hy : y.IsTransitive) :\n (x βͺ y).IsTransitive",
"end": [
84,
14
],
"full_name": "ZFSet.IsTransitive.union",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "protected theorem IsTransitive.powerset (h : x.IsTransitive) : (powerset x).IsTransitive",
"end": [
90,
32
],
"full_name": "ZFSet.IsTransitive.powerset",
"kind": "commanddeclaration",
"start": [
87,
1
]
},
{
"code": "theorem isTransitive_iff_sUnion_subset : x.IsTransitive β (ββ x : ZFSet) β x",
"end": [
96,
79
],
"full_name": "ZFSet.isTransitive_iff_sUnion_subset",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem isTransitive_iff_subset_powerset : x.IsTransitive β x β powerset x",
"end": [
103,
100
],
"full_name": "ZFSet.isTransitive_iff_subset_powerset",
"kind": "commanddeclaration",
"start": [
102,
1
]
}
]
| 6 | ZFSet.IsTransitive.union | [
[
78,
29
],
[
84,
14
]
]
| 2 | 6 | intro | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
β’ β z β {x, y}, z.IsTransitive | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
zβ : ZFSet
β’ zβ β {x, y} β zβ.IsTransitive |
Mathlib/SetTheory/ZFC/Ordinal.lean | [
[
"Mathlib.SetTheory.ZFC.Basic",
"Mathlib/SetTheory/ZFC/Basic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def IsTransitive (x : ZFSet) : Prop :=\n β y β x, y β x",
"end": [
41,
17
],
"full_name": "ZFSet.IsTransitive",
"kind": "commanddeclaration",
"start": [
39,
1
]
},
{
"code": "@[simp]\ntheorem empty_isTransitive : IsTransitive β
",
"end": [
45,
85
],
"full_name": "ZFSet.empty_isTransitive",
"kind": "commanddeclaration",
"start": [
44,
1
]
},
{
"code": "theorem IsTransitive.subset_of_mem (h : x.IsTransitive) : y β x β y β x",
"end": [
49,
6
],
"full_name": "ZFSet.IsTransitive.subset_of_mem",
"kind": "commanddeclaration",
"start": [
48,
1
]
},
{
"code": "theorem isTransitive_iff_mem_trans : z.IsTransitive β β {x y : ZFSet}, x β y β y β z β x β z",
"end": [
53,
73
],
"full_name": "ZFSet.isTransitive_iff_mem_trans",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "protected theorem IsTransitive.inter (hx : x.IsTransitive) (hy : y.IsTransitive) :\n (x β© y).IsTransitive",
"end": [
62,
53
],
"full_name": "ZFSet.IsTransitive.inter",
"kind": "commanddeclaration",
"start": [
59,
1
]
},
{
"code": "protected theorem IsTransitive.sUnion (h : x.IsTransitive) :\n (ββ x : ZFSet).IsTransitive",
"end": [
68,
50
],
"full_name": "ZFSet.IsTransitive.sUnion",
"kind": "commanddeclaration",
"start": [
65,
1
]
},
{
"code": "theorem IsTransitive.sUnion' (H : β y β x, IsTransitive y) :\n (ββ x : ZFSet).IsTransitive",
"end": [
74,
57
],
"full_name": "ZFSet.IsTransitive.sUnion'",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "protected theorem IsTransitive.union (hx : x.IsTransitive) (hy : y.IsTransitive) :\n (x βͺ y).IsTransitive",
"end": [
84,
14
],
"full_name": "ZFSet.IsTransitive.union",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "protected theorem IsTransitive.powerset (h : x.IsTransitive) : (powerset x).IsTransitive",
"end": [
90,
32
],
"full_name": "ZFSet.IsTransitive.powerset",
"kind": "commanddeclaration",
"start": [
87,
1
]
},
{
"code": "theorem isTransitive_iff_sUnion_subset : x.IsTransitive β (ββ x : ZFSet) β x",
"end": [
96,
79
],
"full_name": "ZFSet.isTransitive_iff_sUnion_subset",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem isTransitive_iff_subset_powerset : x.IsTransitive β x β powerset x",
"end": [
103,
100
],
"full_name": "ZFSet.isTransitive_iff_subset_powerset",
"kind": "commanddeclaration",
"start": [
102,
1
]
}
]
| 6 | ZFSet.IsTransitive.union | [
[
78,
29
],
[
84,
14
]
]
| 3 | 6 | rw [mem_pair] | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
zβ : ZFSet
β’ zβ β {x, y} β zβ.IsTransitive | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
zβ : ZFSet
β’ zβ = x β¨ zβ = y β zβ.IsTransitive |
Mathlib/SetTheory/ZFC/Ordinal.lean | [
[
"Mathlib.SetTheory.ZFC.Basic",
"Mathlib/SetTheory/ZFC/Basic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def IsTransitive (x : ZFSet) : Prop :=\n β y β x, y β x",
"end": [
41,
17
],
"full_name": "ZFSet.IsTransitive",
"kind": "commanddeclaration",
"start": [
39,
1
]
},
{
"code": "@[simp]\ntheorem empty_isTransitive : IsTransitive β
",
"end": [
45,
85
],
"full_name": "ZFSet.empty_isTransitive",
"kind": "commanddeclaration",
"start": [
44,
1
]
},
{
"code": "theorem IsTransitive.subset_of_mem (h : x.IsTransitive) : y β x β y β x",
"end": [
49,
6
],
"full_name": "ZFSet.IsTransitive.subset_of_mem",
"kind": "commanddeclaration",
"start": [
48,
1
]
},
{
"code": "theorem isTransitive_iff_mem_trans : z.IsTransitive β β {x y : ZFSet}, x β y β y β z β x β z",
"end": [
53,
73
],
"full_name": "ZFSet.isTransitive_iff_mem_trans",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "protected theorem IsTransitive.inter (hx : x.IsTransitive) (hy : y.IsTransitive) :\n (x β© y).IsTransitive",
"end": [
62,
53
],
"full_name": "ZFSet.IsTransitive.inter",
"kind": "commanddeclaration",
"start": [
59,
1
]
},
{
"code": "protected theorem IsTransitive.sUnion (h : x.IsTransitive) :\n (ββ x : ZFSet).IsTransitive",
"end": [
68,
50
],
"full_name": "ZFSet.IsTransitive.sUnion",
"kind": "commanddeclaration",
"start": [
65,
1
]
},
{
"code": "theorem IsTransitive.sUnion' (H : β y β x, IsTransitive y) :\n (ββ x : ZFSet).IsTransitive",
"end": [
74,
57
],
"full_name": "ZFSet.IsTransitive.sUnion'",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "protected theorem IsTransitive.union (hx : x.IsTransitive) (hy : y.IsTransitive) :\n (x βͺ y).IsTransitive",
"end": [
84,
14
],
"full_name": "ZFSet.IsTransitive.union",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "protected theorem IsTransitive.powerset (h : x.IsTransitive) : (powerset x).IsTransitive",
"end": [
90,
32
],
"full_name": "ZFSet.IsTransitive.powerset",
"kind": "commanddeclaration",
"start": [
87,
1
]
},
{
"code": "theorem isTransitive_iff_sUnion_subset : x.IsTransitive β (ββ x : ZFSet) β x",
"end": [
96,
79
],
"full_name": "ZFSet.isTransitive_iff_sUnion_subset",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem isTransitive_iff_subset_powerset : x.IsTransitive β x β powerset x",
"end": [
103,
100
],
"full_name": "ZFSet.isTransitive_iff_subset_powerset",
"kind": "commanddeclaration",
"start": [
102,
1
]
}
]
| 6 | ZFSet.IsTransitive.union | [
[
78,
29
],
[
84,
14
]
]
| 4 | 6 | rintro (rfl | rfl) | x y z : ZFSet
hx : x.IsTransitive
hy : y.IsTransitive
zβ : ZFSet
β’ zβ = x β¨ zβ = y β zβ.IsTransitive | case inl
y z : ZFSet
hy : y.IsTransitive
zβ : ZFSet
hx : zβ.IsTransitive
β’ zβ.IsTransitive
case inr
x z : ZFSet
hx : x.IsTransitive
zβ : ZFSet
hy : zβ.IsTransitive
β’ zβ.IsTransitive |
Mathlib/SetTheory/ZFC/Ordinal.lean | [
[
"Mathlib.SetTheory.ZFC.Basic",
"Mathlib/SetTheory/ZFC/Basic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def IsTransitive (x : ZFSet) : Prop :=\n β y β x, y β x",
"end": [
41,
17
],
"full_name": "ZFSet.IsTransitive",
"kind": "commanddeclaration",
"start": [
39,
1
]
},
{
"code": "@[simp]\ntheorem empty_isTransitive : IsTransitive β
",
"end": [
45,
85
],
"full_name": "ZFSet.empty_isTransitive",
"kind": "commanddeclaration",
"start": [
44,
1
]
},
{
"code": "theorem IsTransitive.subset_of_mem (h : x.IsTransitive) : y β x β y β x",
"end": [
49,
6
],
"full_name": "ZFSet.IsTransitive.subset_of_mem",
"kind": "commanddeclaration",
"start": [
48,
1
]
},
{
"code": "theorem isTransitive_iff_mem_trans : z.IsTransitive β β {x y : ZFSet}, x β y β y β z β x β z",
"end": [
53,
73
],
"full_name": "ZFSet.isTransitive_iff_mem_trans",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "protected theorem IsTransitive.inter (hx : x.IsTransitive) (hy : y.IsTransitive) :\n (x β© y).IsTransitive",
"end": [
62,
53
],
"full_name": "ZFSet.IsTransitive.inter",
"kind": "commanddeclaration",
"start": [
59,
1
]
},
{
"code": "protected theorem IsTransitive.sUnion (h : x.IsTransitive) :\n (ββ x : ZFSet).IsTransitive",
"end": [
68,
50
],
"full_name": "ZFSet.IsTransitive.sUnion",
"kind": "commanddeclaration",
"start": [
65,
1
]
},
{
"code": "theorem IsTransitive.sUnion' (H : β y β x, IsTransitive y) :\n (ββ x : ZFSet).IsTransitive",
"end": [
74,
57
],
"full_name": "ZFSet.IsTransitive.sUnion'",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "protected theorem IsTransitive.union (hx : x.IsTransitive) (hy : y.IsTransitive) :\n (x βͺ y).IsTransitive",
"end": [
84,
14
],
"full_name": "ZFSet.IsTransitive.union",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "protected theorem IsTransitive.powerset (h : x.IsTransitive) : (powerset x).IsTransitive",
"end": [
90,
32
],
"full_name": "ZFSet.IsTransitive.powerset",
"kind": "commanddeclaration",
"start": [
87,
1
]
},
{
"code": "theorem isTransitive_iff_sUnion_subset : x.IsTransitive β (ββ x : ZFSet) β x",
"end": [
96,
79
],
"full_name": "ZFSet.isTransitive_iff_sUnion_subset",
"kind": "commanddeclaration",
"start": [
93,
1
]
},
{
"code": "theorem isTransitive_iff_subset_powerset : x.IsTransitive β x β powerset x",
"end": [
103,
100
],
"full_name": "ZFSet.isTransitive_iff_subset_powerset",
"kind": "commanddeclaration",
"start": [
102,
1
]
}
]
| 6 | ZFSet.IsTransitive.union | [
[
78,
29
],
[
84,
14
]
]
| 5 | 6 | assumption' | case inl
y z : ZFSet
hy : y.IsTransitive
zβ : ZFSet
hx : zβ.IsTransitive
β’ zβ.IsTransitive
case inr
x z : ZFSet
hx : x.IsTransitive
zβ : ZFSet
hy : zβ.IsTransitive
β’ zβ.IsTransitive | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 0 | Surreal.Multiplication.P3_comm | [
[
96,
52
],
[
98,
34
]
]
| 0 | 2 | rw [P3, P3, add_comm] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ β β¦yβ * xββ§ + β¦yβ * xββ§ < β¦yβ * xββ§ + β¦yβ * xββ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 0 | Surreal.Multiplication.P3_comm | [
[
96,
52
],
[
98,
34
]
]
| 1 | 2 | congr! 2 <;> rw [quot_mul_comm] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ β β¦yβ * xββ§ + β¦yβ * xββ§ < β¦yβ * xββ§ + β¦yβ * xββ§ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 1 | Surreal.Multiplication.P3.trans | [
[
100,
80
],
[
103,
44
]
]
| 0 | 3 | rw [P3] at hβ hβ | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
hβ : P3 xβ xβ yβ yβ
hβ : P3 xβ xβ yβ yβ
β’ P3 xβ xβ yβ yβ | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
hβ : β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§
hβ : β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§
β’ P3 xβ xβ yβ yβ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 1 | Surreal.Multiplication.P3.trans | [
[
100,
80
],
[
103,
44
]
]
| 1 | 3 | rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
hβ : β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§
hβ : β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§
β’ P3 xβ xβ yβ yβ | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
hβ : β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§
hβ : β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§
β’ β¦xβ * yββ§ + β¦xβ * yββ§ + (β¦xβ * yββ§ + β¦xβ * yββ§) < β¦xβ * yββ§ + β¦xβ * yββ§ + (β¦xβ * yββ§ + β¦xβ * yββ§) |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 1 | Surreal.Multiplication.P3.trans | [
[
100,
80
],
[
103,
44
]
]
| 2 | 3 | convert add_lt_add hβ hβ using 1 <;> abel | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
hβ : β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§
hβ : β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§
β’ β¦xβ * yββ§ + β¦xβ * yββ§ + (β¦xβ * yββ§ + β¦xβ * yββ§) < β¦xβ * yββ§ + β¦xβ * yββ§ + (β¦xβ * yββ§ + β¦xβ * yββ§) | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 2 | Surreal.Multiplication.P3_neg | [
[
105,
57
],
[
108,
10
]
]
| 0 | 3 | simp_rw [P3, quot_neg_mul] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ β -β¦xβ * yββ§ + -β¦xβ * yββ§ < -β¦xβ * yββ§ + -β¦xβ * yββ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 2 | Surreal.Multiplication.P3_neg | [
[
105,
57
],
[
108,
10
]
]
| 1 | 3 | rw [β _root_.neg_lt_neg_iff] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ β -β¦xβ * yββ§ + -β¦xβ * yββ§ < -β¦xβ * yββ§ + -β¦xβ * yββ§ | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ -(β¦xβ * yββ§ + β¦xβ * yββ§) < -(β¦xβ * yββ§ + β¦xβ * yββ§) β -β¦xβ * yββ§ + -β¦xβ * yββ§ < -β¦xβ * yββ§ + -β¦xβ * yββ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 2 | Surreal.Multiplication.P3_neg | [
[
105,
57
],
[
108,
10
]
]
| 2 | 3 | abel_nf | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ -(β¦xβ * yββ§ + β¦xβ * yββ§) < -(β¦xβ * yββ§ + β¦xβ * yββ§) β -β¦xβ * yββ§ + -β¦xβ * yββ§ < -β¦xβ * yββ§ + -β¦xβ * yββ§ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 3 | Surreal.Multiplication.P2_neg_left | [
[
110,
54
],
[
116,
16
]
]
| 0 | 8 | rw [P2, P2] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ P2 xβ xβ y β P2 (-xβ) (-xβ) y | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ β -xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 3 | Surreal.Multiplication.P2_neg_left | [
[
110,
54
],
[
116,
16
]
]
| 1 | 8 | constructor | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ β -xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§ | case mp
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§) β -xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§
case mpr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (-xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§) β xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 3 | Surreal.Multiplication.P2_neg_left | [
[
110,
54
],
[
116,
16
]
]
| 2 | 8 | Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]
exact (Β· Β·) | case mp
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§) β -xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§
case mpr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (-xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§) β xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ | case mpr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (-xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§) β xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 3 | Surreal.Multiplication.P2_neg_left | [
[
110,
54
],
[
116,
16
]
]
| 3 | 8 | Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]
exact (Β· Β·) | case mpr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (-xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§) β xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 3 | Surreal.Multiplication.P2_neg_left | [
[
110,
54
],
[
116,
16
]
]
| 4 | 8 | rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm] | case mp
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§) β -xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§ | case mp
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§) β xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 3 | Surreal.Multiplication.P2_neg_left | [
[
110,
54
],
[
116,
16
]
]
| 5 | 8 | exact (Β· Β·) | case mp
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§) β xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 3 | Surreal.Multiplication.P2_neg_left | [
[
110,
54
],
[
116,
16
]
]
| 6 | 8 | rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm] | case mpr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (-xβ β -xβ β β¦-xβ * yβ§ = β¦-xβ * yβ§) β xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ | case mpr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§) β xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 3 | Surreal.Multiplication.P2_neg_left | [
[
110,
54
],
[
116,
16
]
]
| 7 | 8 | exact (Β· Β·) | case mpr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ (xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§) β xβ β xβ β β¦xβ * yβ§ = β¦xβ * yβ§ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 4 | Surreal.Multiplication.P2_neg_right | [
[
118,
52
],
[
119,
51
]
]
| 0 | 1 | rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ P2 xβ xβ y β P2 xβ xβ (-y) | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 5 | Surreal.Multiplication.P4_neg_left | [
[
121,
54
],
[
122,
62
]
]
| 0 | 1 | simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ P4 xβ xβ y β P4 (-xβ) (-xβ) y | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 6 | Surreal.Multiplication.P4_neg_right | [
[
124,
52
],
[
125,
33
]
]
| 0 | 1 | rw [P4, P4, neg_neg, and_comm] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ P4 xβ xβ y β P4 xβ xβ (-y) | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 7 | Surreal.Multiplication.P24_neg_left | [
[
127,
57
],
[
127,
99
]
]
| 0 | 1 | rw [P24, P24, P2_neg_left, P4_neg_left] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ P24 xβ xβ y β P24 (-xβ) (-xβ) y | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 8 | Surreal.Multiplication.P24_neg_right | [
[
128,
55
],
[
128,
99
]
]
| 0 | 1 | rw [P24, P24, P2_neg_right, P4_neg_right] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
β’ P24 xβ xβ y β P24 xβ xβ (-y) | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 9 | Surreal.Multiplication.mulOption_lt_iff_P1 | [
[
134,
79
],
[
136,
53
]
]
| 0 | 2 | dsimp only [P1, mulOption, quot_sub, quot_add] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ β P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
β’ β¦x.moveLeft i * yβ§ + β¦x * y.moveLeft kβ§ - β¦x.moveLeft i * y.moveLeft kβ§ <
-(β¦x.moveLeft j * -yβ§ + β¦x * (-y).moveLeft lβ§ - β¦x.moveLeft j * (-y).moveLeft lβ§) β
β¦x.moveLeft i * yβ§ + β¦x * y.moveLeft kβ§ - β¦x.moveLeft i * y.moveLeft kβ§ <
β¦x.moveLeft j * yβ§ + β¦x * -(-y).moveLeft lβ§ - β¦x.moveLeft j * -(-y).moveLeft lβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 9 | Surreal.Multiplication.mulOption_lt_iff_P1 | [
[
134,
79
],
[
136,
53
]
]
| 1 | 2 | simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
β’ β¦x.moveLeft i * yβ§ + β¦x * y.moveLeft kβ§ - β¦x.moveLeft i * y.moveLeft kβ§ <
-(β¦x.moveLeft j * -yβ§ + β¦x * (-y).moveLeft lβ§ - β¦x.moveLeft j * (-y).moveLeft lβ§) β
β¦x.moveLeft i * yβ§ + β¦x * y.moveLeft kβ§ - β¦x.moveLeft i * y.moveLeft kβ§ <
β¦x.moveLeft j * yβ§ + β¦x * -(-y).moveLeft lβ§ - β¦x.moveLeft j * -(-y).moveLeft lβ§ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 10 | Surreal.Multiplication.mulOption_lt_mul_iff_P3 | [
[
139,
86
],
[
141,
27
]
]
| 0 | 2 | dsimp only [mulOption, quot_sub, quot_add] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
i : x.LeftMoves
j : y.LeftMoves
β’ β¦x.mulOption y i jβ§ < β¦x * yβ§ β P3 (x.moveLeft i) x (y.moveLeft j) y | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
i : x.LeftMoves
j : y.LeftMoves
β’ β¦x.moveLeft i * yβ§ + β¦x * y.moveLeft jβ§ - β¦x.moveLeft i * y.moveLeft jβ§ < β¦x * yβ§ β
P3 (x.moveLeft i) x (y.moveLeft j) y |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 10 | Surreal.Multiplication.mulOption_lt_mul_iff_P3 | [
[
139,
86
],
[
141,
27
]
]
| 1 | 2 | exact sub_lt_iff_lt_add' | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
i : x.LeftMoves
j : y.LeftMoves
β’ β¦x.moveLeft i * yβ§ + β¦x * y.moveLeft jβ§ - β¦x.moveLeft i * y.moveLeft jβ§ < β¦x * yβ§ β
P3 (x.moveLeft i) x (y.moveLeft j) y | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 11 | Surreal.Multiplication.P1_of_eq | [
[
144,
29
],
[
146,
56
]
]
| 0 | 2 | rw [P1, β hβ he, β hβ he, sub_lt_sub_iff] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
he : xβ β xβ
hβ : P2 xβ xβ yβ
hβ : P2 xβ xβ yβ
h3 : P3 xβ xβ yβ yβ
β’ P1 xβ xβ xβ yβ yβ yβ | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
he : xβ β xβ
hβ : P2 xβ xβ yβ
hβ : P2 xβ xβ yβ
h3 : P3 xβ xβ yβ yβ
β’ β¦xβ * yββ§ + β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ + β¦xβ * yββ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 11 | Surreal.Multiplication.P1_of_eq | [
[
144,
29
],
[
146,
56
]
]
| 1 | 2 | convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
he : xβ β xβ
hβ : P2 xβ xβ yβ
hβ : P2 xβ xβ yβ
h3 : P3 xβ xβ yβ yβ
β’ β¦xβ * yββ§ + β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ + β¦xβ * yββ§ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 12 | Surreal.Multiplication.P1_of_lt | [
[
148,
86
],
[
150,
44
]
]
| 0 | 2 | rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
hβ : P3 xβ xβ yβ yβ
hβ : P3 xβ xβ yβ yβ
β’ P1 xβ xβ xβ yβ yβ yβ | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
hβ : P3 xβ xβ yβ yβ
hβ : P3 xβ xβ yβ yβ
β’ β¦xβ * yββ§ + (β¦xβ * yββ§ + β¦xβ * yββ§ + β¦xβ * yββ§) < β¦xβ * yββ§ + (β¦xβ * yββ§ + β¦xβ * yββ§ + β¦xβ * yββ§) |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 12 | Surreal.Multiplication.P1_of_lt | [
[
148,
86
],
[
150,
44
]
]
| 1 | 2 | convert add_lt_add hβ hβ using 1 <;> abel | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
hβ : P3 xβ xβ yβ yβ
hβ : P3 xβ xβ yβ yβ
β’ β¦xβ * yββ§ + (β¦xβ * yββ§ + β¦xβ * yββ§ + β¦xβ * yββ§) < β¦xβ * yββ§ + (β¦xβ * yββ§ + β¦xβ * yββ§ + β¦xβ * yββ§) | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 13 | Surreal.Multiplication.Args.numeric_P1 | [
[
165,
80
],
[
166,
39
]
]
| 0 | 1 | simp [Args.Numeric, Args.toMultiset] | xβ xβ xβ xβ x' yβ yβ yβ yβ y' : PGame
x y : PGame
β’ (P1 x y).Numeric β x.Numeric β§ y.Numeric | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 14 | Surreal.Multiplication.Args.numeric_P24 | [
[
169,
73
],
[
170,
39
]
]
| 0 | 1 | simp [Args.Numeric, Args.toMultiset] | x xββ xββ xβ x' yβ yβ yβ yβ y' : PGame
xβ xβ y : PGame
β’ (P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 22 | Surreal.Multiplication.ih1 | [
[
220,
24
],
[
223,
64
]
]
| 0 | 5 | rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _) | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
β’ IH1 x y | case inl
x xββ xββ xβ x' yβ yβ yβ y'β xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
ih : β (a : Args), ArgsRel a (Args.P1 x y') β P124 a
β’ ArgsRel (Args.P24 xβ xβ y') (Args.P1 x y')
case inr
x xββ xββ xβ x' y yβ yβ yβ y'β : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
hy : y'.IsOption y
β’ ArgsRel (Args.P24 xβ xβ y') (Args.P1 x y) |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 22 | Surreal.Multiplication.ih1 | [
[
220,
24
],
[
223,
64
]
]
| 1 | 5 | on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy) | case inl
x xββ xββ xβ x' yβ yβ yβ y'β xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
ih : β (a : Args), ArgsRel a (Args.P1 x y') β P124 a
β’ ArgsRel (Args.P24 xβ xβ y') (Args.P1 x y')
case inr
x xββ xββ xβ x' y yβ yβ yβ y'β : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
hy : y'.IsOption y
β’ ArgsRel (Args.P24 xβ xβ y') (Args.P1 x y) | case inl
x xββ xββ xβ x' yβ yβ yβ y'β xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
ih : β (a : Args), ArgsRel a (Args.P1 x y') β P124 a
β’ ArgsRel (Args.P24 xβ xβ y') (Args.P1 x y')
case inr
x xββ xββ xβ x' y yβ yβ yβ y'β : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
hy : y'.IsOption y
β’ TransGen (CutExpand IsOption) (Args.P24 xβ xβ y').toMultiset {x, y'} |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 22 | Surreal.Multiplication.ih1 | [
[
220,
24
],
[
223,
64
]
]
| 2 | 5 | all_goals exact TransGen.single (cutExpand_double_left hβ hβ) | case inl
x xββ xββ xβ x' yβ yβ yβ y'β xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
ih : β (a : Args), ArgsRel a (Args.P1 x y') β P124 a
β’ ArgsRel (Args.P24 xβ xβ y') (Args.P1 x y')
case inr
x xββ xββ xβ x' y yβ yβ yβ y'β : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
hy : y'.IsOption y
β’ TransGen (CutExpand IsOption) (Args.P24 xβ xβ y').toMultiset {x, y'} | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 22 | Surreal.Multiplication.ih1 | [
[
220,
24
],
[
223,
64
]
]
| 3 | 5 | refine TransGen.tail ?_ (cutExpand_pair_right hy) | case inr
x xββ xββ xβ x' y yβ yβ yβ y'β : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
hy : y'.IsOption y
β’ ArgsRel (Args.P24 xβ xβ y') (Args.P1 x y) | case inr
x xββ xββ xβ x' y yβ yβ yβ y'β : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
hy : y'.IsOption y
β’ TransGen (CutExpand IsOption) (Args.P24 xβ xβ y').toMultiset {x, y'} |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 22 | Surreal.Multiplication.ih1 | [
[
220,
24
],
[
223,
64
]
]
| 4 | 5 | exact TransGen.single (cutExpand_double_left hβ hβ) | case inr
x xββ xββ xβ x' y yβ yβ yβ y'β : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
xβ xβ y' : PGame
hβ : xβ.IsOption x
hβ : xβ.IsOption x
hy : y'.IsOption y
β’ TransGen (CutExpand IsOption) (Args.P24 xβ xβ y').toMultiset {x, y'} | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 27 | Surreal.Multiplication.mulOption_lt | [
[
247,
90
],
[
254,
87
]
]
| 0 | 9 | obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j) | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ | case inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i < x.moveLeft j
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§
case inr.inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i β x.moveLeft j
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§
case inr.inr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 27 | Surreal.Multiplication.mulOption_lt | [
[
247,
90
],
[
254,
87
]
]
| 1 | 9 | Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h | case inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i < x.moveLeft j
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§
case inr.inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i β x.moveLeft j
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§
case inr.inr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ | case inr.inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i β x.moveLeft j
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§
case inr.inr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 27 | Surreal.Multiplication.mulOption_lt | [
[
247,
90
],
[
254,
87
]
]
| 2 | 9 | Β· have ml := @IsOption.moveLeft
exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1
(ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l) | case inr.inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i β x.moveLeft j
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§
case inr.inr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ | case inr.inr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 27 | Surreal.Multiplication.mulOption_lt | [
[
247,
90
],
[
254,
87
]
]
| 3 | 9 | Β· rw [mulOption_neg_neg, lt_neg]
exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h | case inr.inr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 27 | Surreal.Multiplication.mulOption_lt | [
[
247,
90
],
[
254,
87
]
]
| 4 | 9 | exact mulOption_lt_of_lt hy ihxy ihyx i j k l h | case inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i < x.moveLeft j
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 27 | Surreal.Multiplication.mulOption_lt | [
[
247,
90
],
[
254,
87
]
]
| 5 | 9 | have ml := @IsOption.moveLeft | case inr.inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i β x.moveLeft j
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ | case inr.inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i β x.moveLeft j
ml : β {x : PGame} (i : x.LeftMoves), (x.moveLeft i).IsOption x
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 27 | Surreal.Multiplication.mulOption_lt | [
[
247,
90
],
[
254,
87
]
]
| 6 | 9 | exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1
(ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l) | case inr.inl
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft i β x.moveLeft j
ml : β {x : PGame} (i : x.LeftMoves), (x.moveLeft i).IsOption x
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 27 | Surreal.Multiplication.mulOption_lt | [
[
247,
90
],
[
254,
87
]
]
| 7 | 9 | rw [mulOption_neg_neg, lt_neg] | case inr.inr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
β’ β¦x.mulOption y i kβ§ < -β¦x.mulOption (-y) j lβ§ | case inr.inr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
β’ β¦x.mulOption (-y) j lβ§ < -β¦x.mulOption (- -y) i (toLeftMovesNeg (toRightMovesNeg k))β§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 27 | Surreal.Multiplication.mulOption_lt | [
[
247,
90
],
[
254,
87
]
]
| 8 | 9 | exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h | case inr.inr
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
i j : x.LeftMoves
k : y.LeftMoves
l : (-y).LeftMoves
h : x.moveLeft j < x.moveLeft i
β’ β¦x.mulOption (-y) j lβ§ < -β¦x.mulOption (- -y) i (toLeftMovesNeg (toRightMovesNeg k))β§ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 0 | 29 | have ihxy := ih1 ih | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
β’ (x * y).Numeric | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
β’ (x * y).Numeric |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 1 | 29 | have ihyx := ih1_swap ih | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
β’ (x * y).Numeric | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
β’ (x * y).Numeric |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 2 | 29 | have ihxyn := ih1_neg_left (ih1_neg_right ihxy) | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
β’ (x * y).Numeric | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
β’ (x * y).Numeric |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 3 | 29 | have ihyxn := ih1_neg_left (ih1_neg_right ihyx) | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
β’ (x * y).Numeric | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ (x * y).Numeric |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 4 | 29 | refine numeric_def.mpr β¨?_, ?_, ?_β© | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ (x * y).Numeric | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (i : (x * y).LeftMoves) (j : (x * y).RightMoves), (x * y).moveLeft i < (x * y).moveRight j
case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (i : (x * y).LeftMoves), ((x * y).moveLeft i).Numeric
case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (j : (x * y).RightMoves), ((x * y).moveRight j).Numeric |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 5 | 29 | Β· simp_rw [lt_iff_game_lt]
intro i
rw [rightMoves_mul_iff]
constructor <;> (intro j l; revert i; rw [leftMoves_mul_iff (_ > Β·)]; constructor <;> intro i k)
Β· apply mulOption_lt hx hy ihxy ihyx
Β· simp_rw [β mulOption_symm (-y), mulOption_neg_neg x]
apply mulOption_lt hy.neg hx.neg ihyxn ihxyn
Β· simp only [β mulOption_symm y]
apply mulOption_lt hy hx ihyx ihxy
Β· rw [mulOption_neg_neg y]
apply mulOption_lt hx.neg hy.neg ihxyn ihyxn | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (i : (x * y).LeftMoves) (j : (x * y).RightMoves), (x * y).moveLeft i < (x * y).moveRight j
case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (i : (x * y).LeftMoves), ((x * y).moveLeft i).Numeric
case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (j : (x * y).RightMoves), ((x * y).moveRight j).Numeric | case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (i : (x * y).LeftMoves), ((x * y).moveLeft i).Numeric
case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (j : (x * y).RightMoves), ((x * y).moveRight j).Numeric |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 6 | 29 | all_goals
cases x; cases y
rintro (β¨i,jβ©|β¨i,jβ©) <;>
refine ((numeric_option_mul ih ?_).add <| numeric_mul_option ih ?_).sub
(numeric_option_mul_option ih ?_ ?_) <;>
solve_by_elim [IsOption.mk_left, IsOption.mk_right] | case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (i : (x * y).LeftMoves), ((x * y).moveLeft i).Numeric
case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (j : (x * y).RightMoves), ((x * y).moveRight j).Numeric | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 7 | 29 | simp_rw [lt_iff_game_lt] | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (i : (x * y).LeftMoves) (j : (x * y).RightMoves), (x * y).moveLeft i < (x * y).moveRight j | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (i : (x * y).LeftMoves) (j : (x * y).RightMoves), β¦(x * y).moveLeft iβ§ < β¦(x * y).moveRight jβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 8 | 29 | intro i | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (i : (x * y).LeftMoves) (j : (x * y).RightMoves), β¦(x * y).moveLeft iβ§ < β¦(x * y).moveRight jβ§ | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
i : (x * y).LeftMoves
β’ β (j : (x * y).RightMoves), β¦(x * y).moveLeft iβ§ < β¦(x * y).moveRight jβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 9 | 29 | rw [rightMoves_mul_iff] | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
i : (x * y).LeftMoves
β’ β (j : (x * y).RightMoves), β¦(x * y).moveLeft iβ§ < β¦(x * y).moveRight jβ§ | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
i : (x * y).LeftMoves
β’ (β (i_1 : x.LeftMoves) (j : (-y).LeftMoves), β¦(x * y).moveLeft iβ§ < -β¦x.mulOption (-y) i_1 jβ§) β§
β (i_1 : (-x).LeftMoves) (j : y.LeftMoves), β¦(x * y).moveLeft iβ§ < -β¦(-x).mulOption y i_1 jβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 10 | 29 | constructor <;> (intro j l; revert i; rw [leftMoves_mul_iff (_ > Β·)]; constructor <;> intro i k) | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
i : (x * y).LeftMoves
β’ (β (i_1 : x.LeftMoves) (j : (-y).LeftMoves), β¦(x * y).moveLeft iβ§ < -β¦x.mulOption (-y) i_1 jβ§) β§
β (i_1 : (-x).LeftMoves) (j : y.LeftMoves), β¦(x * y).moveLeft iβ§ < -β¦(-x).mulOption y i_1 jβ§ | case refine_1.left.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦x.mulOption (-y) j lβ§ > β¦x.mulOption y i kβ§
case refine_1.left.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦x.mulOption (-y) j lβ§ > β¦(-x).mulOption (-y) i kβ§
case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i kβ§
case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 11 | 29 | Β· apply mulOption_lt hx hy ihxy ihyx | case refine_1.left.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦x.mulOption (-y) j lβ§ > β¦x.mulOption y i kβ§
case refine_1.left.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦x.mulOption (-y) j lβ§ > β¦(-x).mulOption (-y) i kβ§
case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i kβ§
case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ | case refine_1.left.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦x.mulOption (-y) j lβ§ > β¦(-x).mulOption (-y) i kβ§
case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i kβ§
case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 12 | 29 | Β· simp_rw [β mulOption_symm (-y), mulOption_neg_neg x]
apply mulOption_lt hy.neg hx.neg ihyxn ihxyn | case refine_1.left.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦x.mulOption (-y) j lβ§ > β¦(-x).mulOption (-y) i kβ§
case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i kβ§
case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ | case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i kβ§
case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 13 | 29 | Β· simp only [β mulOption_symm y]
apply mulOption_lt hy hx ihyx ihxy | case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i kβ§
case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ | case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 14 | 29 | Β· rw [mulOption_neg_neg y]
apply mulOption_lt hx.neg hy.neg ihxyn ihyxn | case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 15 | 29 | intro j l | case refine_1.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
i : (x * y).LeftMoves
β’ β (i_1 : (-x).LeftMoves) (j : y.LeftMoves), β¦(x * y).moveLeft iβ§ < -β¦(-x).mulOption y i_1 jβ§ | case refine_1.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
i : (x * y).LeftMoves
j : (-x).LeftMoves
l : y.LeftMoves
β’ β¦(x * y).moveLeft iβ§ < -β¦(-x).mulOption y j lβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 16 | 29 | revert i | case refine_1.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
i : (x * y).LeftMoves
j : (-x).LeftMoves
l : y.LeftMoves
β’ β¦(x * y).moveLeft iβ§ < -β¦(-x).mulOption y j lβ§ | case refine_1.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
β’ β (i : (x * y).LeftMoves), β¦(x * y).moveLeft iβ§ < -β¦(-x).mulOption y j lβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 17 | 29 | rw [leftMoves_mul_iff (_ > Β·)] | case refine_1.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
β’ β (i : (x * y).LeftMoves), β¦(x * y).moveLeft iβ§ < -β¦(-x).mulOption y j lβ§ | case refine_1.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
β’ (β (i : x.LeftMoves) (j_1 : y.LeftMoves), -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i j_1β§) β§
β (i : (-x).LeftMoves) (j_1 : (-y).LeftMoves), -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i j_1β§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 18 | 29 | constructor <;> intro i k | case refine_1.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
β’ (β (i : x.LeftMoves) (j_1 : y.LeftMoves), -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i j_1β§) β§
β (i : (-x).LeftMoves) (j_1 : (-y).LeftMoves), -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i j_1β§ | case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i kβ§
case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 19 | 29 | apply mulOption_lt hx hy ihxy ihyx | case refine_1.left.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦x.mulOption (-y) j lβ§ > β¦x.mulOption y i kβ§ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 20 | 29 | simp_rw [β mulOption_symm (-y), mulOption_neg_neg x] | case refine_1.left.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦x.mulOption (-y) j lβ§ > β¦(-x).mulOption (-y) i kβ§ | case refine_1.left.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-y).mulOption (- -x) l (toLeftMovesNeg (toRightMovesNeg j))β§ > β¦(-y).mulOption (-x) k iβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 21 | 29 | apply mulOption_lt hy.neg hx.neg ihyxn ihxyn | case refine_1.left.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : x.LeftMoves
l : (-y).LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-y).mulOption (- -x) l (toLeftMovesNeg (toRightMovesNeg j))β§ > β¦(-y).mulOption (-x) k iβ§ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 22 | 29 | simp only [β mulOption_symm y] | case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦x.mulOption y i kβ§ | case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦y.mulOption (-x) l jβ§ > β¦y.mulOption x k iβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 23 | 29 | apply mulOption_lt hy hx ihyx ihxy | case refine_1.right.left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : x.LeftMoves
k : y.LeftMoves
β’ -β¦y.mulOption (-x) l jβ§ > β¦y.mulOption x k iβ§ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 24 | 29 | rw [mulOption_neg_neg y] | case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption y j lβ§ > β¦(-x).mulOption (-y) i kβ§ | case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption (- -y) j (toLeftMovesNeg (toRightMovesNeg l))β§ > β¦(-x).mulOption (-y) i kβ§ |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 25 | 29 | apply mulOption_lt hx.neg hy.neg ihxyn ihyxn | case refine_1.right.right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
j : (-x).LeftMoves
l : y.LeftMoves
i : (-x).LeftMoves
k : (-y).LeftMoves
β’ -β¦(-x).mulOption (- -y) j (toLeftMovesNeg (toRightMovesNeg l))β§ > β¦(-x).mulOption (-y) i kβ§ | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 26 | 29 | cases x | case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ihxy : IH1 x y
ihyx : IH1 y x
ihxyn : IH1 (-x) (-y)
ihyxn : IH1 (-y) (-x)
β’ β (j : (x * y).RightMoves), ((x * y).moveRight j).Numeric | case refine_3.mk
xβ xβ xβ x' y yβ yβ yβ y' : PGame
hy : y.Numeric
Ξ±β Ξ²β : Type u
aβΒΉ : Ξ±β β PGame
aβ : Ξ²β β PGame
ih : β (a : Args), ArgsRel a (Args.P1 (PGame.mk Ξ±β Ξ²β aβΒΉ aβ) y) β P124 a
hx : (PGame.mk Ξ±β Ξ²β aβΒΉ aβ).Numeric
ihxy : IH1 (PGame.mk Ξ±β Ξ²β aβΒΉ aβ) y
ihyx : IH1 y (PGame.mk Ξ±β Ξ²β aβΒΉ aβ)
ihxyn : IH1 (-PGame.mk Ξ±β Ξ²β aβΒΉ aβ) (-y)
ihyxn : IH1 (-y) (-PGame.mk Ξ±β Ξ²β aβΒΉ aβ)
β’ β (j : (PGame.mk Ξ±β Ξ²β aβΒΉ aβ * y).RightMoves), ((PGame.mk Ξ±β Ξ²β aβΒΉ aβ * y).moveRight j).Numeric |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 27 | 29 | cases y | case refine_3.mk
xβ xβ xβ x' y yβ yβ yβ y' : PGame
hy : y.Numeric
Ξ±β Ξ²β : Type u
aβΒΉ : Ξ±β β PGame
aβ : Ξ²β β PGame
ih : β (a : Args), ArgsRel a (Args.P1 (PGame.mk Ξ±β Ξ²β aβΒΉ aβ) y) β P124 a
hx : (PGame.mk Ξ±β Ξ²β aβΒΉ aβ).Numeric
ihxy : IH1 (PGame.mk Ξ±β Ξ²β aβΒΉ aβ) y
ihyx : IH1 y (PGame.mk Ξ±β Ξ²β aβΒΉ aβ)
ihxyn : IH1 (-PGame.mk Ξ±β Ξ²β aβΒΉ aβ) (-y)
ihyxn : IH1 (-y) (-PGame.mk Ξ±β Ξ²β aβΒΉ aβ)
β’ β (j : (PGame.mk Ξ±β Ξ²β aβΒΉ aβ * y).RightMoves), ((PGame.mk Ξ±β Ξ²β aβΒΉ aβ * y).moveRight j).Numeric | case refine_3.mk.mk
xβ xβ xβ x' yβ yβ yβ y' : PGame
Ξ±βΒΉ Ξ²βΒΉ : Type u
aβΒ³ : Ξ±βΒΉ β PGame
aβΒ² : Ξ²βΒΉ β PGame
hx : (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²).Numeric
Ξ±β Ξ²β : Type u
aβΒΉ : Ξ±β β PGame
aβ : Ξ²β β PGame
hy : (PGame.mk Ξ±β Ξ²β aβΒΉ aβ).Numeric
ih : β (a : Args), ArgsRel a (Args.P1 (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²) (PGame.mk Ξ±β Ξ²β aβΒΉ aβ)) β P124 a
ihxy : IH1 (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²) (PGame.mk Ξ±β Ξ²β aβΒΉ aβ)
ihyx : IH1 (PGame.mk Ξ±β Ξ²β aβΒΉ aβ) (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²)
ihxyn : IH1 (-PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²) (-PGame.mk Ξ±β Ξ²β aβΒΉ aβ)
ihyxn : IH1 (-PGame.mk Ξ±β Ξ²β aβΒΉ aβ) (-PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²)
β’ β (j : (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ² * PGame.mk Ξ±β Ξ²β aβΒΉ aβ).RightMoves),
((PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ² * PGame.mk Ξ±β Ξ²β aβΒΉ aβ).moveRight j).Numeric |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 28 | Surreal.Multiplication.P1_of_ih | [
[
259,
39
],
[
281,
56
]
]
| 28 | 29 | rintro (β¨i,jβ©|β¨i,jβ©) <;>
refine ((numeric_option_mul ih ?_).add <| numeric_mul_option ih ?_).sub
(numeric_option_mul_option ih ?_ ?_) <;>
solve_by_elim [IsOption.mk_left, IsOption.mk_right] | case refine_3.mk.mk
xβ xβ xβ x' yβ yβ yβ y' : PGame
Ξ±βΒΉ Ξ²βΒΉ : Type u
aβΒ³ : Ξ±βΒΉ β PGame
aβΒ² : Ξ²βΒΉ β PGame
hx : (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²).Numeric
Ξ±β Ξ²β : Type u
aβΒΉ : Ξ±β β PGame
aβ : Ξ²β β PGame
hy : (PGame.mk Ξ±β Ξ²β aβΒΉ aβ).Numeric
ih : β (a : Args), ArgsRel a (Args.P1 (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²) (PGame.mk Ξ±β Ξ²β aβΒΉ aβ)) β P124 a
ihxy : IH1 (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²) (PGame.mk Ξ±β Ξ²β aβΒΉ aβ)
ihyx : IH1 (PGame.mk Ξ±β Ξ²β aβΒΉ aβ) (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²)
ihxyn : IH1 (-PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²) (-PGame.mk Ξ±β Ξ²β aβΒΉ aβ)
ihyxn : IH1 (-PGame.mk Ξ±β Ξ²β aβΒΉ aβ) (-PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ²)
β’ β (j : (PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ² * PGame.mk Ξ±β Ξ²β aβΒΉ aβ).RightMoves),
((PGame.mk Ξ±βΒΉ Ξ²βΒΉ aβΒ³ aβΒ² * PGame.mk Ξ±β Ξ²β aβΒΉ aβ).moveRight j).Numeric | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 29 | Surreal.Multiplication.ihββ | [
[
295,
30
],
[
301,
63
]
]
| 0 | 8 | rw [IH24] | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ IH24 xβ xβ y | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ β β¦z : PGameβ¦, (z.IsOption xβ β P24 z xβ y) β§ (z.IsOption xβ β P24 xβ z y) β§ (z.IsOption y β P24 xβ xβ z) |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 29 | Surreal.Multiplication.ihββ | [
[
295,
30
],
[
301,
63
]
]
| 1 | 8 | refine fun z β¦ β¨?_, ?_, ?_β© <;>
refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_) | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ β β¦z : PGameβ¦, (z.IsOption xβ β P24 z xβ y) β§ (z.IsOption xβ β P24 xβ z y) β§ (z.IsOption y β P24 xβ xβ z) | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 z xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset
case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 xβ z y).toMultiset (Args.P24 xβ xβ y).toMultiset
case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption y
β’ CutExpand IsOption (Args.P24 xβ xβ z).toMultiset (Args.P24 xβ xβ y).toMultiset |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 29 | Surreal.Multiplication.ihββ | [
[
295,
30
],
[
301,
63
]
]
| 2 | 8 | Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h) | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 z xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset
case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 xβ z y).toMultiset (Args.P24 xβ xβ y).toMultiset
case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption y
β’ CutExpand IsOption (Args.P24 xβ xβ z).toMultiset (Args.P24 xβ xβ y).toMultiset | case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 xβ z y).toMultiset (Args.P24 xβ xβ y).toMultiset
case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption y
β’ CutExpand IsOption (Args.P24 xβ xβ z).toMultiset (Args.P24 xβ xβ y).toMultiset |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 29 | Surreal.Multiplication.ihββ | [
[
295,
30
],
[
301,
63
]
]
| 3 | 8 | Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h) | case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 xβ z y).toMultiset (Args.P24 xβ xβ y).toMultiset
case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption y
β’ CutExpand IsOption (Args.P24 xβ xβ z).toMultiset (Args.P24 xβ xβ y).toMultiset | case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption y
β’ CutExpand IsOption (Args.P24 xβ xβ z).toMultiset (Args.P24 xβ xβ y).toMultiset |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 29 | Surreal.Multiplication.ihββ | [
[
295,
30
],
[
301,
63
]
]
| 4 | 8 | Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h) | case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption y
β’ CutExpand IsOption (Args.P24 xβ xβ z).toMultiset (Args.P24 xβ xβ y).toMultiset | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 29 | Surreal.Multiplication.ihββ | [
[
295,
30
],
[
301,
63
]
]
| 5 | 8 | exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h) | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 z xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 29 | Surreal.Multiplication.ihββ | [
[
295,
30
],
[
301,
63
]
]
| 6 | 8 | exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h) | case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 xβ z y).toMultiset (Args.P24 xβ xβ y).toMultiset | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 29 | Surreal.Multiplication.ihββ | [
[
295,
30
],
[
301,
63
]
]
| 7 | 8 | exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h) | case refine_3
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z : PGame
h : z.IsOption y
β’ CutExpand IsOption (Args.P24 xβ xβ z).toMultiset (Args.P24 xβ xβ y).toMultiset | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 31 | Surreal.Multiplication.ih4 | [
[
309,
28
],
[
316,
69
]
]
| 0 | 8 | refine fun z w h β¦ β¨?_, ?_β© | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ IH4 xβ xβ y | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
β’ z.IsOption xβ β P2 z xβ w
case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
β’ z.IsOption xβ β P2 xβ z w |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 31 | Surreal.Multiplication.ih4 | [
[
309,
28
],
[
316,
69
]
]
| 1 | 8 | all_goals
intro h'
apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|
(cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1
try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'
try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h' | case refine_1
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
β’ z.IsOption xβ β P2 z xβ w
case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
β’ z.IsOption xβ β P2 xβ z w | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 31 | Surreal.Multiplication.ih4 | [
[
309,
28
],
[
316,
69
]
]
| 2 | 8 | intro h' | case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
β’ z.IsOption xβ β P2 xβ z w | case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
h' : z.IsOption xβ
β’ P2 xβ z w |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 31 | Surreal.Multiplication.ih4 | [
[
309,
28
],
[
316,
69
]
]
| 3 | 8 | apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|
(cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1 | case refine_2
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
h' : z.IsOption xβ
β’ P2 xβ z w | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
h' : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 xβ z w).toMultiset ({xβ} + {xβ, w}) |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 31 | Surreal.Multiplication.ih4 | [
[
309,
28
],
[
316,
69
]
]
| 4 | 8 | try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h' | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
h' : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 z xβ w).toMultiset ({xβ} + {xβ, w}) | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 31 | Surreal.Multiplication.ih4 | [
[
309,
28
],
[
316,
69
]
]
| 5 | 8 | try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h' | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
h' : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 xβ z w).toMultiset ({xβ} + {xβ, w}) | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 31 | Surreal.Multiplication.ih4 | [
[
309,
28
],
[
316,
69
]
]
| 6 | 8 | exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h' | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
h' : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 z xβ w).toMultiset ({xβ} + {xβ, w}) | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 31 | Surreal.Multiplication.ih4 | [
[
309,
28
],
[
316,
69
]
]
| 7 | 8 | exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h' | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
z w : PGame
h : w.IsOption y
h' : z.IsOption xβ
β’ CutExpand IsOption (Args.P24 xβ z w).toMultiset ({xβ} + {xβ, w}) | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 32 | Surreal.Multiplication.numeric_of_ih | [
[
318,
62
],
[
321,
57
]
]
| 0 | 5 | constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_) | x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ (xβ * y).Numeric β§ (xβ * y).Numeric | case left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ CutExpand IsOption (Args.P1 xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset
case right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ CutExpand IsOption (Args.P1 xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 32 | Surreal.Multiplication.numeric_of_ih | [
[
318,
62
],
[
321,
57
]
]
| 1 | 5 | Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero | case left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ CutExpand IsOption (Args.P1 xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset
case right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ CutExpand IsOption (Args.P1 xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset | case right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ CutExpand IsOption (Args.P1 xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 32 | Surreal.Multiplication.numeric_of_ih | [
[
318,
62
],
[
321,
57
]
]
| 2 | 5 | Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero | case right
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ CutExpand IsOption (Args.P1 xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset | no goals |
Mathlib/SetTheory/Surreal/Multiplication.lean | [
[
"Mathlib.SetTheory.Surreal.Basic",
"Mathlib/SetTheory/Surreal/Basic.lean"
],
[
"Mathlib.Logic.Hydra",
"Mathlib/Logic/Hydra.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
]
| [
{
"code": "def P1 (xβ xβ xβ yβ yβ yβ : PGame) :=\n β¦xβ * yββ§ + β¦xβ * yββ§ - β¦xβ * yββ§ < β¦xβ * yββ§ + β¦xβ * yββ§ - (β¦xβ * yββ§ : Game)",
"end": [
74,
81
],
"full_name": "Surreal.Multiplication.P1",
"kind": "commanddeclaration",
"start": [
71,
1
]
},
{
"code": "def P2 (xβ xβ y : PGame) := xβ β xβ β β¦xβ * yβ§ = (β¦xβ * yβ§ : Game)",
"end": [
77,
67
],
"full_name": "Surreal.Multiplication.P2",
"kind": "commanddeclaration",
"start": [
76,
1
]
},
{
"code": "def P3 (xβ xβ yβ yβ : PGame) := β¦xβ * yββ§ + β¦xβ * yββ§ < β¦xβ * yββ§ + (β¦xβ * yββ§ : Game)",
"end": [
80,
87
],
"full_name": "Surreal.Multiplication.P3",
"kind": "commanddeclaration",
"start": [
79,
1
]
},
{
"code": "def P4 (xβ xβ y : PGame) :=\n xβ < xβ β (β i, P3 xβ xβ (y.moveLeft i) y) β§ β j, P3 xβ xβ ((-y).moveLeft j) (-y)",
"end": [
87,
84
],
"full_name": "Surreal.Multiplication.P4",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "def P24 (xβ xβ y : PGame) : Prop := P2 xβ xβ y β§ P4 xβ xβ y",
"end": [
90,
60
],
"full_name": "Surreal.Multiplication.P24",
"kind": "commanddeclaration",
"start": [
89,
1
]
},
{
"code": "lemma P3_comm : P3 xβ xβ yβ yβ β P3 yβ yβ xβ xβ := by\n rw [P3, P3, add_comm]\n congr! 2 <;> rw [quot_mul_comm]",
"end": [
98,
34
],
"full_name": "Surreal.Multiplication.P3_comm",
"kind": "lemma",
"start": [
96,
1
]
},
{
"code": "lemma P3.trans (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P3 xβ xβ yβ yβ := by\n rw [P3] at hβ hβ\n rw [P3, β add_lt_add_iff_left (β¦xβ * yββ§ + β¦xβ * yββ§)]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
103,
44
],
"full_name": "Surreal.Multiplication.P3.trans",
"kind": "lemma",
"start": [
100,
1
]
},
{
"code": "lemma P3_neg : P3 xβ xβ yβ yβ β P3 (-xβ) (-xβ) yβ yβ := by\n simp_rw [P3, quot_neg_mul]\n rw [β _root_.neg_lt_neg_iff]\n abel_nf",
"end": [
108,
10
],
"full_name": "Surreal.Multiplication.P3_neg",
"kind": "lemma",
"start": [
105,
1
]
},
{
"code": "lemma P2_neg_left : P2 xβ xβ y β P2 (-xβ) (-xβ) y := by\n rw [P2, P2]\n constructor\n Β· rw [quot_neg_mul, quot_neg_mul, eq_comm, neg_inj, neg_equiv_neg_iff, PGame.equiv_comm]\n exact (Β· Β·)\n Β· rw [PGame.equiv_comm, neg_equiv_neg_iff, quot_neg_mul, quot_neg_mul, neg_inj, eq_comm]\n exact (Β· Β·)",
"end": [
116,
16
],
"full_name": "Surreal.Multiplication.P2_neg_left",
"kind": "lemma",
"start": [
110,
1
]
},
{
"code": "lemma P2_neg_right : P2 xβ xβ y β P2 xβ xβ (-y) := by\n rw [P2, P2, quot_mul_neg, quot_mul_neg, neg_inj]",
"end": [
119,
51
],
"full_name": "Surreal.Multiplication.P2_neg_right",
"kind": "lemma",
"start": [
118,
1
]
},
{
"code": "lemma P4_neg_left : P4 xβ xβ y β P4 (-xβ) (-xβ) y := by\n simp_rw [P4, PGame.neg_lt_neg_iff, moveLeft_neg', β P3_neg]",
"end": [
122,
62
],
"full_name": "Surreal.Multiplication.P4_neg_left",
"kind": "lemma",
"start": [
121,
1
]
},
{
"code": "lemma P4_neg_right : P4 xβ xβ y β P4 xβ xβ (-y) := by\n rw [P4, P4, neg_neg, and_comm]",
"end": [
125,
33
],
"full_name": "Surreal.Multiplication.P4_neg_right",
"kind": "lemma",
"start": [
124,
1
]
},
{
"code": "lemma P24_neg_left : P24 xβ xβ y β P24 (-xβ) (-xβ) y := by rw [P24, P24, P2_neg_left, P4_neg_left]",
"end": [
127,
99
],
"full_name": "Surreal.Multiplication.P24_neg_left",
"kind": "lemma",
"start": [
127,
1
]
},
{
"code": "lemma P24_neg_right : P24 xβ xβ y β P24 xβ xβ (-y) := by rw [P24, P24, P2_neg_right, P4_neg_right]",
"end": [
128,
99
],
"full_name": "Surreal.Multiplication.P24_neg_right",
"kind": "lemma",
"start": [
128,
1
]
},
{
"code": "lemma mulOption_lt_iff_P1 {i j k l} :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ β\n P1 (x.moveLeft i) x (x.moveLeft j) y (y.moveLeft k) (-(-y).moveLeft l) := by\n dsimp only [P1, mulOption, quot_sub, quot_add]\n simp_rw [neg_sub', neg_add, quot_mul_neg, neg_neg]",
"end": [
136,
53
],
"full_name": "Surreal.Multiplication.mulOption_lt_iff_P1",
"kind": "lemma",
"start": [
132,
1
]
},
{
"code": "lemma mulOption_lt_mul_iff_P3 {i j} :\n β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game) β P3 (x.moveLeft i) x (y.moveLeft j) y := by\n dsimp only [mulOption, quot_sub, quot_add]\n exact sub_lt_iff_lt_add'",
"end": [
141,
27
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_iff_P3",
"kind": "lemma",
"start": [
138,
1
]
},
{
"code": "lemma P1_of_eq (he : xβ β xβ) (hβ : P2 xβ xβ yβ) (hβ : P2 xβ xβ yβ) (h3 : P3 xβ xβ yβ yβ) :\n P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, β hβ he, β hβ he, sub_lt_sub_iff]\n convert add_lt_add_left h3 β¦xβ * yββ§ using 1 <;> abel",
"end": [
146,
56
],
"full_name": "Surreal.Multiplication.P1_of_eq",
"kind": "lemma",
"start": [
143,
1
]
},
{
"code": "lemma P1_of_lt (hβ : P3 xβ xβ yβ yβ) (hβ : P3 xβ xβ yβ yβ) : P1 xβ xβ xβ yβ yβ yβ := by\n rw [P1, sub_lt_sub_iff, β add_lt_add_iff_left β¦xβ * yββ§]\n convert add_lt_add hβ hβ using 1 <;> abel",
"end": [
150,
44
],
"full_name": "Surreal.Multiplication.P1_of_lt",
"kind": "lemma",
"start": [
148,
1
]
},
{
"code": "inductive Args : Type (u+1)\n | P1 (x y : PGame.{u}) : Args\n | P24 (xβ xβ y : PGame.{u}) : Args",
"end": [
155,
37
],
"full_name": "Surreal.Multiplication.Args",
"kind": "commanddeclaration",
"start": [
152,
1
]
},
{
"code": "def Args.toMultiset : Args β Multiset PGame\n | (Args.P1 x y) => {x, y}\n | (Args.P24 xβ xβ y) => {xβ, xβ, y}",
"end": [
160,
38
],
"full_name": "Surreal.Multiplication.Args.toMultiset",
"kind": "commanddeclaration",
"start": [
157,
1
]
},
{
"code": "def Args.Numeric (a : Args) := β x β a.toMultiset, SetTheory.PGame.Numeric x",
"end": [
163,
77
],
"full_name": "Surreal.Multiplication.Args.Numeric",
"kind": "commanddeclaration",
"start": [
162,
1
]
},
{
"code": "lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric β x.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
166,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P1",
"kind": "lemma",
"start": [
165,
1
]
},
{
"code": "lemma Args.numeric_P24 {xβ xβ y} :\n (Args.P24 xβ xβ y).Numeric β xβ.Numeric β§ xβ.Numeric β§ y.Numeric := by\n simp [Args.Numeric, Args.toMultiset]",
"end": [
170,
39
],
"full_name": "Surreal.Multiplication.Args.numeric_P24",
"kind": "lemma",
"start": [
168,
1
]
},
{
"code": "def ArgsRel := InvImage (TransGen <| CutExpand IsOption) Args.toMultiset",
"end": [
177,
73
],
"full_name": "Surreal.Multiplication.ArgsRel",
"kind": "commanddeclaration",
"start": [
174,
1
]
},
{
"code": "theorem argsRel_wf : WellFounded ArgsRel",
"end": [
180,
89
],
"full_name": "Surreal.Multiplication.argsRel_wf",
"kind": "commanddeclaration",
"start": [
179,
1
]
},
{
"code": "def P124 : Args β Prop\n | (Args.P1 x y) => Numeric (x * y)\n | (Args.P24 xβ xβ y) => P24 xβ xβ y",
"end": [
185,
38
],
"full_name": "Surreal.Multiplication.P124",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "lemma ArgsRel.numeric_closed {a' a} : ArgsRel a' a β a.Numeric β a'.Numeric :=\n TransGen.closed' <| @cutExpand_closed _ IsOption β¨wf_isOption.isIrrefl.1β© _ Numeric.isOption",
"end": [
189,
95
],
"full_name": "Surreal.Multiplication.ArgsRel.numeric_closed",
"kind": "lemma",
"start": [
187,
1
]
},
{
"code": "def IH1 (x y : PGame) : Prop :=\n β β¦xβ xβ y'β¦, IsOption xβ x β IsOption xβ x β (y' = y β¨ IsOption y' y) β P24 xβ xβ y'",
"end": [
193,
88
],
"full_name": "Surreal.Multiplication.IH1",
"kind": "commanddeclaration",
"start": [
191,
1
]
},
{
"code": "lemma ih1_neg_left : IH1 x y β IH1 (-x) y :=\n fun h xβ xβ y' hβ hβ hy β¦ by\n rw [isOption_neg] at hβ hβ\n exact P24_neg_left.2 (h hβ hβ hy)",
"end": [
200,
38
],
"full_name": "Surreal.Multiplication.ih1_neg_left",
"kind": "lemma",
"start": [
197,
1
]
},
{
"code": "lemma ih1_neg_right : IH1 x y β IH1 x (-y) :=\n fun h xβ xβ y' β¦ by\n rw [β neg_eq_iff_eq_neg, isOption_neg, P24_neg_right]\n apply h",
"end": [
205,
12
],
"full_name": "Surreal.Multiplication.ih1_neg_right",
"kind": "lemma",
"start": [
202,
1
]
},
{
"code": "lemma numeric_option_mul (h : IsOption x' x) : (x' * y).Numeric :=\n ih (Args.P1 x' y) (TransGen.single <| cutExpand_pair_left h)",
"end": [
212,
63
],
"full_name": "Surreal.Multiplication.numeric_option_mul",
"kind": "lemma",
"start": [
211,
1
]
},
{
"code": "lemma numeric_mul_option (h : IsOption y' y) : (x * y').Numeric :=\n ih (Args.P1 x y') (TransGen.single <| cutExpand_pair_right h)",
"end": [
215,
64
],
"full_name": "Surreal.Multiplication.numeric_mul_option",
"kind": "lemma",
"start": [
214,
1
]
},
{
"code": "lemma numeric_option_mul_option (hx : IsOption x' x) (hy : IsOption y' y) : (x' * y').Numeric :=\n ih (Args.P1 x' y') ((TransGen.single <| cutExpand_pair_right hy).tail <| cutExpand_pair_left hx)",
"end": [
218,
99
],
"full_name": "Surreal.Multiplication.numeric_option_mul_option",
"kind": "lemma",
"start": [
217,
1
]
},
{
"code": "lemma ih1 : IH1 x y := by\n rintro xβ xβ y' hβ hβ (rfl|hy) <;> apply ih (Args.P24 _ _ _)\n on_goal 2 => refine TransGen.tail ?_ (cutExpand_pair_right hy)\n all_goals exact TransGen.single (cutExpand_double_left hβ hβ)",
"end": [
223,
64
],
"full_name": "Surreal.Multiplication.ih1",
"kind": "lemma",
"start": [
220,
1
]
},
{
"code": "lemma ih1_swap : IH1 y x := ih1 <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih β’\n exact ih",
"end": [
227,
11
],
"full_name": "Surreal.Multiplication.ih1_swap",
"kind": "lemma",
"start": [
225,
1
]
},
{
"code": "lemma P3_of_ih (hy : Numeric y) (ihyx : IH1 y x) (i k l) :\n P3 (x.moveLeft i) x (y.moveLeft k) (-(-y).moveLeft l) :=\n P3_comm.2 <| ((ihyx (IsOption.moveLeft k) (isOption_neg.1 <| .moveLeft l) <| Or.inl rfl).2\n (by rw [β moveRight_neg_symm]; apply hy.left_lt_right)).1 i",
"end": [
232,
64
],
"full_name": "Surreal.Multiplication.P3_of_ih",
"kind": "lemma",
"start": [
229,
1
]
},
{
"code": "lemma P24_of_ih (ihxy : IH1 x y) (i j) : P24 (x.moveLeft i) (x.moveLeft j) y :=\n ihxy (IsOption.moveLeft i) (IsOption.moveLeft j) (Or.inl rfl)",
"end": [
235,
64
],
"full_name": "Surreal.Multiplication.P24_of_ih",
"kind": "lemma",
"start": [
234,
1
]
},
{
"code": "lemma mulOption_lt_of_lt (i j k l) (h : x.moveLeft i < x.moveLeft j) :\n (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ :=\n mulOption_lt_iff_P1.2 <| P1_of_lt (P3_of_ih hy ihyx j k l) <| ((P24_of_ih ihxy i j).2 h).1 k",
"end": [
245,
95
],
"full_name": "Surreal.Multiplication.mulOption_lt_of_lt",
"kind": "lemma",
"start": [
243,
1
]
},
{
"code": "lemma mulOption_lt (i j k l) : (β¦mulOption x y i kβ§ : Game) < -β¦mulOption x (-y) j lβ§ := by\n obtain (h|h|h) := lt_or_equiv_or_gt (hx.moveLeft i) (hx.moveLeft j)\n Β· exact mulOption_lt_of_lt hy ihxy ihyx i j k l h\n Β· have ml := @IsOption.moveLeft\n exact mulOption_lt_iff_P1.2 (P1_of_eq h (P24_of_ih ihxy i j).1\n (ihxy (ml i) (ml j) <| Or.inr <| isOption_neg.1 <| ml l).1 <| P3_of_ih hy ihyx i k l)\n Β· rw [mulOption_neg_neg, lt_neg]\n exact mulOption_lt_of_lt hy.neg (ih1_neg_right ihxy) (ih1_neg_left ihyx) j i l _ h",
"end": [
254,
87
],
"full_name": "Surreal.Multiplication.mulOption_lt",
"kind": "lemma",
"start": [
247,
1
]
},
{
"code": "theorem P1_of_ih : (x * y).Numeric",
"end": [
281,
56
],
"full_name": "Surreal.Multiplication.P1_of_ih",
"kind": "commanddeclaration",
"start": [
258,
1
]
},
{
"code": "def IH24 (xβ xβ y : PGame) : Prop :=\n β β¦zβ¦, (IsOption z xβ β P24 z xβ y) β§ (IsOption z xβ β P24 xβ z y) β§ (IsOption z y β P24 xβ xβ z)",
"end": [
285,
100
],
"full_name": "Surreal.Multiplication.IH24",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "def IH4 (xβ xβ y : PGame) : Prop :=\n β β¦z wβ¦, IsOption w y β (IsOption z xβ β P2 z xβ w) β§ (IsOption z xβ β P2 xβ z w)",
"end": [
289,
84
],
"full_name": "Surreal.Multiplication.IH4",
"kind": "commanddeclaration",
"start": [
287,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := by\n rw [IH24]\n refine fun z β¦ β¨?_, ?_, ?_β© <;>\n refine fun h β¦ ih' (Args.P24 _ _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_left h)\n Β· exact (cutExpand_add_left {xβ}).2 (cutExpand_pair_right h)",
"end": [
301,
63
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
295,
1
]
},
{
"code": "lemma ihββ : IH24 xβ xβ y := ihββ <| by\n simp_rw [ArgsRel, InvImage, Args.toMultiset, Multiset.pair_comm] at ih' β’\n suffices {xβ, y, xβ} = {xβ, y, xβ} by rwa [β this]\n dsimp only [Multiset.insert_eq_cons, β Multiset.singleton_add] at ih' β’\n abel",
"end": [
307,
7
],
"full_name": "Surreal.Multiplication.ihββ",
"kind": "lemma",
"start": [
303,
1
]
},
{
"code": "lemma ih4 : IH4 xβ xβ y := by\n refine fun z w h β¦ β¨?_, ?_β©\n all_goals\n intro h'\n apply (ih' (Args.P24 _ _ _) <| (TransGen.single _).tail <|\n (cutExpand_add_left {xβ}).2 <| cutExpand_pair_right h).1\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_left h'\n try exact (cutExpand_add_right {w}).2 <| cutExpand_pair_right h'",
"end": [
316,
69
],
"full_name": "Surreal.Multiplication.ih4",
"kind": "lemma",
"start": [
309,
1
]
},
{
"code": "lemma numeric_of_ih : (xβ * y).Numeric β§ (xβ * y).Numeric := by\n constructor <;> refine ih' (Args.P1 _ _) (TransGen.single ?_)\n Β· exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero\n Β· exact (cutExpand_add_right {xβ, y}).2 cutExpand_zero",
"end": [
321,
57
],
"full_name": "Surreal.Multiplication.numeric_of_ih",
"kind": "lemma",
"start": [
318,
1
]
},
{
"code": "lemma ih24_neg : IH24 xβ xβ y β IH24 (-xβ) (-xβ) y β§ IH24 xβ xβ (-y) := by\n simp_rw [IH24, β P24_neg_right, isOption_neg]\n refine fun h β¦ β¨fun z β¦ β¨?_, ?_, ?_β©,\n fun z β¦ β¨(@h z).1, (@h z).2.1, P24_neg_right.2 β (@h <| -z).2.2β©β©\n all_goals\n rw [P24_neg_left]\n simp only [neg_neg]\n first | exact (@h <| -z).2.1 | exact (@h <| -z).1 | exact (@h z).2.2",
"end": [
331,
73
],
"full_name": "Surreal.Multiplication.ih24_neg",
"kind": "lemma",
"start": [
323,
1
]
},
{
"code": "lemma ih4_neg : IH4 xβ xβ y β IH4 (-xβ) (-xβ) y β§ IH4 xβ xβ (-y) := by\n simp_rw [IH4, isOption_neg]\n refine fun h β¦ β¨fun z w h' β¦ ?_, fun z w h' β¦ ?_β©\n Β· convert (h h').symm using 2 <;> rw [P2_neg_left, neg_neg]\n Β· convert h h' using 2 <;> rw [P2_neg_right]",
"end": [
338,
47
],
"full_name": "Surreal.Multiplication.ih4_neg",
"kind": "lemma",
"start": [
333,
1
]
},
{
"code": "lemma mulOption_lt_mul_of_equiv (hn : xβ.Numeric) (h : IH24 xβ xβ y) (he : xβ β xβ) (i j) :\n β¦mulOption xβ y i jβ§ < (β¦xβ * yβ§ : Game) := by\n convert sub_lt_iff_lt_add'.2 ((((@h _).1 <| IsOption.moveLeft i).2 _).1 j) using 1\n Β· rw [β ((@h _).2.2 <| IsOption.moveLeft j).1 he]\n rfl\n Β· rw [β lt_congr_right he]\n apply hn.moveLeft_lt",
"end": [
346,
25
],
"full_name": "Surreal.Multiplication.mulOption_lt_mul_of_equiv",
"kind": "lemma",
"start": [
340,
1
]
},
{
"code": "theorem mul_right_le_of_equiv (hβ : xβ.Numeric) (hβ : xβ.Numeric)\n (hββ : IH24 xβ xβ y) (hββ : IH24 xβ xβ y) (he : xβ β xβ) : xβ * y β€ xβ * y",
"end": [
362,
73
],
"full_name": "Surreal.Multiplication.mul_right_le_of_equiv",
"kind": "commanddeclaration",
"start": [
348,
1
]
},
{
"code": "def MulOptionsLTMul (x y : PGame) : Prop := β β¦i jβ¦, β¦mulOption x y i jβ§ < (β¦x * yβ§ : Game)",
"end": [
365,
92
],
"full_name": "Surreal.Multiplication.MulOptionsLTMul",
"kind": "commanddeclaration",
"start": [
364,
1
]
},
{
"code": "lemma mulOptionsLTMul_of_numeric (hn : (x * y).Numeric) :\n (MulOptionsLTMul x y β§ MulOptionsLTMul (-x) (-y)) β§\n (MulOptionsLTMul x (-y) β§ MulOptionsLTMul (-x) y) := by\n constructor\n Β· have h := hn.moveLeft_lt\n simp_rw [lt_iff_game_lt] at h\n convert (leftMoves_mul_iff <| GT.gt _).1 h\n rw [β quot_neg_mul_neg]\n rfl\n Β· have h := hn.lt_moveRight\n simp_rw [lt_iff_game_lt, rightMoves_mul_iff] at h\n refine h.imp ?_ ?_ <;> refine forallβ_imp fun a b β¦ ?_\n all_goals\n rw [lt_neg]\n first | rw [quot_mul_neg] | rw [quot_neg_mul]\n exact id",
"end": [
386,
15
],
"full_name": "Surreal.Multiplication.mulOptionsLTMul_of_numeric",
"kind": "lemma",
"start": [
367,
1
]
},
{
"code": "def IH3 (xβ x' xβ yβ yβ : PGame) : Prop :=\n P2 xβ x' yβ β§ P2 xβ x' yβ β§ P3 x' xβ yβ yβ β§ (xβ < x' β P3 xβ x' yβ yβ)",
"end": [
395,
76
],
"full_name": "Surreal.Multiplication.IH3",
"kind": "commanddeclaration",
"start": [
388,
1
]
},
{
"code": "lemma ih3_of_ih (h24 : IH24 xβ xβ y) (h4 : IH4 xβ xβ y) (hl : MulOptionsLTMul xβ y) (i j) :\n IH3 xβ (xβ.moveLeft i) xβ (y.moveLeft j) y :=\n have ml := @IsOption.moveLeft\n have h24 := (@h24 _).2.1 (ml i)\n β¨(h4 <| ml j).2 (ml i), h24.1, mulOption_lt_mul_iff_P3.1 (@hl i j), fun l β¦ (h24.2 l).1 _β©",
"end": [
401,
93
],
"full_name": "Surreal.Multiplication.ih3_of_ih",
"kind": "lemma",
"start": [
397,
1
]
},
{
"code": "lemma P3_of_le_left {yβ yβ} (i) (h : IH3 xβ (xβ.moveLeft i) xβ yβ yβ) (hl : xβ β€ xβ.moveLeft i) :\n P3 xβ xβ yβ yβ := by\n obtain (hl|he) := lt_or_equiv_of_le hl\n Β· exact (h.2.2.2 hl).trans h.2.2.1\n Β· rw [P3, h.1 he, h.2.1 he]\n exact h.2.2.1",
"end": [
408,
18
],
"full_name": "Surreal.Multiplication.P3_of_le_left",
"kind": "lemma",
"start": [
403,
1
]
},
{
"code": "theorem P3_of_lt {yβ yβ} (h : β i, IH3 xβ (xβ.moveLeft i) xβ yβ yβ)\n (hs : β i, IH3 (-xβ) ((-xβ).moveLeft i) (-xβ) yβ yβ) (hl : xβ < xβ) :\n P3 xβ xβ yβ yβ",
"end": [
419,
45
],
"full_name": "Surreal.Multiplication.P3_of_lt",
"kind": "commanddeclaration",
"start": [
410,
1
]
},
{
"code": "theorem main (a : Args) : a.Numeric β P124 a",
"end": [
448,
50
],
"full_name": "Surreal.Multiplication.main",
"kind": "commanddeclaration",
"start": [
421,
1
]
},
{
"code": "theorem Numeric.mul : Numeric (x * y)",
"end": [
459,
80
],
"full_name": "SetTheory.PGame.Numeric.mul",
"kind": "commanddeclaration",
"start": [
459,
1
]
},
{
"code": "theorem P24 : P24 xβ xβ y",
"end": [
461,
75
],
"full_name": "SetTheory.PGame.P24",
"kind": "commanddeclaration",
"start": [
461,
1
]
},
{
"code": "theorem Equiv.mul_congr_left (he : xβ β xβ) : xβ * y β xβ * y",
"end": [
464,
47
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_left",
"kind": "commanddeclaration",
"start": [
463,
1
]
},
{
"code": "theorem Equiv.mul_congr_right (he : yβ β yβ) : x * yβ β x * yβ",
"end": [
467,
92
],
"full_name": "SetTheory.PGame.Equiv.mul_congr_right",
"kind": "commanddeclaration",
"start": [
466,
1
]
},
{
"code": "theorem Equiv.mul_congr (hx : xβ β xβ) (hy : yβ β yβ) : xβ * yβ β xβ * yβ",
"end": [
470,
74
],
"full_name": "SetTheory.PGame.Equiv.mul_congr",
"kind": "commanddeclaration",
"start": [
469,
1
]
},
{
"code": "theorem P3_of_lt_of_lt (hx : xβ < xβ) (hy : yβ < yβ) : P3 xβ xβ yβ yβ",
"end": [
489,
72
],
"full_name": "SetTheory.PGame.P3_of_lt_of_lt",
"kind": "commanddeclaration",
"start": [
474,
1
]
},
{
"code": "theorem Numeric.mul_pos (hpβ : 0 < xβ) (hpβ : 0 < xβ) : 0 < xβ * xβ",
"end": [
495,
13
],
"full_name": "SetTheory.PGame.Numeric.mul_pos",
"kind": "commanddeclaration",
"start": [
491,
1
]
}
]
| 32 | Surreal.Multiplication.numeric_of_ih | [
[
318,
62
],
[
321,
57
]
]
| 3 | 5 | exact (cutExpand_add_right {y}).2 <| (cutExpand_add_left {xβ}).2 cutExpand_zero | case left
x xβ xβ xβ x' y yβ yβ yβ y' : PGame
ih : β (a : Args), ArgsRel a (Args.P1 x y) β P124 a
hx : x.Numeric
hy : y.Numeric
ih' : β (a : Args), ArgsRel a (Args.P24 xβ xβ y) β P124 a
β’ CutExpand IsOption (Args.P1 xβ y).toMultiset (Args.P24 xβ xβ y).toMultiset | no goals |
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