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Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 17 | Ordinal.opow_isLimit_left | [
[
131,
93
],
[
136,
35
]
] | 2 | 8 | Β· rw [opow_succ]
exact mul_isLimit (opow_pos _ l.pos) l | case inr.inl.intro
a : Ordinal.{u_1}
l : a.IsLimit
b : Ordinal.{u_1}
hb : succ b β 0
β’ (a ^ succ b).IsLimit
case inr.inr
a b : Ordinal.{u_1}
l : a.IsLimit
hb : b β 0
l' : b.IsLimit
β’ (a ^ b).IsLimit | case inr.inr
a b : Ordinal.{u_1}
l : a.IsLimit
hb : b β 0
l' : b.IsLimit
β’ (a ^ b).IsLimit |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 17 | Ordinal.opow_isLimit_left | [
[
131,
93
],
[
136,
35
]
] | 3 | 8 | Β· exact opow_isLimit l.one_lt l' | case inr.inr
a b : Ordinal.{u_1}
l : a.IsLimit
hb : b β 0
l' : b.IsLimit
β’ (a ^ b).IsLimit | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 17 | Ordinal.opow_isLimit_left | [
[
131,
93
],
[
136,
35
]
] | 4 | 8 | exact absurd e hb | case inl
a b : Ordinal.{u_1}
l : a.IsLimit
hb : b β 0
e : b = 0
β’ (a ^ b).IsLimit | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 17 | Ordinal.opow_isLimit_left | [
[
131,
93
],
[
136,
35
]
] | 5 | 8 | rw [opow_succ] | case inr.inl.intro
a : Ordinal.{u_1}
l : a.IsLimit
b : Ordinal.{u_1}
hb : succ b β 0
β’ (a ^ succ b).IsLimit | case inr.inl.intro
a : Ordinal.{u_1}
l : a.IsLimit
b : Ordinal.{u_1}
hb : succ b β 0
β’ (a ^ b * a).IsLimit |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 17 | Ordinal.opow_isLimit_left | [
[
131,
93
],
[
136,
35
]
] | 6 | 8 | exact mul_isLimit (opow_pos _ l.pos) l | case inr.inl.intro
a : Ordinal.{u_1}
l : a.IsLimit
b : Ordinal.{u_1}
hb : succ b β 0
β’ (a ^ b * a).IsLimit | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 17 | Ordinal.opow_isLimit_left | [
[
131,
93
],
[
136,
35
]
] | 7 | 8 | exact opow_isLimit l.one_lt l' | case inr.inr
a b : Ordinal.{u_1}
l : a.IsLimit
hb : b β 0
l' : b.IsLimit
β’ (a ^ b).IsLimit | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 18 | Ordinal.opow_le_opow_right | [
[
139,
91
],
[
144,
34
]
] | 0 | 6 | rcases lt_or_eq_of_le (one_le_iff_pos.2 hβ) with hβ | hβ | a b c : Ordinal.{u_1}
hβ : 0 < a
hβ : b β€ c
β’ a ^ b β€ a ^ c | case inl
a b c : Ordinal.{u_1}
hββ : 0 < a
hβ : b β€ c
hβ : 1 < a
β’ a ^ b β€ a ^ c
case inr
a b c : Ordinal.{u_1}
hββ : 0 < a
hβ : b β€ c
hβ : 1 = a
β’ a ^ b β€ a ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 18 | Ordinal.opow_le_opow_right | [
[
139,
91
],
[
144,
34
]
] | 1 | 6 | Β· exact (opow_le_opow_iff_right hβ).2 hβ | case inl
a b c : Ordinal.{u_1}
hββ : 0 < a
hβ : b β€ c
hβ : 1 < a
β’ a ^ b β€ a ^ c
case inr
a b c : Ordinal.{u_1}
hββ : 0 < a
hβ : b β€ c
hβ : 1 = a
β’ a ^ b β€ a ^ c | case inr
a b c : Ordinal.{u_1}
hββ : 0 < a
hβ : b β€ c
hβ : 1 = a
β’ a ^ b β€ a ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 18 | Ordinal.opow_le_opow_right | [
[
139,
91
],
[
144,
34
]
] | 2 | 6 | Β· subst a
simp only [one_opow, le_refl] | case inr
a b c : Ordinal.{u_1}
hββ : 0 < a
hβ : b β€ c
hβ : 1 = a
β’ a ^ b β€ a ^ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 18 | Ordinal.opow_le_opow_right | [
[
139,
91
],
[
144,
34
]
] | 3 | 6 | exact (opow_le_opow_iff_right hβ).2 hβ | case inl
a b c : Ordinal.{u_1}
hββ : 0 < a
hβ : b β€ c
hβ : 1 < a
β’ a ^ b β€ a ^ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 18 | Ordinal.opow_le_opow_right | [
[
139,
91
],
[
144,
34
]
] | 4 | 6 | subst a | case inr
a b c : Ordinal.{u_1}
hββ : 0 < a
hβ : b β€ c
hβ : 1 = a
β’ a ^ b β€ a ^ c | case inr
b c : Ordinal.{u_1}
hβ : b β€ c
hβ : 0 < 1
β’ 1 ^ b β€ 1 ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 18 | Ordinal.opow_le_opow_right | [
[
139,
91
],
[
144,
34
]
] | 5 | 6 | simp only [one_opow, le_refl] | case inr
b c : Ordinal.{u_1}
hβ : b β€ c
hβ : 0 < 1
β’ 1 ^ b β€ 1 ^ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 19 | Ordinal.opow_le_opow_left | [
[
147,
89
],
[
162,
96
]
] | 0 | 14 | by_cases a0 : a = 0 | a b c : Ordinal.{u_1}
ab : a β€ b
β’ a ^ c β€ b ^ c | case pos
a b c : Ordinal.{u_1}
ab : a β€ b
a0 : a = 0
β’ a ^ c β€ b ^ c
case neg
a b c : Ordinal.{u_1}
ab : a β€ b
a0 : Β¬a = 0
β’ a ^ c β€ b ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 19 | Ordinal.opow_le_opow_left | [
[
147,
89
],
[
162,
96
]
] | 1 | 14 | Β· subst a
by_cases c0 : c = 0
Β· subst c
simp only [opow_zero, le_refl]
Β· simp only [zero_opow c0, Ordinal.zero_le] | case pos
a b c : Ordinal.{u_1}
ab : a β€ b
a0 : a = 0
β’ a ^ c β€ b ^ c
case neg
a b c : Ordinal.{u_1}
ab : a β€ b
a0 : Β¬a = 0
β’ a ^ c β€ b ^ c | case neg
a b c : Ordinal.{u_1}
ab : a β€ b
a0 : Β¬a = 0
β’ a ^ c β€ b ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 19 | Ordinal.opow_le_opow_left | [
[
147,
89
],
[
162,
96
]
] | 2 | 14 | Β· induction c using limitRecOn with
| Hβ => simp only [opow_zero, le_refl]
| Hβ c IH =>
simpa only [opow_succ] using mul_le_mul' IH ab
| Hβ c l IH =>
exact
(opow_le_of_limit a0 l).2 fun b' h =>
(IH _ h).trans (opow_le_opow_right ((Ordinal.pos_iff_ne_zero.2 a0).trans_le ab) h.le) | case neg
a b c : Ordinal.{u_1}
ab : a β€ b
a0 : Β¬a = 0
β’ a ^ c β€ b ^ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 19 | Ordinal.opow_le_opow_left | [
[
147,
89
],
[
162,
96
]
] | 3 | 14 | subst a | case pos
a b c : Ordinal.{u_1}
ab : a β€ b
a0 : a = 0
β’ a ^ c β€ b ^ c | case pos
b c : Ordinal.{u_1}
ab : 0 β€ b
β’ 0 ^ c β€ b ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 19 | Ordinal.opow_le_opow_left | [
[
147,
89
],
[
162,
96
]
] | 4 | 14 | by_cases c0 : c = 0 | case pos
b c : Ordinal.{u_1}
ab : 0 β€ b
β’ 0 ^ c β€ b ^ c | case pos
b c : Ordinal.{u_1}
ab : 0 β€ b
c0 : c = 0
β’ 0 ^ c β€ b ^ c
case neg
b c : Ordinal.{u_1}
ab : 0 β€ b
c0 : Β¬c = 0
β’ 0 ^ c β€ b ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 19 | Ordinal.opow_le_opow_left | [
[
147,
89
],
[
162,
96
]
] | 5 | 14 | Β· subst c
simp only [opow_zero, le_refl] | case pos
b c : Ordinal.{u_1}
ab : 0 β€ b
c0 : c = 0
β’ 0 ^ c β€ b ^ c
case neg
b c : Ordinal.{u_1}
ab : 0 β€ b
c0 : Β¬c = 0
β’ 0 ^ c β€ b ^ c | case neg
b c : Ordinal.{u_1}
ab : 0 β€ b
c0 : Β¬c = 0
β’ 0 ^ c β€ b ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 19 | Ordinal.opow_le_opow_left | [
[
147,
89
],
[
162,
96
]
] | 6 | 14 | Β· simp only [zero_opow c0, Ordinal.zero_le] | case neg
b c : Ordinal.{u_1}
ab : 0 β€ b
c0 : Β¬c = 0
β’ 0 ^ c β€ b ^ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 19 | Ordinal.opow_le_opow_left | [
[
147,
89
],
[
162,
96
]
] | 7 | 14 | subst c | case pos
b c : Ordinal.{u_1}
ab : 0 β€ b
c0 : c = 0
β’ 0 ^ c β€ b ^ c | case pos
b : Ordinal.{u_1}
ab : 0 β€ b
β’ 0 ^ 0 β€ b ^ 0 |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 19 | Ordinal.opow_le_opow_left | [
[
147,
89
],
[
162,
96
]
] | 8 | 14 | simp only [opow_zero, le_refl] | case pos
b : Ordinal.{u_1}
ab : 0 β€ b
β’ 0 ^ 0 β€ b ^ 0 | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 19 | Ordinal.opow_le_opow_left | [
[
147,
89
],
[
162,
96
]
] | 9 | 14 | simp only [zero_opow c0, Ordinal.zero_le] | case neg
b c : Ordinal.{u_1}
ab : 0 β€ b
c0 : Β¬c = 0
β’ 0 ^ c β€ b ^ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 19 | Ordinal.opow_le_opow_left | [
[
147,
89
],
[
162,
96
]
] | 10 | 14 | induction c using limitRecOn with
| Hβ => simp only [opow_zero, le_refl]
| Hβ c IH =>
simpa only [opow_succ] using mul_le_mul' IH ab
| Hβ c l IH =>
exact
(opow_le_of_limit a0 l).2 fun b' h =>
(IH _ h).trans (opow_le_opow_right ((Ordinal.pos_iff_ne_zero.2 a0).trans_le ab) h.le) | case neg
a b c : Ordinal.{u_1}
ab : a β€ b
a0 : Β¬a = 0
β’ a ^ c β€ b ^ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 19 | Ordinal.opow_le_opow_left | [
[
147,
89
],
[
162,
96
]
] | 11 | 14 | simp only [opow_zero, le_refl] | case neg.Hβ
a b : Ordinal.{u_1}
ab : a β€ b
a0 : Β¬a = 0
β’ a ^ 0 β€ b ^ 0 | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 19 | Ordinal.opow_le_opow_left | [
[
147,
89
],
[
162,
96
]
] | 12 | 14 | simpa only [opow_succ] using mul_le_mul' IH ab | case neg.Hβ
a b : Ordinal.{u_1}
ab : a β€ b
a0 : Β¬a = 0
c : Ordinal.{u_1}
IH : a ^ c β€ b ^ c
β’ a ^ succ c β€ b ^ succ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 19 | Ordinal.opow_le_opow_left | [
[
147,
89
],
[
162,
96
]
] | 13 | 14 | exact
(opow_le_of_limit a0 l).2 fun b' h =>
(IH _ h).trans (opow_le_opow_right ((Ordinal.pos_iff_ne_zero.2 a0).trans_le ab) h.le) | case neg.Hβ
a b : Ordinal.{u_1}
ab : a β€ b
a0 : Β¬a = 0
c : Ordinal.{u_1}
l : c.IsLimit
IH : β o' < c, a ^ o' β€ b ^ o'
β’ a ^ c β€ b ^ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 20 | Ordinal.left_le_opow | [
[
165,
78
],
[
173,
50
]
] | 0 | 10 | nth_rw 1 [β opow_one a] | a b : Ordinal.{u_1}
b1 : 0 < b
β’ a β€ a ^ b | a b : Ordinal.{u_1}
b1 : 0 < b
β’ a ^ 1 β€ a ^ b |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 20 | Ordinal.left_le_opow | [
[
165,
78
],
[
173,
50
]
] | 1 | 10 | cases' le_or_gt a 1 with a1 a1 | a b : Ordinal.{u_1}
b1 : 0 < b
β’ a ^ 1 β€ a ^ b | case inl
a b : Ordinal.{u_1}
b1 : 0 < b
a1 : a β€ 1
β’ a ^ 1 β€ a ^ b
case inr
a b : Ordinal.{u_1}
b1 : 0 < b
a1 : a > 1
β’ a ^ 1 β€ a ^ b |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 20 | Ordinal.left_le_opow | [
[
165,
78
],
[
173,
50
]
] | 2 | 10 | Β· rcases lt_or_eq_of_le a1 with a0 | a1
Β· rw [lt_one_iff_zero] at a0
rw [a0, zero_opow Ordinal.one_ne_zero]
exact Ordinal.zero_le _
rw [a1, one_opow, one_opow] | case inl
a b : Ordinal.{u_1}
b1 : 0 < b
a1 : a β€ 1
β’ a ^ 1 β€ a ^ b
case inr
a b : Ordinal.{u_1}
b1 : 0 < b
a1 : a > 1
β’ a ^ 1 β€ a ^ b | case inr
a b : Ordinal.{u_1}
b1 : 0 < b
a1 : a > 1
β’ a ^ 1 β€ a ^ b |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 20 | Ordinal.left_le_opow | [
[
165,
78
],
[
173,
50
]
] | 3 | 10 | rwa [opow_le_opow_iff_right a1, one_le_iff_pos] | case inr
a b : Ordinal.{u_1}
b1 : 0 < b
a1 : a > 1
β’ a ^ 1 β€ a ^ b | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 20 | Ordinal.left_le_opow | [
[
165,
78
],
[
173,
50
]
] | 4 | 10 | rcases lt_or_eq_of_le a1 with a0 | a1 | case inl
a b : Ordinal.{u_1}
b1 : 0 < b
a1 : a β€ 1
β’ a ^ 1 β€ a ^ b | case inl.inl
a b : Ordinal.{u_1}
b1 : 0 < b
a1 : a β€ 1
a0 : a < 1
β’ a ^ 1 β€ a ^ b
case inl.inr
a b : Ordinal.{u_1}
b1 : 0 < b
a1β : a β€ 1
a1 : a = 1
β’ a ^ 1 β€ a ^ b |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 20 | Ordinal.left_le_opow | [
[
165,
78
],
[
173,
50
]
] | 5 | 10 | Β· rw [lt_one_iff_zero] at a0
rw [a0, zero_opow Ordinal.one_ne_zero]
exact Ordinal.zero_le _ | case inl.inl
a b : Ordinal.{u_1}
b1 : 0 < b
a1 : a β€ 1
a0 : a < 1
β’ a ^ 1 β€ a ^ b
case inl.inr
a b : Ordinal.{u_1}
b1 : 0 < b
a1β : a β€ 1
a1 : a = 1
β’ a ^ 1 β€ a ^ b | case inl.inr
a b : Ordinal.{u_1}
b1 : 0 < b
a1β : a β€ 1
a1 : a = 1
β’ a ^ 1 β€ a ^ b |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 20 | Ordinal.left_le_opow | [
[
165,
78
],
[
173,
50
]
] | 6 | 10 | rw [a1, one_opow, one_opow] | case inl.inr
a b : Ordinal.{u_1}
b1 : 0 < b
a1β : a β€ 1
a1 : a = 1
β’ a ^ 1 β€ a ^ b | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 20 | Ordinal.left_le_opow | [
[
165,
78
],
[
173,
50
]
] | 7 | 10 | rw [lt_one_iff_zero] at a0 | case inl.inl
a b : Ordinal.{u_1}
b1 : 0 < b
a1 : a β€ 1
a0 : a < 1
β’ a ^ 1 β€ a ^ b | case inl.inl
a b : Ordinal.{u_1}
b1 : 0 < b
a1 : a β€ 1
a0 : a = 0
β’ a ^ 1 β€ a ^ b |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 20 | Ordinal.left_le_opow | [
[
165,
78
],
[
173,
50
]
] | 8 | 10 | rw [a0, zero_opow Ordinal.one_ne_zero] | case inl.inl
a b : Ordinal.{u_1}
b1 : 0 < b
a1 : a β€ 1
a0 : a = 0
β’ a ^ 1 β€ a ^ b | case inl.inl
a b : Ordinal.{u_1}
b1 : 0 < b
a1 : a β€ 1
a0 : a = 0
β’ 0 β€ 0 ^ b |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 20 | Ordinal.left_le_opow | [
[
165,
78
],
[
173,
50
]
] | 9 | 10 | exact Ordinal.zero_le _ | case inl.inl
a b : Ordinal.{u_1}
b1 : 0 < b
a1 : a β€ 1
a0 : a = 0
β’ 0 β€ 0 ^ b | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 22 | Ordinal.opow_lt_opow_left_of_succ | [
[
180,
95
],
[
184,
81
]
] | 0 | 2 | rw [opow_succ, opow_succ] | a b c : Ordinal.{u_1}
ab : a < b
β’ a ^ succ c < b ^ succ c | a b c : Ordinal.{u_1}
ab : a < b
β’ a ^ c * a < b ^ c * b |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 22 | Ordinal.opow_lt_opow_left_of_succ | [
[
180,
95
],
[
184,
81
]
] | 1 | 2 | exact
(mul_le_mul_right' (opow_le_opow_left c ab.le) a).trans_lt
(mul_lt_mul_of_pos_left ab (opow_pos c ((Ordinal.zero_le a).trans_lt ab))) | a b c : Ordinal.{u_1}
ab : a < b
β’ a ^ c * a < b ^ c * b | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 23 | Ordinal.opow_add | [
[
187,
69
],
[
208,
18
]
] | 0 | 17 | rcases eq_or_ne a 0 with (rfl | a0) | a b c : Ordinal.{u_1}
β’ a ^ (b + c) = a ^ b * a ^ c | case inl
b c : Ordinal.{u_1}
β’ 0 ^ (b + c) = 0 ^ b * 0 ^ c
case inr
a b c : Ordinal.{u_1}
a0 : a β 0
β’ a ^ (b + c) = a ^ b * a ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 23 | Ordinal.opow_add | [
[
187,
69
],
[
208,
18
]
] | 1 | 17 | Β· rcases eq_or_ne c 0 with (rfl | c0)
Β· simp
have : b + c β 0 := ((Ordinal.pos_iff_ne_zero.2 c0).trans_le (le_add_left _ _)).ne'
simp only [zero_opow c0, zero_opow this, mul_zero] | case inl
b c : Ordinal.{u_1}
β’ 0 ^ (b + c) = 0 ^ b * 0 ^ c
case inr
a b c : Ordinal.{u_1}
a0 : a β 0
β’ a ^ (b + c) = a ^ b * a ^ c | case inr
a b c : Ordinal.{u_1}
a0 : a β 0
β’ a ^ (b + c) = a ^ b * a ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 23 | Ordinal.opow_add | [
[
187,
69
],
[
208,
18
]
] | 2 | 17 | rcases eq_or_lt_of_le (one_le_iff_ne_zero.2 a0) with (rfl | a1) | case inr
a b c : Ordinal.{u_1}
a0 : a β 0
β’ a ^ (b + c) = a ^ b * a ^ c | case inr.inl
b c : Ordinal.{u_1}
a0 : 1 β 0
β’ 1 ^ (b + c) = 1 ^ b * 1 ^ c
case inr.inr
a b c : Ordinal.{u_1}
a0 : a β 0
a1 : 1 < a
β’ a ^ (b + c) = a ^ b * a ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 23 | Ordinal.opow_add | [
[
187,
69
],
[
208,
18
]
] | 3 | 17 | Β· simp only [one_opow, mul_one] | case inr.inl
b c : Ordinal.{u_1}
a0 : 1 β 0
β’ 1 ^ (b + c) = 1 ^ b * 1 ^ c
case inr.inr
a b c : Ordinal.{u_1}
a0 : a β 0
a1 : 1 < a
β’ a ^ (b + c) = a ^ b * a ^ c | case inr.inr
a b c : Ordinal.{u_1}
a0 : a β 0
a1 : 1 < a
β’ a ^ (b + c) = a ^ b * a ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 23 | Ordinal.opow_add | [
[
187,
69
],
[
208,
18
]
] | 4 | 17 | induction c using limitRecOn with
| Hβ => simp
| Hβ c IH =>
rw [add_succ, opow_succ, IH, opow_succ, mul_assoc]
| Hβ c l IH =>
refine
eq_of_forall_ge_iff fun d =>
(((opow_isNormal a1).trans (add_isNormal b)).limit_le l).trans ?_
dsimp only [Function.comp_def]
simp (config := { contextual := true }) only [IH]
exact
(((mul_isNormal <| opow_pos b (Ordinal.pos_iff_ne_zero.2 a0)).trans
(opow_isNormal a1)).limit_le
l).symm | case inr.inr
a b c : Ordinal.{u_1}
a0 : a β 0
a1 : 1 < a
β’ a ^ (b + c) = a ^ b * a ^ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 23 | Ordinal.opow_add | [
[
187,
69
],
[
208,
18
]
] | 5 | 17 | rcases eq_or_ne c 0 with (rfl | c0) | case inl
b c : Ordinal.{u_1}
β’ 0 ^ (b + c) = 0 ^ b * 0 ^ c | case inl.inl
b : Ordinal.{u_1}
β’ 0 ^ (b + 0) = 0 ^ b * 0 ^ 0
case inl.inr
b c : Ordinal.{u_1}
c0 : c β 0
β’ 0 ^ (b + c) = 0 ^ b * 0 ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 23 | Ordinal.opow_add | [
[
187,
69
],
[
208,
18
]
] | 6 | 17 | Β· simp | case inl.inl
b : Ordinal.{u_1}
β’ 0 ^ (b + 0) = 0 ^ b * 0 ^ 0
case inl.inr
b c : Ordinal.{u_1}
c0 : c β 0
β’ 0 ^ (b + c) = 0 ^ b * 0 ^ c | case inl.inr
b c : Ordinal.{u_1}
c0 : c β 0
β’ 0 ^ (b + c) = 0 ^ b * 0 ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 23 | Ordinal.opow_add | [
[
187,
69
],
[
208,
18
]
] | 7 | 17 | have : b + c β 0 := ((Ordinal.pos_iff_ne_zero.2 c0).trans_le (le_add_left _ _)).ne' | case inl.inr
b c : Ordinal.{u_1}
c0 : c β 0
β’ 0 ^ (b + c) = 0 ^ b * 0 ^ c | case inl.inr
b c : Ordinal.{u_1}
c0 : c β 0
this : b + c β 0
β’ 0 ^ (b + c) = 0 ^ b * 0 ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 23 | Ordinal.opow_add | [
[
187,
69
],
[
208,
18
]
] | 8 | 17 | simp only [zero_opow c0, zero_opow this, mul_zero] | case inl.inr
b c : Ordinal.{u_1}
c0 : c β 0
this : b + c β 0
β’ 0 ^ (b + c) = 0 ^ b * 0 ^ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 23 | Ordinal.opow_add | [
[
187,
69
],
[
208,
18
]
] | 9 | 17 | simp | case inl.inl
b : Ordinal.{u_1}
β’ 0 ^ (b + 0) = 0 ^ b * 0 ^ 0 | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 23 | Ordinal.opow_add | [
[
187,
69
],
[
208,
18
]
] | 10 | 17 | simp only [one_opow, mul_one] | case inr.inl
b c : Ordinal.{u_1}
a0 : 1 β 0
β’ 1 ^ (b + c) = 1 ^ b * 1 ^ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 23 | Ordinal.opow_add | [
[
187,
69
],
[
208,
18
]
] | 11 | 17 | simp | case inr.inr.Hβ
a b : Ordinal.{u_1}
a0 : a β 0
a1 : 1 < a
β’ a ^ (b + 0) = a ^ b * a ^ 0 | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 23 | Ordinal.opow_add | [
[
187,
69
],
[
208,
18
]
] | 12 | 17 | rw [add_succ, opow_succ, IH, opow_succ, mul_assoc] | case inr.inr.Hβ
a b : Ordinal.{u_1}
a0 : a β 0
a1 : 1 < a
c : Ordinal.{u_1}
IH : a ^ (b + c) = a ^ b * a ^ c
β’ a ^ (b + succ c) = a ^ b * a ^ succ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 23 | Ordinal.opow_add | [
[
187,
69
],
[
208,
18
]
] | 13 | 17 | refine
eq_of_forall_ge_iff fun d =>
(((opow_isNormal a1).trans (add_isNormal b)).limit_le l).trans ?_ | case inr.inr.Hβ
a b : Ordinal.{u_1}
a0 : a β 0
a1 : 1 < a
c : Ordinal.{u_1}
l : c.IsLimit
IH : β o' < c, a ^ (b + o') = a ^ b * a ^ o'
β’ a ^ (b + c) = a ^ b * a ^ c | case inr.inr.Hβ
a b : Ordinal.{u_1}
a0 : a β 0
a1 : 1 < a
c : Ordinal.{u_1}
l : c.IsLimit
IH : β o' < c, a ^ (b + o') = a ^ b * a ^ o'
d : Ordinal.{u_1}
β’ (β b_1 < c, ((fun x => a ^ x) β fun x => b + x) b_1 β€ d) β a ^ b * a ^ c β€ d |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 23 | Ordinal.opow_add | [
[
187,
69
],
[
208,
18
]
] | 14 | 17 | dsimp only [Function.comp_def] | case inr.inr.Hβ
a b : Ordinal.{u_1}
a0 : a β 0
a1 : 1 < a
c : Ordinal.{u_1}
l : c.IsLimit
IH : β o' < c, a ^ (b + o') = a ^ b * a ^ o'
d : Ordinal.{u_1}
β’ (β b_1 < c, ((fun x => a ^ x) β fun x => b + x) b_1 β€ d) β a ^ b * a ^ c β€ d | case inr.inr.Hβ
a b : Ordinal.{u_1}
a0 : a β 0
a1 : 1 < a
c : Ordinal.{u_1}
l : c.IsLimit
IH : β o' < c, a ^ (b + o') = a ^ b * a ^ o'
d : Ordinal.{u_1}
β’ (β b_1 < c, a ^ (b + b_1) β€ d) β a ^ b * a ^ c β€ d |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 23 | Ordinal.opow_add | [
[
187,
69
],
[
208,
18
]
] | 15 | 17 | simp (config := { contextual := true }) only [IH] | case inr.inr.Hβ
a b : Ordinal.{u_1}
a0 : a β 0
a1 : 1 < a
c : Ordinal.{u_1}
l : c.IsLimit
IH : β o' < c, a ^ (b + o') = a ^ b * a ^ o'
d : Ordinal.{u_1}
β’ (β b_1 < c, a ^ (b + b_1) β€ d) β a ^ b * a ^ c β€ d | case inr.inr.Hβ
a b : Ordinal.{u_1}
a0 : a β 0
a1 : 1 < a
c : Ordinal.{u_1}
l : c.IsLimit
IH : β o' < c, a ^ (b + o') = a ^ b * a ^ o'
d : Ordinal.{u_1}
β’ (β b_1 < c, a ^ b * a ^ b_1 β€ d) β a ^ b * a ^ c β€ d |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 23 | Ordinal.opow_add | [
[
187,
69
],
[
208,
18
]
] | 16 | 17 | exact
(((mul_isNormal <| opow_pos b (Ordinal.pos_iff_ne_zero.2 a0)).trans
(opow_isNormal a1)).limit_le
l).symm | case inr.inr.Hβ
a b : Ordinal.{u_1}
a0 : a β 0
a1 : 1 < a
c : Ordinal.{u_1}
l : c.IsLimit
IH : β o' < c, a ^ (b + o') = a ^ b * a ^ o'
d : Ordinal.{u_1}
β’ (β b_1 < c, a ^ b * a ^ b_1 β€ d) β a ^ b * a ^ c β€ d | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 24 | Ordinal.opow_one_add | [
[
211,
67
],
[
211,
93
]
] | 0 | 1 | rw [opow_add, opow_one] | a b : Ordinal.{u_1}
β’ a ^ (1 + b) = a * a ^ b | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 0 | 21 | by_cases b0 : b = 0 | a b c : Ordinal.{u_1}
β’ a ^ (b * c) = (a ^ b) ^ c | case pos
a b c : Ordinal.{u_1}
b0 : b = 0
β’ a ^ (b * c) = (a ^ b) ^ c
case neg
a b c : Ordinal.{u_1}
b0 : Β¬b = 0
β’ a ^ (b * c) = (a ^ b) ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 1 | 21 | Β· simp only [b0, zero_mul, opow_zero, one_opow] | case pos
a b c : Ordinal.{u_1}
b0 : b = 0
β’ a ^ (b * c) = (a ^ b) ^ c
case neg
a b c : Ordinal.{u_1}
b0 : Β¬b = 0
β’ a ^ (b * c) = (a ^ b) ^ c | case neg
a b c : Ordinal.{u_1}
b0 : Β¬b = 0
β’ a ^ (b * c) = (a ^ b) ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 2 | 21 | by_cases a0 : a = 0 | case neg
a b c : Ordinal.{u_1}
b0 : Β¬b = 0
β’ a ^ (b * c) = (a ^ b) ^ c | case pos
a b c : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : a = 0
β’ a ^ (b * c) = (a ^ b) ^ c
case neg
a b c : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
β’ a ^ (b * c) = (a ^ b) ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 3 | 21 | Β· subst a
by_cases c0 : c = 0
Β· simp only [c0, mul_zero, opow_zero]
simp only [zero_opow b0, zero_opow c0, zero_opow (mul_ne_zero b0 c0)] | case pos
a b c : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : a = 0
β’ a ^ (b * c) = (a ^ b) ^ c
case neg
a b c : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
β’ a ^ (b * c) = (a ^ b) ^ c | case neg
a b c : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
β’ a ^ (b * c) = (a ^ b) ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 4 | 21 | cases' eq_or_lt_of_le (one_le_iff_ne_zero.2 a0) with a1 a1 | case neg
a b c : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
β’ a ^ (b * c) = (a ^ b) ^ c | case neg.inl
a b c : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
a1 : 1 = a
β’ a ^ (b * c) = (a ^ b) ^ c
case neg.inr
a b c : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
a1 : 1 < a
β’ a ^ (b * c) = (a ^ b) ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 5 | 21 | Β· subst a1
simp only [one_opow] | case neg.inl
a b c : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
a1 : 1 = a
β’ a ^ (b * c) = (a ^ b) ^ c
case neg.inr
a b c : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
a1 : 1 < a
β’ a ^ (b * c) = (a ^ b) ^ c | case neg.inr
a b c : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
a1 : 1 < a
β’ a ^ (b * c) = (a ^ b) ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 6 | 21 | induction c using limitRecOn with
| Hβ => simp only [mul_zero, opow_zero]
| Hβ c IH =>
rw [mul_succ, opow_add, IH, opow_succ]
| Hβ c l IH =>
refine
eq_of_forall_ge_iff fun d =>
(((opow_isNormal a1).trans (mul_isNormal (Ordinal.pos_iff_ne_zero.2 b0))).limit_le
l).trans
?_
dsimp only [Function.comp_def]
simp (config := { contextual := true }) only [IH]
exact (opow_le_of_limit (opow_ne_zero _ a0) l).symm | case neg.inr
a b c : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
a1 : 1 < a
β’ a ^ (b * c) = (a ^ b) ^ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 7 | 21 | simp only [b0, zero_mul, opow_zero, one_opow] | case pos
a b c : Ordinal.{u_1}
b0 : b = 0
β’ a ^ (b * c) = (a ^ b) ^ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 8 | 21 | subst a | case pos
a b c : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : a = 0
β’ a ^ (b * c) = (a ^ b) ^ c | case pos
b c : Ordinal.{u_1}
b0 : Β¬b = 0
β’ 0 ^ (b * c) = (0 ^ b) ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 9 | 21 | by_cases c0 : c = 0 | case pos
b c : Ordinal.{u_1}
b0 : Β¬b = 0
β’ 0 ^ (b * c) = (0 ^ b) ^ c | case pos
b c : Ordinal.{u_1}
b0 : Β¬b = 0
c0 : c = 0
β’ 0 ^ (b * c) = (0 ^ b) ^ c
case neg
b c : Ordinal.{u_1}
b0 : Β¬b = 0
c0 : Β¬c = 0
β’ 0 ^ (b * c) = (0 ^ b) ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 10 | 21 | Β· simp only [c0, mul_zero, opow_zero] | case pos
b c : Ordinal.{u_1}
b0 : Β¬b = 0
c0 : c = 0
β’ 0 ^ (b * c) = (0 ^ b) ^ c
case neg
b c : Ordinal.{u_1}
b0 : Β¬b = 0
c0 : Β¬c = 0
β’ 0 ^ (b * c) = (0 ^ b) ^ c | case neg
b c : Ordinal.{u_1}
b0 : Β¬b = 0
c0 : Β¬c = 0
β’ 0 ^ (b * c) = (0 ^ b) ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 11 | 21 | simp only [zero_opow b0, zero_opow c0, zero_opow (mul_ne_zero b0 c0)] | case neg
b c : Ordinal.{u_1}
b0 : Β¬b = 0
c0 : Β¬c = 0
β’ 0 ^ (b * c) = (0 ^ b) ^ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 12 | 21 | simp only [c0, mul_zero, opow_zero] | case pos
b c : Ordinal.{u_1}
b0 : Β¬b = 0
c0 : c = 0
β’ 0 ^ (b * c) = (0 ^ b) ^ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 13 | 21 | subst a1 | case neg.inl
a b c : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
a1 : 1 = a
β’ a ^ (b * c) = (a ^ b) ^ c | case neg.inl
b c : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬1 = 0
β’ 1 ^ (b * c) = (1 ^ b) ^ c |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 14 | 21 | simp only [one_opow] | case neg.inl
b c : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬1 = 0
β’ 1 ^ (b * c) = (1 ^ b) ^ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 15 | 21 | simp only [mul_zero, opow_zero] | case neg.inr.Hβ
a b : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
a1 : 1 < a
β’ a ^ (b * 0) = (a ^ b) ^ 0 | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 16 | 21 | rw [mul_succ, opow_add, IH, opow_succ] | case neg.inr.Hβ
a b : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
a1 : 1 < a
c : Ordinal.{u_1}
IH : a ^ (b * c) = (a ^ b) ^ c
β’ a ^ (b * succ c) = (a ^ b) ^ succ c | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 17 | 21 | refine
eq_of_forall_ge_iff fun d =>
(((opow_isNormal a1).trans (mul_isNormal (Ordinal.pos_iff_ne_zero.2 b0))).limit_le
l).trans
?_ | case neg.inr.Hβ
a b : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
a1 : 1 < a
c : Ordinal.{u_1}
l : c.IsLimit
IH : β o' < c, a ^ (b * o') = (a ^ b) ^ o'
β’ a ^ (b * c) = (a ^ b) ^ c | case neg.inr.Hβ
a b : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
a1 : 1 < a
c : Ordinal.{u_1}
l : c.IsLimit
IH : β o' < c, a ^ (b * o') = (a ^ b) ^ o'
d : Ordinal.{u_1}
β’ (β b_1 < c, ((fun x => a ^ x) β fun x => b * x) b_1 β€ d) β (a ^ b) ^ c β€ d |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 18 | 21 | dsimp only [Function.comp_def] | case neg.inr.Hβ
a b : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
a1 : 1 < a
c : Ordinal.{u_1}
l : c.IsLimit
IH : β o' < c, a ^ (b * o') = (a ^ b) ^ o'
d : Ordinal.{u_1}
β’ (β b_1 < c, ((fun x => a ^ x) β fun x => b * x) b_1 β€ d) β (a ^ b) ^ c β€ d | case neg.inr.Hβ
a b : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
a1 : 1 < a
c : Ordinal.{u_1}
l : c.IsLimit
IH : β o' < c, a ^ (b * o') = (a ^ b) ^ o'
d : Ordinal.{u_1}
β’ (β b_1 < c, a ^ (b * b_1) β€ d) β (a ^ b) ^ c β€ d |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 19 | 21 | simp (config := { contextual := true }) only [IH] | case neg.inr.Hβ
a b : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
a1 : 1 < a
c : Ordinal.{u_1}
l : c.IsLimit
IH : β o' < c, a ^ (b * o') = (a ^ b) ^ o'
d : Ordinal.{u_1}
β’ (β b_1 < c, a ^ (b * b_1) β€ d) β (a ^ b) ^ c β€ d | case neg.inr.Hβ
a b : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
a1 : 1 < a
c : Ordinal.{u_1}
l : c.IsLimit
IH : β o' < c, a ^ (b * o') = (a ^ b) ^ o'
d : Ordinal.{u_1}
β’ (β b_1 < c, (a ^ b) ^ b_1 β€ d) β (a ^ b) ^ c β€ d |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 27 | Ordinal.opow_mul | [
[
226,
67
],
[
248,
56
]
] | 20 | 21 | exact (opow_le_of_limit (opow_ne_zero _ a0) l).symm | case neg.inr.Hβ
a b : Ordinal.{u_1}
b0 : Β¬b = 0
a0 : Β¬a = 0
a1 : 1 < a
c : Ordinal.{u_1}
l : c.IsLimit
IH : β o' < c, a ^ (b * o') = (a ^ b) ^ o'
d : Ordinal.{u_1}
β’ (β b_1 < c, (a ^ b) ^ b_1 β€ d) β (a ^ b) ^ c β€ d | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 29 | Ordinal.log_def | [
[
267,
48
],
[
267,
77
]
] | 0 | 1 | simp only [log, dif_pos h] | b : Ordinal.{u_1}
h : 1 < b
x : Ordinal.{u_1}
β’ log b x = (sInf {o | x < b ^ o}).pred | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 30 | Ordinal.log_of_not_one_lt_left | [
[
270,
90
],
[
271,
29
]
] | 0 | 1 | simp only [log, dif_neg h] | b : Ordinal.{u_1}
h : Β¬1 < b
x : Ordinal.{u_1}
β’ log b x = 0 | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 35 | Ordinal.succ_log_def | [
[
300,
58
],
[
308,
76
]
] | 0 | 11 | let t := sInf { o : Ordinal | x < b ^ o } | b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
β’ succ (log b x) = sInf {o | x < b ^ o} | b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
β’ succ (log b x) = sInf {o | x < b ^ o} |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 35 | Ordinal.succ_log_def | [
[
300,
58
],
[
308,
76
]
] | 1 | 11 | have : x < (b^t) := csInf_mem (log_nonempty hb) | b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
β’ succ (log b x) = sInf {o | x < b ^ o} | b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
β’ succ (log b x) = sInf {o | x < b ^ o} |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 35 | Ordinal.succ_log_def | [
[
300,
58
],
[
308,
76
]
] | 2 | 11 | rcases zero_or_succ_or_limit t with (h | h | h) | b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
β’ succ (log b x) = sInf {o | x < b ^ o} | case inl
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : t = 0
β’ succ (log b x) = sInf {o | x < b ^ o}
case inr.inl
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : β a, t = succ a
β’ succ (log b x) = sInf {o | x < b ^ o}
case inr.inr
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : t.IsLimit
β’ succ (log b x) = sInf {o | x < b ^ o} |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 35 | Ordinal.succ_log_def | [
[
300,
58
],
[
308,
76
]
] | 3 | 11 | Β· refine ((one_le_iff_ne_zero.2 hx).not_lt ?_).elim
simpa only [h, opow_zero] using this | case inl
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : t = 0
β’ succ (log b x) = sInf {o | x < b ^ o}
case inr.inl
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : β a, t = succ a
β’ succ (log b x) = sInf {o | x < b ^ o}
case inr.inr
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : t.IsLimit
β’ succ (log b x) = sInf {o | x < b ^ o} | case inr.inl
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : β a, t = succ a
β’ succ (log b x) = sInf {o | x < b ^ o}
case inr.inr
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : t.IsLimit
β’ succ (log b x) = sInf {o | x < b ^ o} |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 35 | Ordinal.succ_log_def | [
[
300,
58
],
[
308,
76
]
] | 4 | 11 | Β· rw [show log b x = pred t from log_def hb x, succ_pred_iff_is_succ.2 h] | case inr.inl
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : β a, t = succ a
β’ succ (log b x) = sInf {o | x < b ^ o}
case inr.inr
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : t.IsLimit
β’ succ (log b x) = sInf {o | x < b ^ o} | case inr.inr
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : t.IsLimit
β’ succ (log b x) = sInf {o | x < b ^ o} |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 35 | Ordinal.succ_log_def | [
[
300,
58
],
[
308,
76
]
] | 5 | 11 | Β· rcases (lt_opow_of_limit (zero_lt_one.trans hb).ne' h).1 this with β¨a, hβ, hββ©
exact hβ.not_le.elim ((le_csInf_iff'' (log_nonempty hb)).1 le_rfl a hβ) | case inr.inr
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : t.IsLimit
β’ succ (log b x) = sInf {o | x < b ^ o} | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 35 | Ordinal.succ_log_def | [
[
300,
58
],
[
308,
76
]
] | 6 | 11 | refine ((one_le_iff_ne_zero.2 hx).not_lt ?_).elim | case inl
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : t = 0
β’ succ (log b x) = sInf {o | x < b ^ o} | case inl
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : t = 0
β’ x < 1 |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 35 | Ordinal.succ_log_def | [
[
300,
58
],
[
308,
76
]
] | 7 | 11 | simpa only [h, opow_zero] using this | case inl
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : t = 0
β’ x < 1 | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 35 | Ordinal.succ_log_def | [
[
300,
58
],
[
308,
76
]
] | 8 | 11 | rw [show log b x = pred t from log_def hb x, succ_pred_iff_is_succ.2 h] | case inr.inl
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : β a, t = succ a
β’ succ (log b x) = sInf {o | x < b ^ o} | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 35 | Ordinal.succ_log_def | [
[
300,
58
],
[
308,
76
]
] | 9 | 11 | rcases (lt_opow_of_limit (zero_lt_one.trans hb).ne' h).1 this with β¨a, hβ, hββ© | case inr.inr
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : t.IsLimit
β’ succ (log b x) = sInf {o | x < b ^ o} | case inr.inr.intro.intro
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : t.IsLimit
a : Ordinal.{u_1}
hβ : a < t
hβ : x < b ^ a
β’ succ (log b x) = sInf {o | x < b ^ o} |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 35 | Ordinal.succ_log_def | [
[
300,
58
],
[
308,
76
]
] | 10 | 11 | exact hβ.not_le.elim ((le_csInf_iff'' (log_nonempty hb)).1 le_rfl a hβ) | case inr.inr.intro.intro
b x : Ordinal.{u_1}
hb : 1 < b
hx : x β 0
t : Ordinal.{u_1} := sInf {o | x < b ^ o}
this : x < b ^ t
h : t.IsLimit
a : Ordinal.{u_1}
hβ : a < t
hβ : x < b ^ a
β’ succ (log b x) = sInf {o | x < b ^ o} | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 36 | Ordinal.lt_opow_succ_log_self | [
[
312,
31
],
[
316,
38
]
] | 0 | 6 | rcases eq_or_ne x 0 with (rfl | hx) | b : Ordinal.{u_1}
hb : 1 < b
x : Ordinal.{u_1}
β’ x < b ^ succ (log b x) | case inl
b : Ordinal.{u_1}
hb : 1 < b
β’ 0 < b ^ succ (log b 0)
case inr
b : Ordinal.{u_1}
hb : 1 < b
x : Ordinal.{u_1}
hx : x β 0
β’ x < b ^ succ (log b x) |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 36 | Ordinal.lt_opow_succ_log_self | [
[
312,
31
],
[
316,
38
]
] | 1 | 6 | Β· apply opow_pos _ (zero_lt_one.trans hb) | case inl
b : Ordinal.{u_1}
hb : 1 < b
β’ 0 < b ^ succ (log b 0)
case inr
b : Ordinal.{u_1}
hb : 1 < b
x : Ordinal.{u_1}
hx : x β 0
β’ x < b ^ succ (log b x) | case inr
b : Ordinal.{u_1}
hb : 1 < b
x : Ordinal.{u_1}
hx : x β 0
β’ x < b ^ succ (log b x) |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 36 | Ordinal.lt_opow_succ_log_self | [
[
312,
31
],
[
316,
38
]
] | 2 | 6 | Β· rw [succ_log_def hb hx]
exact csInf_mem (log_nonempty hb) | case inr
b : Ordinal.{u_1}
hb : 1 < b
x : Ordinal.{u_1}
hx : x β 0
β’ x < b ^ succ (log b x) | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 36 | Ordinal.lt_opow_succ_log_self | [
[
312,
31
],
[
316,
38
]
] | 3 | 6 | apply opow_pos _ (zero_lt_one.trans hb) | case inl
b : Ordinal.{u_1}
hb : 1 < b
β’ 0 < b ^ succ (log b 0) | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 36 | Ordinal.lt_opow_succ_log_self | [
[
312,
31
],
[
316,
38
]
] | 4 | 6 | rw [succ_log_def hb hx] | case inr
b : Ordinal.{u_1}
hb : 1 < b
x : Ordinal.{u_1}
hx : x β 0
β’ x < b ^ succ (log b x) | case inr
b : Ordinal.{u_1}
hb : 1 < b
x : Ordinal.{u_1}
hx : x β 0
β’ x < b ^ sInf {o | x < b ^ o} |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 36 | Ordinal.lt_opow_succ_log_self | [
[
312,
31
],
[
316,
38
]
] | 5 | 6 | exact csInf_mem (log_nonempty hb) | case inr
b : Ordinal.{u_1}
hb : 1 < b
x : Ordinal.{u_1}
hx : x β 0
β’ x < b ^ sInf {o | x < b ^ o} | no goals |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 37 | Ordinal.opow_log_le_self | [
[
319,
88
],
[
327,
39
]
] | 0 | 11 | rcases eq_or_ne b 0 with (rfl | b0) | b x : Ordinal.{u_1}
hx : x β 0
β’ b ^ log b x β€ x | case inl
x : Ordinal.{u_1}
hx : x β 0
β’ 0 ^ log 0 x β€ x
case inr
b x : Ordinal.{u_1}
hx : x β 0
b0 : b β 0
β’ b ^ log b x β€ x |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 37 | Ordinal.opow_log_le_self | [
[
319,
88
],
[
327,
39
]
] | 1 | 11 | Β· rw [zero_opow']
exact (sub_le_self _ _).trans (one_le_iff_ne_zero.2 hx) | case inl
x : Ordinal.{u_1}
hx : x β 0
β’ 0 ^ log 0 x β€ x
case inr
b x : Ordinal.{u_1}
hx : x β 0
b0 : b β 0
β’ b ^ log b x β€ x | case inr
b x : Ordinal.{u_1}
hx : x β 0
b0 : b β 0
β’ b ^ log b x β€ x |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 37 | Ordinal.opow_log_le_self | [
[
319,
88
],
[
327,
39
]
] | 2 | 11 | rcases lt_or_eq_of_le (one_le_iff_ne_zero.2 b0) with (hb | rfl) | case inr
b x : Ordinal.{u_1}
hx : x β 0
b0 : b β 0
β’ b ^ log b x β€ x | case inr.inl
b x : Ordinal.{u_1}
hx : x β 0
b0 : b β 0
hb : 1 < b
β’ b ^ log b x β€ x
case inr.inr
x : Ordinal.{u_1}
hx : x β 0
b0 : 1 β 0
β’ 1 ^ log 1 x β€ x |
Mathlib/SetTheory/Ordinal/Exponential.lean | [
[
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib/SetTheory/Ordinal/Arithmetic.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "instance pow : Pow Ordinal Ordinal :=\n β¨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} bβ©",
"end": [
31,
101
],
"full_name": "Ordinal.pow",
"kind": "commanddeclaration",
"start": [
29,
1
]
},
{
"code": "theorem opow_def (a b : Ordinal) :\n a ^ b = if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b",
"end": [
38,
6
],
"full_name": "Ordinal.opow_def",
"kind": "commanddeclaration",
"start": [
36,
1
]
},
{
"code": "theorem zero_opow' (a : Ordinal) : 0 ^ a = 1 - a",
"end": [
42,
85
],
"full_name": "Ordinal.zero_opow'",
"kind": "commanddeclaration",
"start": [
42,
1
]
},
{
"code": "@[simp]\ntheorem zero_opow {a : Ordinal} (a0 : a β 0) : (0 : Ordinal) ^ a = 0",
"end": [
47,
67
],
"full_name": "Ordinal.zero_opow",
"kind": "commanddeclaration",
"start": [
45,
1
]
},
{
"code": "@[simp]\ntheorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1",
"end": [
54,
52
],
"full_name": "Ordinal.opow_zero",
"kind": "commanddeclaration",
"start": [
50,
1
]
},
{
"code": "@[simp]\ntheorem opow_succ (a b : Ordinal) : a ^ succ b = a ^ b * a",
"end": [
60,
58
],
"full_name": "Ordinal.opow_succ",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "theorem opow_limit {a b : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b = bsup.{u, u} b fun c _ => a ^ c",
"end": [
65,
67
],
"full_name": "Ordinal.opow_limit",
"kind": "commanddeclaration",
"start": [
63,
1
]
},
{
"code": "theorem opow_le_of_limit {a b c : Ordinal} (a0 : a β 0) (h : IsLimit b) :\n a ^ b β€ c β β b' < b, a ^ b' β€ c",
"end": [
69,
77
],
"full_name": "Ordinal.opow_le_of_limit",
"kind": "commanddeclaration",
"start": [
68,
1
]
},
{
"code": "theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b β 0) (h : IsLimit c) :\n a < b ^ c β β c' < c, a < b ^ c'",
"end": [
74,
98
],
"full_name": "Ordinal.lt_opow_of_limit",
"kind": "commanddeclaration",
"start": [
72,
1
]
},
{
"code": "@[simp]\ntheorem opow_one (a : Ordinal) : a ^ (1 : Ordinal) = a",
"end": [
79,
62
],
"full_name": "Ordinal.opow_one",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "@[simp]\ntheorem one_opow (a : Ordinal) : (1 : Ordinal) ^ a = 1",
"end": [
91,
94
],
"full_name": "Ordinal.one_opow",
"kind": "commanddeclaration",
"start": [
82,
1
]
},
{
"code": "theorem opow_pos {a : Ordinal} (b : Ordinal) (a0 : 0 < a) : 0 < a ^ b",
"end": [
102,
79
],
"full_name": "Ordinal.opow_pos",
"kind": "commanddeclaration",
"start": [
94,
1
]
},
{
"code": "theorem opow_ne_zero {a : Ordinal} (b : Ordinal) (a0 : a β 0) : a ^ b β 0",
"end": [
106,
74
],
"full_name": "Ordinal.opow_ne_zero",
"kind": "commanddeclaration",
"start": [
105,
1
]
},
{
"code": "theorem opow_isNormal {a : Ordinal} (h : 1 < a) : IsNormal (a ^ Β·)",
"end": [
112,
51
],
"full_name": "Ordinal.opow_isNormal",
"kind": "commanddeclaration",
"start": [
109,
1
]
},
{
"code": "theorem opow_lt_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b < a ^ c β b < c",
"end": [
116,
28
],
"full_name": "Ordinal.opow_lt_opow_iff_right",
"kind": "commanddeclaration",
"start": [
115,
1
]
},
{
"code": "theorem opow_le_opow_iff_right {a b c : Ordinal} (a1 : 1 < a) : a ^ b β€ a ^ c β b β€ c",
"end": [
120,
28
],
"full_name": "Ordinal.opow_le_opow_iff_right",
"kind": "commanddeclaration",
"start": [
119,
1
]
},
{
"code": "theorem opow_right_inj {a b c : Ordinal} (a1 : 1 < a) : a ^ b = a ^ c β b = c",
"end": [
124,
25
],
"full_name": "Ordinal.opow_right_inj",
"kind": "commanddeclaration",
"start": [
123,
1
]
},
{
"code": "theorem opow_isLimit {a b : Ordinal} (a1 : 1 < a) : IsLimit b β IsLimit (a ^ b)",
"end": [
128,
29
],
"full_name": "Ordinal.opow_isLimit",
"kind": "commanddeclaration",
"start": [
127,
1
]
},
{
"code": "theorem opow_isLimit_left {a b : Ordinal} (l : IsLimit a) (hb : b β 0) : IsLimit (a ^ b)",
"end": [
136,
35
],
"full_name": "Ordinal.opow_isLimit_left",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem opow_le_opow_right {a b c : Ordinal} (hβ : 0 < a) (hβ : b β€ c) : a ^ b β€ a ^ c",
"end": [
144,
34
],
"full_name": "Ordinal.opow_le_opow_right",
"kind": "commanddeclaration",
"start": [
139,
1
]
},
{
"code": "theorem opow_le_opow_left {a b : Ordinal} (c : Ordinal) (ab : a β€ b) : a ^ c β€ b ^ c",
"end": [
162,
96
],
"full_name": "Ordinal.opow_le_opow_left",
"kind": "commanddeclaration",
"start": [
147,
1
]
},
{
"code": "theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a β€ a ^ b",
"end": [
173,
50
],
"full_name": "Ordinal.left_le_opow",
"kind": "commanddeclaration",
"start": [
165,
1
]
},
{
"code": "theorem right_le_opow {a : Ordinal} (b : Ordinal) (a1 : 1 < a) : b β€ a ^ b",
"end": [
177,
31
],
"full_name": "Ordinal.right_le_opow",
"kind": "commanddeclaration",
"start": [
176,
1
]
},
{
"code": "theorem opow_lt_opow_left_of_succ {a b c : Ordinal} (ab : a < b) : a ^ succ c < b ^ succ c",
"end": [
184,
81
],
"full_name": "Ordinal.opow_lt_opow_left_of_succ",
"kind": "commanddeclaration",
"start": [
180,
1
]
},
{
"code": "theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c",
"end": [
208,
18
],
"full_name": "Ordinal.opow_add",
"kind": "commanddeclaration",
"start": [
187,
1
]
},
{
"code": "theorem opow_one_add (a b : Ordinal) : a ^ (1 + b) = a * a ^ b",
"end": [
211,
93
],
"full_name": "Ordinal.opow_one_add",
"kind": "commanddeclaration",
"start": [
211,
1
]
},
{
"code": "theorem opow_dvd_opow (a : Ordinal) {b c : Ordinal} (h : b β€ c) : a ^ b β£ a ^ c",
"end": [
215,
68
],
"full_name": "Ordinal.opow_dvd_opow",
"kind": "commanddeclaration",
"start": [
214,
1
]
},
{
"code": "theorem opow_dvd_opow_iff {a b c : Ordinal} (a1 : 1 < a) : a ^ b β£ a ^ c β b β€ c",
"end": [
223,
21
],
"full_name": "Ordinal.opow_dvd_opow_iff",
"kind": "commanddeclaration",
"start": [
218,
1
]
},
{
"code": "theorem opow_mul (a b c : Ordinal) : a ^ (b * c) = (a ^ b) ^ c",
"end": [
248,
56
],
"full_name": "Ordinal.opow_mul",
"kind": "commanddeclaration",
"start": [
226,
1
]
},
{
"code": "@[pp_nodot]\ndef log (b : Ordinal) (x : Ordinal) : Ordinal :=\n if _h : 1 < b then pred (sInf { o | x < b ^ o }) else 0",
"end": [
258,
58
],
"full_name": "Ordinal.log",
"kind": "commanddeclaration",
"start": [
254,
1
]
},
{
"code": "theorem log_nonempty {b x : Ordinal} (h : 1 < b) : { o : Ordinal | x < b ^ o }.Nonempty",
"end": [
263,
41
],
"full_name": "Ordinal.log_nonempty",
"kind": "commanddeclaration",
"start": [
261,
1
]
},
{
"code": "theorem log_def {b : Ordinal} (h : 1 < b) (x : Ordinal) :\n log b x = pred (sInf { o | x < b ^ o })",
"end": [
267,
77
],
"full_name": "Ordinal.log_def",
"kind": "commanddeclaration",
"start": [
266,
1
]
},
{
"code": "theorem log_of_not_one_lt_left {b : Ordinal} (h : Β¬1 < b) (x : Ordinal) : log b x = 0",
"end": [
271,
29
],
"full_name": "Ordinal.log_of_not_one_lt_left",
"kind": "commanddeclaration",
"start": [
270,
1
]
},
{
"code": "theorem log_of_left_le_one {b : Ordinal} (h : b β€ 1) : β x, log b x = 0",
"end": [
275,
34
],
"full_name": "Ordinal.log_of_left_le_one",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_left : β b, log 0 b = 0",
"end": [
280,
33
],
"full_name": "Ordinal.log_zero_left",
"kind": "commanddeclaration",
"start": [
278,
1
]
},
{
"code": "@[simp]\ntheorem log_zero_right (b : Ordinal) : log b 0 = 0",
"end": [
291,
48
],
"full_name": "Ordinal.log_zero_right",
"kind": "commanddeclaration",
"start": [
283,
1
]
},
{
"code": "@[simp]\ntheorem log_one_left : β b, log 1 b = 0",
"end": [
296,
28
],
"full_name": "Ordinal.log_one_left",
"kind": "commanddeclaration",
"start": [
294,
1
]
},
{
"code": "theorem succ_log_def {b x : Ordinal} (hb : 1 < b) (hx : x β 0) :\n succ (log b x) = sInf { o : Ordinal | x < b ^ o }",
"end": [
308,
76
],
"full_name": "Ordinal.succ_log_def",
"kind": "commanddeclaration",
"start": [
299,
1
]
},
{
"code": "theorem lt_opow_succ_log_self {b : Ordinal} (hb : 1 < b) (x : Ordinal) :\n x < b ^ succ (log b x)",
"end": [
316,
38
],
"full_name": "Ordinal.lt_opow_succ_log_self",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem opow_log_le_self (b : Ordinal) {x : Ordinal} (hx : x β 0) : b ^ log b x β€ x",
"end": [
327,
39
],
"full_name": "Ordinal.opow_log_le_self",
"kind": "commanddeclaration",
"start": [
319,
1
]
},
{
"code": "theorem opow_le_iff_le_log {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : b ^ c β€ x β c β€ log b x",
"end": [
336,
78
],
"full_name": "Ordinal.opow_le_iff_le_log",
"kind": "commanddeclaration",
"start": [
330,
1
]
},
{
"code": "theorem lt_opow_iff_log_lt {b x c : Ordinal} (hb : 1 < b) (hx : x β 0) : x < b ^ c β log b x < c",
"end": [
340,
52
],
"full_name": "Ordinal.lt_opow_iff_log_lt",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem log_pos {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) : 0 < log b o",
"end": [
344,
71
],
"full_name": "Ordinal.log_pos",
"kind": "commanddeclaration",
"start": [
343,
1
]
},
{
"code": "theorem log_eq_zero {b o : Ordinal} (hbo : o < b) : log b o = 0",
"end": [
354,
92
],
"full_name": "Ordinal.log_eq_zero",
"kind": "commanddeclaration",
"start": [
347,
1
]
},
{
"code": "@[mono]\ntheorem log_mono_right (b : Ordinal) {x y : Ordinal} (xy : x β€ y) : log b x β€ log b y",
"end": [
364,
67
],
"full_name": "Ordinal.log_mono_right",
"kind": "commanddeclaration",
"start": [
357,
1
]
},
{
"code": "theorem log_le_self (b x : Ordinal) : log b x β€ x",
"end": [
371,
67
],
"full_name": "Ordinal.log_le_self",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "@[simp]\ntheorem log_one_right (b : Ordinal) : log b 1 = 0",
"end": [
376,
69
],
"full_name": "Ordinal.log_one_right",
"kind": "commanddeclaration",
"start": [
374,
1
]
},
{
"code": "theorem mod_opow_log_lt_self (b : Ordinal) {o : Ordinal} (ho : o β 0) : o % (b ^ log b o) < o",
"end": [
382,
75
],
"full_name": "Ordinal.mod_opow_log_lt_self",
"kind": "commanddeclaration",
"start": [
379,
1
]
},
{
"code": "theorem log_mod_opow_log_lt_log_self {b o : Ordinal} (hb : 1 < b) (ho : o β 0) (hbo : b β€ o) :\n log b (o % (b ^ log b o)) < log b o",
"end": [
394,
44
],
"full_name": "Ordinal.log_mod_opow_log_lt_log_self",
"kind": "commanddeclaration",
"start": [
385,
1
]
},
{
"code": "theorem opow_mul_add_pos {b v : Ordinal} (hb : b β 0) (u : Ordinal) (hv : v β 0) (w : Ordinal) :\n 0 < b ^ u * v + w",
"end": [
400,
78
],
"full_name": "Ordinal.opow_mul_add_pos",
"kind": "commanddeclaration",
"start": [
397,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_mul_succ {b u w : Ordinal} (v : Ordinal) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ u * succ v",
"end": [
404,
77
],
"full_name": "Ordinal.opow_mul_add_lt_opow_mul_succ",
"kind": "commanddeclaration",
"start": [
403,
1
]
},
{
"code": "theorem opow_mul_add_lt_opow_succ {b u v w : Ordinal} (hvb : v < b) (hw : w < b ^ u) :\n b ^ u * v + w < b ^ succ u",
"end": [
411,
22
],
"full_name": "Ordinal.opow_mul_add_lt_opow_succ",
"kind": "commanddeclaration",
"start": [
407,
1
]
},
{
"code": "theorem log_opow_mul_add {b u v w : Ordinal} (hb : 1 < b) (hv : v β 0) (hvb : v < b)\n (hw : w < b ^ u) : log b (b ^ u * v + w) = u",
"end": [
423,
62
],
"full_name": "Ordinal.log_opow_mul_add",
"kind": "commanddeclaration",
"start": [
414,
1
]
},
{
"code": "theorem log_opow {b : Ordinal} (hb : 1 < b) (x : Ordinal) : log b (b ^ x) = x",
"end": [
429,
25
],
"full_name": "Ordinal.log_opow",
"kind": "commanddeclaration",
"start": [
426,
1
]
},
{
"code": "theorem div_opow_log_pos (b : Ordinal) {o : Ordinal} (ho : o β 0) : 0 < o / (b ^ log b o)",
"end": [
436,
32
],
"full_name": "Ordinal.div_opow_log_pos",
"kind": "commanddeclaration",
"start": [
432,
1
]
},
{
"code": "theorem div_opow_log_lt {b : Ordinal} (o : Ordinal) (hb : 1 < b) : o / (b ^ log b o) < b",
"end": [
441,
35
],
"full_name": "Ordinal.div_opow_log_lt",
"kind": "commanddeclaration",
"start": [
439,
1
]
},
{
"code": "theorem add_log_le_log_mul {x y : Ordinal} (b : Ordinal) (hx : x β 0) (hy : y β 0) :\n log b x + log b y β€ log b (x * y)",
"end": [
450,
59
],
"full_name": "Ordinal.add_log_le_log_mul",
"kind": "commanddeclaration",
"start": [
444,
1
]
},
{
"code": "@[simp, norm_cast]\ntheorem natCast_opow (m : β) : β n : β, β(m ^ n : β) = (m : Ordinal) ^ (n : Ordinal)",
"end": [
459,
92
],
"full_name": "Ordinal.natCast_opow",
"kind": "commanddeclaration",
"start": [
455,
1
]
},
{
"code": "theorem sup_opow_nat {o : Ordinal} (ho : 0 < o) : (sup fun n : β => o ^ (n : Ordinal)) = o ^ Ο",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 37 | Ordinal.opow_log_le_self | [
[
319,
88
],
[
327,
39
]
] | 3 | 11 | Β· refine le_of_not_lt fun h => (lt_succ (log b x)).not_le ?_
have := @csInf_le' _ _ { o | x < b ^ o } _ h
rwa [β succ_log_def hb hx] at this | case inr.inl
b x : Ordinal.{u_1}
hx : x β 0
b0 : b β 0
hb : 1 < b
β’ b ^ log b x β€ x
case inr.inr
x : Ordinal.{u_1}
hx : x β 0
b0 : 1 β 0
β’ 1 ^ log 1 x β€ x | case inr.inr
x : Ordinal.{u_1}
hx : x β 0
b0 : 1 β 0
β’ 1 ^ log 1 x β€ x |
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