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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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 37 | Ordinal.opow_log_le_self | [
[
319,
88
],
[
327,
39
]
] | 4 | 11 | · rwa [one_opow, one_le_iff_ne_zero] | case inr.inr
x : Ordinal.{u_1}
hx : x ≠ 0
b0 : 1 ≠ 0
⊢ 1 ^ log 1 x ≤ 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 37 | Ordinal.opow_log_le_self | [
[
319,
88
],
[
327,
39
]
] | 5 | 11 | rw [zero_opow'] | case inl
x : Ordinal.{u_1}
hx : x ≠ 0
⊢ 0 ^ log 0 x ≤ x | case inl
x : Ordinal.{u_1}
hx : x ≠ 0
⊢ 1 - log 0 x ≤ 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 37 | Ordinal.opow_log_le_self | [
[
319,
88
],
[
327,
39
]
] | 6 | 11 | exact (sub_le_self _ _).trans (one_le_iff_ne_zero.2 hx) | case inl
x : Ordinal.{u_1}
hx : x ≠ 0
⊢ 1 - log 0 x ≤ 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 37 | Ordinal.opow_log_le_self | [
[
319,
88
],
[
327,
39
]
] | 7 | 11 | refine le_of_not_lt fun h => (lt_succ (log b x)).not_le ?_ | case inr.inl
b x : Ordinal.{u_1}
hx : x ≠ 0
b0 : b ≠ 0
hb : 1 < b
⊢ b ^ log b x ≤ x | case inr.inl
b x : Ordinal.{u_1}
hx : x ≠ 0
b0 : b ≠ 0
hb : 1 < b
h : x < b ^ log b x
⊢ succ (log b x) ≤ 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 37 | Ordinal.opow_log_le_self | [
[
319,
88
],
[
327,
39
]
] | 8 | 11 | have := @csInf_le' _ _ { o | x < b ^ o } _ h | case inr.inl
b x : Ordinal.{u_1}
hx : x ≠ 0
b0 : b ≠ 0
hb : 1 < b
h : x < b ^ log b x
⊢ succ (log b x) ≤ log b x | case inr.inl
b x : Ordinal.{u_1}
hx : x ≠ 0
b0 : b ≠ 0
hb : 1 < b
h : x < b ^ log b x
this : sInf {o | x < b ^ o} ≤ log b x
⊢ succ (log b x) ≤ 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 37 | Ordinal.opow_log_le_self | [
[
319,
88
],
[
327,
39
]
] | 9 | 11 | rwa [← succ_log_def hb hx] at this | case inr.inl
b x : Ordinal.{u_1}
hx : x ≠ 0
b0 : b ≠ 0
hb : 1 < b
h : x < b ^ log b x
this : sInf {o | x < b ^ o} ≤ log b x
⊢ succ (log b x) ≤ 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 37 | Ordinal.opow_log_le_self | [
[
319,
88
],
[
327,
39
]
] | 10 | 11 | rwa [one_opow, one_le_iff_ne_zero] | case inr.inr
x : Ordinal.{u_1}
hx : x ≠ 0
b0 : 1 ≠ 0
⊢ 1 ^ log 1 x ≤ 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 40 | Ordinal.log_pos | [
[
343,
90
],
[
344,
71
]
] | 0 | 1 | rwa [← succ_le_iff, succ_zero, ← opow_le_iff_le_log hb ho, opow_one] | b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
⊢ 0 < log 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 41 | Ordinal.log_eq_zero | [
[
347,
68
],
[
354,
92
]
] | 0 | 12 | rcases eq_or_ne o 0 with (rfl | ho) | b o : Ordinal.{u_1}
hbo : o < b
⊢ log b o = 0 | case inl
b : Ordinal.{u_1}
hbo : 0 < b
⊢ log b 0 = 0
case inr
b o : Ordinal.{u_1}
hbo : o < b
ho : o ≠ 0
⊢ log b o = 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 41 | Ordinal.log_eq_zero | [
[
347,
68
],
[
354,
92
]
] | 1 | 12 | · exact log_zero_right b | case inl
b : Ordinal.{u_1}
hbo : 0 < b
⊢ log b 0 = 0
case inr
b o : Ordinal.{u_1}
hbo : o < b
ho : o ≠ 0
⊢ log b o = 0 | case inr
b o : Ordinal.{u_1}
hbo : o < b
ho : o ≠ 0
⊢ log b o = 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 41 | Ordinal.log_eq_zero | [
[
347,
68
],
[
354,
92
]
] | 2 | 12 | rcases le_or_lt b 1 with hb | hb | case inr
b o : Ordinal.{u_1}
hbo : o < b
ho : o ≠ 0
⊢ log b o = 0 | case inr.inl
b o : Ordinal.{u_1}
hbo : o < b
ho : o ≠ 0
hb : b ≤ 1
⊢ log b o = 0
case inr.inr
b o : Ordinal.{u_1}
hbo : o < b
ho : o ≠ 0
hb : 1 < b
⊢ log b o = 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 41 | Ordinal.log_eq_zero | [
[
347,
68
],
[
354,
92
]
] | 3 | 12 | · rcases le_one_iff.1 hb with (rfl | rfl)
· exact log_zero_left o
· exact log_one_left o | case inr.inl
b o : Ordinal.{u_1}
hbo : o < b
ho : o ≠ 0
hb : b ≤ 1
⊢ log b o = 0
case inr.inr
b o : Ordinal.{u_1}
hbo : o < b
ho : o ≠ 0
hb : 1 < b
⊢ log b o = 0 | case inr.inr
b o : Ordinal.{u_1}
hbo : o < b
ho : o ≠ 0
hb : 1 < b
⊢ log b o = 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 41 | Ordinal.log_eq_zero | [
[
347,
68
],
[
354,
92
]
] | 4 | 12 | · rwa [← Ordinal.le_zero, ← lt_succ_iff, succ_zero, ← lt_opow_iff_log_lt hb ho, opow_one] | case inr.inr
b o : Ordinal.{u_1}
hbo : o < b
ho : o ≠ 0
hb : 1 < b
⊢ log b o = 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 41 | Ordinal.log_eq_zero | [
[
347,
68
],
[
354,
92
]
] | 5 | 12 | exact log_zero_right b | case inl
b : Ordinal.{u_1}
hbo : 0 < b
⊢ log 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 41 | Ordinal.log_eq_zero | [
[
347,
68
],
[
354,
92
]
] | 6 | 12 | rcases le_one_iff.1 hb with (rfl | rfl) | case inr.inl
b o : Ordinal.{u_1}
hbo : o < b
ho : o ≠ 0
hb : b ≤ 1
⊢ log b o = 0 | case inr.inl.inl
o : Ordinal.{u_1}
ho : o ≠ 0
hbo : o < 0
hb : 0 ≤ 1
⊢ log 0 o = 0
case inr.inl.inr
o : Ordinal.{u_1}
ho : o ≠ 0
hbo : o < 1
hb : 1 ≤ 1
⊢ log 1 o = 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 41 | Ordinal.log_eq_zero | [
[
347,
68
],
[
354,
92
]
] | 7 | 12 | · exact log_zero_left o | case inr.inl.inl
o : Ordinal.{u_1}
ho : o ≠ 0
hbo : o < 0
hb : 0 ≤ 1
⊢ log 0 o = 0
case inr.inl.inr
o : Ordinal.{u_1}
ho : o ≠ 0
hbo : o < 1
hb : 1 ≤ 1
⊢ log 1 o = 0 | case inr.inl.inr
o : Ordinal.{u_1}
ho : o ≠ 0
hbo : o < 1
hb : 1 ≤ 1
⊢ log 1 o = 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 41 | Ordinal.log_eq_zero | [
[
347,
68
],
[
354,
92
]
] | 8 | 12 | · exact log_one_left o | case inr.inl.inr
o : Ordinal.{u_1}
ho : o ≠ 0
hbo : o < 1
hb : 1 ≤ 1
⊢ log 1 o = 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 41 | Ordinal.log_eq_zero | [
[
347,
68
],
[
354,
92
]
] | 9 | 12 | exact log_zero_left o | case inr.inl.inl
o : Ordinal.{u_1}
ho : o ≠ 0
hbo : o < 0
hb : 0 ≤ 1
⊢ log 0 o = 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 41 | Ordinal.log_eq_zero | [
[
347,
68
],
[
354,
92
]
] | 10 | 12 | exact log_one_left o | case inr.inl.inr
o : Ordinal.{u_1}
ho : o ≠ 0
hbo : o < 1
hb : 1 ≤ 1
⊢ log 1 o = 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 41 | Ordinal.log_eq_zero | [
[
347,
68
],
[
354,
92
]
] | 11 | 12 | rwa [← Ordinal.le_zero, ← lt_succ_iff, succ_zero, ← lt_opow_iff_log_lt hb ho, opow_one] | case inr.inr
b o : Ordinal.{u_1}
hbo : o < b
ho : o ≠ 0
hb : 1 < b
⊢ log b o = 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 45 | Ordinal.mod_opow_log_lt_self | [
[
379,
98
],
[
382,
75
]
] | 0 | 5 | rcases eq_or_ne b 0 with (rfl | hb) | b o : Ordinal.{u_1}
ho : o ≠ 0
⊢ o % b ^ log b o < o | case inl
o : Ordinal.{u_1}
ho : o ≠ 0
⊢ o % 0 ^ log 0 o < o
case inr
b o : Ordinal.{u_1}
ho : o ≠ 0
hb : b ≠ 0
⊢ o % b ^ log b o < 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 45 | Ordinal.mod_opow_log_lt_self | [
[
379,
98
],
[
382,
75
]
] | 1 | 5 | · simpa using Ordinal.pos_iff_ne_zero.2 ho | case inl
o : Ordinal.{u_1}
ho : o ≠ 0
⊢ o % 0 ^ log 0 o < o
case inr
b o : Ordinal.{u_1}
ho : o ≠ 0
hb : b ≠ 0
⊢ o % b ^ log b o < o | case inr
b o : Ordinal.{u_1}
ho : o ≠ 0
hb : b ≠ 0
⊢ o % b ^ log b o < 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 45 | Ordinal.mod_opow_log_lt_self | [
[
379,
98
],
[
382,
75
]
] | 2 | 5 | · exact (mod_lt _ <| opow_ne_zero _ hb).trans_le (opow_log_le_self _ ho) | case inr
b o : Ordinal.{u_1}
ho : o ≠ 0
hb : b ≠ 0
⊢ o % b ^ log b o < 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 45 | Ordinal.mod_opow_log_lt_self | [
[
379,
98
],
[
382,
75
]
] | 3 | 5 | simpa using Ordinal.pos_iff_ne_zero.2 ho | case inl
o : Ordinal.{u_1}
ho : o ≠ 0
⊢ o % 0 ^ log 0 o < 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 45 | Ordinal.mod_opow_log_lt_self | [
[
379,
98
],
[
382,
75
]
] | 4 | 5 | exact (mod_lt _ <| opow_ne_zero _ hb).trans_le (opow_log_le_self _ ho) | case inr
b o : Ordinal.{u_1}
ho : o ≠ 0
hb : b ≠ 0
⊢ o % b ^ log b o < 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 46 | Ordinal.log_mod_opow_log_lt_log_self | [
[
386,
44
],
[
394,
44
]
] | 0 | 10 | rcases eq_or_ne (o % (b ^ log b o)) 0 with h | h | b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
⊢ log b (o % b ^ log b o) < log b o | case inl
b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
h : o % b ^ log b o = 0
⊢ log b (o % b ^ log b o) < log b o
case inr
b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
h : o % b ^ log b o ≠ 0
⊢ log b (o % b ^ log b o) < log 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 46 | Ordinal.log_mod_opow_log_lt_log_self | [
[
386,
44
],
[
394,
44
]
] | 1 | 10 | · rw [h, log_zero_right]
apply log_pos hb ho hbo | case inl
b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
h : o % b ^ log b o = 0
⊢ log b (o % b ^ log b o) < log b o
case inr
b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
h : o % b ^ log b o ≠ 0
⊢ log b (o % b ^ log b o) < log b o | case inr
b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
h : o % b ^ log b o ≠ 0
⊢ log b (o % b ^ log b o) < log 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 46 | Ordinal.log_mod_opow_log_lt_log_self | [
[
386,
44
],
[
394,
44
]
] | 2 | 10 | · rw [← succ_le_iff, succ_log_def hb h]
apply csInf_le'
apply mod_lt
rw [← Ordinal.pos_iff_ne_zero]
exact opow_pos _ (zero_lt_one.trans hb) | case inr
b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
h : o % b ^ log b o ≠ 0
⊢ log b (o % b ^ log b o) < log 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 46 | Ordinal.log_mod_opow_log_lt_log_self | [
[
386,
44
],
[
394,
44
]
] | 3 | 10 | rw [h, log_zero_right] | case inl
b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
h : o % b ^ log b o = 0
⊢ log b (o % b ^ log b o) < log b o | case inl
b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
h : o % b ^ log b o = 0
⊢ 0 < log 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 46 | Ordinal.log_mod_opow_log_lt_log_self | [
[
386,
44
],
[
394,
44
]
] | 4 | 10 | apply log_pos hb ho hbo | case inl
b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
h : o % b ^ log b o = 0
⊢ 0 < log 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 46 | Ordinal.log_mod_opow_log_lt_log_self | [
[
386,
44
],
[
394,
44
]
] | 5 | 10 | rw [← succ_le_iff, succ_log_def hb h] | case inr
b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
h : o % b ^ log b o ≠ 0
⊢ log b (o % b ^ log b o) < log b o | case inr
b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
h : o % b ^ log b o ≠ 0
⊢ sInf {o_1 | o % b ^ log b o < b ^ o_1} ≤ log 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 46 | Ordinal.log_mod_opow_log_lt_log_self | [
[
386,
44
],
[
394,
44
]
] | 6 | 10 | apply csInf_le' | case inr
b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
h : o % b ^ log b o ≠ 0
⊢ sInf {o_1 | o % b ^ log b o < b ^ o_1} ≤ log b o | case inr.h
b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
h : o % b ^ log b o ≠ 0
⊢ log b o ∈ {o_1 | o % b ^ log b o < b ^ o_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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 46 | Ordinal.log_mod_opow_log_lt_log_self | [
[
386,
44
],
[
394,
44
]
] | 7 | 10 | apply mod_lt | case inr.h
b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
h : o % b ^ log b o ≠ 0
⊢ log b o ∈ {o_1 | o % b ^ log b o < b ^ o_1} | case inr.h.h
b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
h : o % b ^ log b o ≠ 0
⊢ b ^ log b o ≠ 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 46 | Ordinal.log_mod_opow_log_lt_log_self | [
[
386,
44
],
[
394,
44
]
] | 8 | 10 | rw [← Ordinal.pos_iff_ne_zero] | case inr.h.h
b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
h : o % b ^ log b o ≠ 0
⊢ b ^ log b o ≠ 0 | case inr.h.h
b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
h : o % b ^ log b o ≠ 0
⊢ 0 < b ^ log 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 46 | Ordinal.log_mod_opow_log_lt_log_self | [
[
386,
44
],
[
394,
44
]
] | 9 | 10 | exact opow_pos _ (zero_lt_one.trans hb) | case inr.h.h
b o : Ordinal.{u_1}
hb : 1 < b
ho : o ≠ 0
hbo : b ≤ o
h : o % b ^ log b o ≠ 0
⊢ 0 < b ^ log 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 48 | Ordinal.opow_mul_add_lt_opow_mul_succ | [
[
404,
39
],
[
404,
77
]
] | 0 | 1 | rwa [mul_succ, add_lt_add_iff_left] | b u w v : Ordinal.{u_1}
hw : w < b ^ u
⊢ b ^ u * v + w < b ^ u * succ v | 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 49 | Ordinal.opow_mul_add_lt_opow_succ | [
[
408,
35
],
[
411,
22
]
] | 0 | 2 | convert (opow_mul_add_lt_opow_mul_succ v hw).trans_le (mul_le_mul_left' (succ_le_of_lt hvb) _)
using 1 | b u v w : Ordinal.{u_1}
hvb : v < b
hw : w < b ^ u
⊢ b ^ u * v + w < b ^ succ u | case h.e'_4
b u v w : Ordinal.{u_1}
hvb : v < b
hw : w < b ^ u
⊢ b ^ succ u = b ^ u * 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 49 | Ordinal.opow_mul_add_lt_opow_succ | [
[
408,
35
],
[
411,
22
]
] | 1 | 2 | exact opow_succ b u | case h.e'_4
b u v w : Ordinal.{u_1}
hvb : v < b
hw : w < b ^ u
⊢ b ^ succ u = b ^ u * 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 50 | Ordinal.log_opow_mul_add | [
[
415,
53
],
[
423,
62
]
] | 0 | 10 | have hne' := (opow_mul_add_pos (zero_lt_one.trans hb).ne' u hv w).ne' | b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
⊢ log b (b ^ u * v + w) = u | b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
hne' : b ^ u * v + w ≠ 0
⊢ log b (b ^ u * v + w) = u |
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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 50 | Ordinal.log_opow_mul_add | [
[
415,
53
],
[
423,
62
]
] | 1 | 10 | by_contra! hne | b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
hne' : b ^ u * v + w ≠ 0
⊢ log b (b ^ u * v + w) = u | b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
hne' : b ^ u * v + w ≠ 0
hne : log b (b ^ u * v + w) ≠ u
⊢ False |
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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 50 | Ordinal.log_opow_mul_add | [
[
415,
53
],
[
423,
62
]
] | 2 | 10 | cases' lt_or_gt_of_ne hne with h h | b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
hne' : b ^ u * v + w ≠ 0
hne : log b (b ^ u * v + w) ≠ u
⊢ False | case inl
b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
hne' : b ^ u * v + w ≠ 0
hne : log b (b ^ u * v + w) ≠ u
h : log b (b ^ u * v + w) < u
⊢ False
case inr
b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
hne' : b ^ u * v + w ≠ 0
hne : log b (b ^ u * v + w) ≠ u
h : log b (b ^ u * v + w) > u
⊢ False |
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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 50 | Ordinal.log_opow_mul_add | [
[
415,
53
],
[
423,
62
]
] | 3 | 10 | · rw [← lt_opow_iff_log_lt hb hne'] at h
exact h.not_le ((le_mul_left _ (Ordinal.pos_iff_ne_zero.2 hv)).trans (le_add_right _ _)) | case inl
b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
hne' : b ^ u * v + w ≠ 0
hne : log b (b ^ u * v + w) ≠ u
h : log b (b ^ u * v + w) < u
⊢ False
case inr
b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
hne' : b ^ u * v + w ≠ 0
hne : log b (b ^ u * v + w) ≠ u
h : log b (b ^ u * v + w) > u
⊢ False | case inr
b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
hne' : b ^ u * v + w ≠ 0
hne : log b (b ^ u * v + w) ≠ u
h : log b (b ^ u * v + w) > u
⊢ False |
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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 50 | Ordinal.log_opow_mul_add | [
[
415,
53
],
[
423,
62
]
] | 4 | 10 | · conv at h => change u < log b (b ^ u * v + w)
rw [← succ_le_iff, ← opow_le_iff_le_log hb hne'] at h
exact (not_lt_of_le h) (opow_mul_add_lt_opow_succ hvb hw) | case inr
b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
hne' : b ^ u * v + w ≠ 0
hne : log b (b ^ u * v + w) ≠ u
h : log b (b ^ u * v + w) > u
⊢ False | 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 50 | Ordinal.log_opow_mul_add | [
[
415,
53
],
[
423,
62
]
] | 5 | 10 | rw [← lt_opow_iff_log_lt hb hne'] at h | case inl
b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
hne' : b ^ u * v + w ≠ 0
hne : log b (b ^ u * v + w) ≠ u
h : log b (b ^ u * v + w) < u
⊢ False | case inl
b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
hne' : b ^ u * v + w ≠ 0
hne : log b (b ^ u * v + w) ≠ u
h : b ^ u * v + w < b ^ u
⊢ False |
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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 50 | Ordinal.log_opow_mul_add | [
[
415,
53
],
[
423,
62
]
] | 6 | 10 | exact h.not_le ((le_mul_left _ (Ordinal.pos_iff_ne_zero.2 hv)).trans (le_add_right _ _)) | case inl
b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
hne' : b ^ u * v + w ≠ 0
hne : log b (b ^ u * v + w) ≠ u
h : b ^ u * v + w < b ^ u
⊢ False | 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 50 | Ordinal.log_opow_mul_add | [
[
415,
53
],
[
423,
62
]
] | 7 | 10 | conv at h => change u < log b (b ^ u * v + w) | case inr
b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
hne' : b ^ u * v + w ≠ 0
hne : log b (b ^ u * v + w) ≠ u
h : log b (b ^ u * v + w) > u
⊢ False | case inr
b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
hne' : b ^ u * v + w ≠ 0
hne : log b (b ^ u * v + w) ≠ u
h : u < log b (b ^ u * v + w)
⊢ False |
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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 50 | Ordinal.log_opow_mul_add | [
[
415,
53
],
[
423,
62
]
] | 8 | 10 | rw [← succ_le_iff, ← opow_le_iff_le_log hb hne'] at h | case inr
b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
hne' : b ^ u * v + w ≠ 0
hne : log b (b ^ u * v + w) ≠ u
h : u < log b (b ^ u * v + w)
⊢ False | case inr
b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
hne' : b ^ u * v + w ≠ 0
hne : log b (b ^ u * v + w) ≠ u
h : b ^ succ u ≤ b ^ u * v + w
⊢ False |
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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 50 | Ordinal.log_opow_mul_add | [
[
415,
53
],
[
423,
62
]
] | 9 | 10 | exact (not_lt_of_le h) (opow_mul_add_lt_opow_succ hvb hw) | case inr
b u v w : Ordinal.{u_1}
hb : 1 < b
hv : v ≠ 0
hvb : v < b
hw : w < b ^ u
hne' : b ^ u * v + w ≠ 0
hne : log b (b ^ u * v + w) ≠ u
h : b ^ succ u ≤ b ^ u * v + w
⊢ False | 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 51 | Ordinal.log_opow | [
[
426,
82
],
[
429,
25
]
] | 0 | 2 | convert log_opow_mul_add hb zero_ne_one.symm hb (opow_pos x (zero_lt_one.trans hb))
using 1 | b : Ordinal.{u_1}
hb : 1 < b
x : Ordinal.{u_1}
⊢ log b (b ^ x) = x | case h.e'_2
b : Ordinal.{u_1}
hb : 1 < b
x : Ordinal.{u_1}
⊢ log b (b ^ x) = log b (b ^ x * 1 + 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 51 | Ordinal.log_opow | [
[
426,
82
],
[
429,
25
]
] | 1 | 2 | rw [add_zero, mul_one] | case h.e'_2
b : Ordinal.{u_1}
hb : 1 < b
x : Ordinal.{u_1}
⊢ log b (b ^ x) = log b (b ^ x * 1 + 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 52 | Ordinal.div_opow_log_pos | [
[
432,
94
],
[
436,
32
]
] | 0 | 6 | rcases eq_zero_or_pos b with (rfl | hb) | b o : Ordinal.{u_1}
ho : o ≠ 0
⊢ 0 < o / b ^ log b o | case inl
o : Ordinal.{u_1}
ho : o ≠ 0
⊢ 0 < o / 0 ^ log 0 o
case inr
b o : Ordinal.{u_1}
ho : o ≠ 0
hb : 0 < b
⊢ 0 < o / b ^ log 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 52 | Ordinal.div_opow_log_pos | [
[
432,
94
],
[
436,
32
]
] | 1 | 6 | · simpa using Ordinal.pos_iff_ne_zero.2 ho | case inl
o : Ordinal.{u_1}
ho : o ≠ 0
⊢ 0 < o / 0 ^ log 0 o
case inr
b o : Ordinal.{u_1}
ho : o ≠ 0
hb : 0 < b
⊢ 0 < o / b ^ log b o | case inr
b o : Ordinal.{u_1}
ho : o ≠ 0
hb : 0 < b
⊢ 0 < o / b ^ log 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 52 | Ordinal.div_opow_log_pos | [
[
432,
94
],
[
436,
32
]
] | 2 | 6 | · rw [div_pos (opow_ne_zero _ hb.ne')]
exact opow_log_le_self b ho | case inr
b o : Ordinal.{u_1}
ho : o ≠ 0
hb : 0 < b
⊢ 0 < o / b ^ log 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 52 | Ordinal.div_opow_log_pos | [
[
432,
94
],
[
436,
32
]
] | 3 | 6 | simpa using Ordinal.pos_iff_ne_zero.2 ho | case inl
o : Ordinal.{u_1}
ho : o ≠ 0
⊢ 0 < o / 0 ^ log 0 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 52 | Ordinal.div_opow_log_pos | [
[
432,
94
],
[
436,
32
]
] | 4 | 6 | rw [div_pos (opow_ne_zero _ hb.ne')] | case inr
b o : Ordinal.{u_1}
ho : o ≠ 0
hb : 0 < b
⊢ 0 < o / b ^ log b o | case inr
b o : Ordinal.{u_1}
ho : o ≠ 0
hb : 0 < b
⊢ b ^ log b o ≤ 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 52 | Ordinal.div_opow_log_pos | [
[
432,
94
],
[
436,
32
]
] | 5 | 6 | exact opow_log_le_self b ho | case inr
b o : Ordinal.{u_1}
ho : o ≠ 0
hb : 0 < b
⊢ b ^ log b o ≤ 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 53 | Ordinal.div_opow_log_lt | [
[
439,
93
],
[
441,
35
]
] | 0 | 2 | rw [div_lt (opow_pos _ (zero_lt_one.trans hb)).ne', ← opow_succ] | b o : Ordinal.{u_1}
hb : 1 < b
⊢ o / b ^ log b o < b | b o : Ordinal.{u_1}
hb : 1 < b
⊢ o < b ^ succ (log 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 53 | Ordinal.div_opow_log_lt | [
[
439,
93
],
[
441,
35
]
] | 1 | 2 | exact lt_opow_succ_log_self hb o | b o : Ordinal.{u_1}
hb : 1 < b
⊢ o < b ^ succ (log 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 54 | Ordinal.add_log_le_log_mul | [
[
445,
42
],
[
450,
59
]
] | 0 | 5 | by_cases hb : 1 < b | x y b : Ordinal.{u_1}
hx : x ≠ 0
hy : y ≠ 0
⊢ log b x + log b y ≤ log b (x * y) | case pos
x y b : Ordinal.{u_1}
hx : x ≠ 0
hy : y ≠ 0
hb : 1 < b
⊢ log b x + log b y ≤ log b (x * y)
case neg
x y b : Ordinal.{u_1}
hx : x ≠ 0
hy : y ≠ 0
hb : ¬1 < b
⊢ log b x + log b y ≤ log b (x * y) |
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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 54 | Ordinal.add_log_le_log_mul | [
[
445,
42
],
[
450,
59
]
] | 1 | 5 | · rw [← opow_le_iff_le_log hb (mul_ne_zero hx hy), opow_add]
exact mul_le_mul' (opow_log_le_self b hx) (opow_log_le_self b hy) | case pos
x y b : Ordinal.{u_1}
hx : x ≠ 0
hy : y ≠ 0
hb : 1 < b
⊢ log b x + log b y ≤ log b (x * y)
case neg
x y b : Ordinal.{u_1}
hx : x ≠ 0
hy : y ≠ 0
hb : ¬1 < b
⊢ log b x + log b y ≤ log b (x * y) | case neg
x y b : Ordinal.{u_1}
hx : x ≠ 0
hy : y ≠ 0
hb : ¬1 < b
⊢ log b x + log b y ≤ log b (x * y) |
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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 54 | Ordinal.add_log_le_log_mul | [
[
445,
42
],
[
450,
59
]
] | 2 | 5 | simp only [log_of_not_one_lt_left hb, zero_add, le_refl] | case neg
x y b : Ordinal.{u_1}
hx : x ≠ 0
hy : y ≠ 0
hb : ¬1 < b
⊢ log b x + log b y ≤ log b (x * y) | 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 54 | Ordinal.add_log_le_log_mul | [
[
445,
42
],
[
450,
59
]
] | 3 | 5 | rw [← opow_le_iff_le_log hb (mul_ne_zero hx hy), opow_add] | case pos
x y b : Ordinal.{u_1}
hx : x ≠ 0
hy : y ≠ 0
hb : 1 < b
⊢ log b x + log b y ≤ log b (x * y) | case pos
x y b : Ordinal.{u_1}
hx : x ≠ 0
hy : y ≠ 0
hb : 1 < b
⊢ b ^ log b x * b ^ log b y ≤ x * y |
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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 54 | Ordinal.add_log_le_log_mul | [
[
445,
42
],
[
450,
59
]
] | 4 | 5 | exact mul_le_mul' (opow_log_le_self b hx) (opow_log_le_self b hy) | case pos
x y b : Ordinal.{u_1}
hx : x ≠ 0
hy : y ≠ 0
hb : 1 < b
⊢ b ^ log b x * b ^ log b y ≤ x * y | 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 56 | Ordinal.sup_opow_nat | [
[
465,
99
],
[
471,
34
]
] | 0 | 9 | rcases lt_or_eq_of_le (one_le_iff_pos.2 ho) with (ho₁ | rfl) | o : Ordinal.{u_1}
ho : 0 < o
⊢ (sup fun n => o ^ ↑n) = o ^ ω | case inl
o : Ordinal.{u_1}
ho : 0 < o
ho₁ : 1 < o
⊢ (sup fun n => o ^ ↑n) = o ^ ω
case inr
ho : 0 < 1
⊢ (sup fun n => 1 ^ ↑n) = 1 ^ ω |
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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 56 | Ordinal.sup_opow_nat | [
[
465,
99
],
[
471,
34
]
] | 1 | 9 | · exact (opow_isNormal ho₁).apply_omega | case inl
o : Ordinal.{u_1}
ho : 0 < o
ho₁ : 1 < o
⊢ (sup fun n => o ^ ↑n) = o ^ ω
case inr
ho : 0 < 1
⊢ (sup fun n => 1 ^ ↑n) = 1 ^ ω | case inr
ho : 0 < 1
⊢ (sup fun n => 1 ^ ↑n) = 1 ^ ω |
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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 56 | Ordinal.sup_opow_nat | [
[
465,
99
],
[
471,
34
]
] | 2 | 9 | · rw [one_opow]
refine le_antisymm (sup_le fun n => by rw [one_opow]) ?_
convert le_sup (fun n : ℕ => 1 ^ (n : Ordinal)) 0
rw [Nat.cast_zero, opow_zero] | case inr
ho : 0 < 1
⊢ (sup fun n => 1 ^ ↑n) = 1 ^ ω | 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 56 | Ordinal.sup_opow_nat | [
[
465,
99
],
[
471,
34
]
] | 3 | 9 | exact (opow_isNormal ho₁).apply_omega | case inl
o : Ordinal.{u_1}
ho : 0 < o
ho₁ : 1 < o
⊢ (sup fun n => o ^ ↑n) = o ^ ω | 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 56 | Ordinal.sup_opow_nat | [
[
465,
99
],
[
471,
34
]
] | 4 | 9 | rw [one_opow] | case inr
ho : 0 < 1
⊢ (sup fun n => 1 ^ ↑n) = 1 ^ ω | case inr
ho : 0 < 1
⊢ (sup fun n => 1 ^ ↑n) = 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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 56 | Ordinal.sup_opow_nat | [
[
465,
99
],
[
471,
34
]
] | 5 | 9 | refine le_antisymm (sup_le fun n => by rw [one_opow]) ?_ | case inr
ho : 0 < 1
⊢ (sup fun n => 1 ^ ↑n) = 1 | case inr
ho : 0 < 1
⊢ 1 ≤ sup fun n => 1 ^ ↑n |
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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 56 | Ordinal.sup_opow_nat | [
[
465,
99
],
[
471,
34
]
] | 6 | 9 | convert le_sup (fun n : ℕ => 1 ^ (n : Ordinal)) 0 | case inr
ho : 0 < 1
⊢ 1 ≤ sup fun n => 1 ^ ↑n | case h.e'_3
ho : 0 < 1
⊢ 1 = 1 ^ ↑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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 56 | Ordinal.sup_opow_nat | [
[
465,
99
],
[
471,
34
]
] | 7 | 9 | rw [Nat.cast_zero, opow_zero] | case h.e'_3
ho : 0 < 1
⊢ 1 = 1 ^ ↑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 ⟨fun a b => if a = 0 then 1 - b else limitRecOn b 1 (fun _ IH => IH * a) fun b _ => bsup.{u, u} b⟩",
"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 ≠ 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 ≠ 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 ≠ 0) (h : IsLimit b) :\n a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ 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 ≠ 0) (h : IsLimit c) :\n a < b ^ c ↔ ∃ 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 ≠ 0) : a ^ b ≠ 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 ^ ·)",
"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 ↔ 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 ≤ a ^ c ↔ b ≤ 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 ↔ 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 → 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 ≠ 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} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ 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 ≤ b) : a ^ c ≤ 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 ≤ 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 ≤ 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 ≤ c) : a ^ b ∣ 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 ∣ a ^ c ↔ b ≤ 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 : ¬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 ≤ 1) : ∀ 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 : ∀ 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 : ∀ 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 ≠ 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 ≠ 0) : b ^ log b x ≤ 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 ≠ 0) : b ^ c ≤ x ↔ c ≤ 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 ≠ 0) : x < b ^ c ↔ 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 ≠ 0) (hbo : b ≤ 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 ≤ y) : log b x ≤ 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 ≤ 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 ≠ 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 ≠ 0) (hbo : b ≤ 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 ≠ 0) (u : Ordinal) (hv : v ≠ 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 ≠ 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 ≠ 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 ≠ 0) (hy : y ≠ 0) :\n log b x + log b y ≤ 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 : ℕ) : ∀ n : ℕ, ↑(m ^ n : ℕ) = (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 : ℕ => o ^ (n : Ordinal)) = o ^ ω",
"end": [
471,
34
],
"full_name": "Ordinal.sup_opow_nat",
"kind": "commanddeclaration",
"start": [
465,
1
]
}
] | 56 | Ordinal.sup_opow_nat | [
[
465,
99
],
[
471,
34
]
] | 8 | 9 | rw [one_opow] | ho : 0 < 1
n : ℕ
⊢ 1 ^ ↑n ≤ 1 | no goals |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 0 | Ordinal.principal_iff_principal_swap | [
[
53,
56
],
[
54,
51
]
] | 0 | 1 | constructor <;> exact fun h a b ha hb => h hb ha | op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
o : Ordinal.{u_1}
⊢ Principal op o ↔ Principal (Function.swap op) o | no goals |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 2 | Ordinal.principal_one_iff | [
[
62,
95
],
[
66,
39
]
] | 0 | 6 | refine ⟨fun h => ?_, fun h a b ha hb => ?_⟩ | op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
⊢ Principal op 1 ↔ op 0 0 = 0 | case refine_1
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
h : Principal op 1
⊢ op 0 0 = 0
case refine_2
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
h : op 0 0 = 0
a b : Ordinal.{u_1}
ha : a < 1
hb : b < 1
⊢ op a b < 1 |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 2 | Ordinal.principal_one_iff | [
[
62,
95
],
[
66,
39
]
] | 1 | 6 | · rw [← lt_one_iff_zero]
exact h zero_lt_one zero_lt_one | case refine_1
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
h : Principal op 1
⊢ op 0 0 = 0
case refine_2
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
h : op 0 0 = 0
a b : Ordinal.{u_1}
ha : a < 1
hb : b < 1
⊢ op a b < 1 | case refine_2
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
h : op 0 0 = 0
a b : Ordinal.{u_1}
ha : a < 1
hb : b < 1
⊢ op a b < 1 |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 2 | Ordinal.principal_one_iff | [
[
62,
95
],
[
66,
39
]
] | 2 | 6 | · rwa [lt_one_iff_zero, ha, hb] at * | case refine_2
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
h : op 0 0 = 0
a b : Ordinal.{u_1}
ha : a < 1
hb : b < 1
⊢ op a b < 1 | no goals |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 2 | Ordinal.principal_one_iff | [
[
62,
95
],
[
66,
39
]
] | 3 | 6 | rw [← lt_one_iff_zero] | case refine_1
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
h : Principal op 1
⊢ op 0 0 = 0 | case refine_1
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
h : Principal op 1
⊢ op 0 0 < 1 |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 2 | Ordinal.principal_one_iff | [
[
62,
95
],
[
66,
39
]
] | 4 | 6 | exact h zero_lt_one zero_lt_one | case refine_1
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
h : Principal op 1
⊢ op 0 0 < 1 | no goals |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 2 | Ordinal.principal_one_iff | [
[
62,
95
],
[
66,
39
]
] | 5 | 6 | rwa [lt_one_iff_zero, ha, hb] at * | case refine_2
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
h : op 0 0 = 0
a b : Ordinal.{u_1}
ha : a < 1
hb : b < 1
⊢ op a b < 1 | no goals |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 3 | Ordinal.Principal.iterate_lt | [
[
70,
57
],
[
74,
20
]
] | 0 | 6 | induction' n with n hn | op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
a o : Ordinal.{u_1}
hao : a < o
ho : Principal op o
n : ℕ
⊢ (op a)^[n] a < o | case zero
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
a o : Ordinal.{u_1}
hao : a < o
ho : Principal op o
⊢ (op a)^[0] a < o
case succ
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
a o : Ordinal.{u_1}
hao : a < o
ho : Principal op o
n : ℕ
hn : (op a)^[n] a < o
⊢ (op a)^[n + 1] a < o |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 3 | Ordinal.Principal.iterate_lt | [
[
70,
57
],
[
74,
20
]
] | 1 | 6 | · rwa [Function.iterate_zero] | case zero
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
a o : Ordinal.{u_1}
hao : a < o
ho : Principal op o
⊢ (op a)^[0] a < o
case succ
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
a o : Ordinal.{u_1}
hao : a < o
ho : Principal op o
n : ℕ
hn : (op a)^[n] a < o
⊢ (op a)^[n + 1] a < o | case succ
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
a o : Ordinal.{u_1}
hao : a < o
ho : Principal op o
n : ℕ
hn : (op a)^[n] a < o
⊢ (op a)^[n + 1] a < o |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 3 | Ordinal.Principal.iterate_lt | [
[
70,
57
],
[
74,
20
]
] | 2 | 6 | · rw [Function.iterate_succ']
exact ho hao hn | case succ
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
a o : Ordinal.{u_1}
hao : a < o
ho : Principal op o
n : ℕ
hn : (op a)^[n] a < o
⊢ (op a)^[n + 1] a < o | no goals |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 3 | Ordinal.Principal.iterate_lt | [
[
70,
57
],
[
74,
20
]
] | 3 | 6 | rwa [Function.iterate_zero] | case zero
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
a o : Ordinal.{u_1}
hao : a < o
ho : Principal op o
⊢ (op a)^[0] a < o | no goals |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 3 | Ordinal.Principal.iterate_lt | [
[
70,
57
],
[
74,
20
]
] | 4 | 6 | rw [Function.iterate_succ'] | case succ
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
a o : Ordinal.{u_1}
hao : a < o
ho : Principal op o
n : ℕ
hn : (op a)^[n] a < o
⊢ (op a)^[n + 1] a < o | case succ
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
a o : Ordinal.{u_1}
hao : a < o
ho : Principal op o
n : ℕ
hn : (op a)^[n] a < o
⊢ (op a ∘ (op a)^[n]) a < o |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 3 | Ordinal.Principal.iterate_lt | [
[
70,
57
],
[
74,
20
]
] | 5 | 6 | exact ho hao hn | case succ
op : Ordinal.{u_1} → Ordinal.{u_1} → Ordinal.{u_1}
a o : Ordinal.{u_1}
hao : a < o
ho : Principal op o
n : ℕ
hn : (op a)^[n] a < o
⊢ (op a ∘ (op a)^[n]) a < o | no goals |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 4 | Ordinal.op_eq_self_of_principal | [
[
78,
83
],
[
81,
37
]
] | 0 | 3 | refine le_antisymm ?_ (H.self_le _) | op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u}
a o : Ordinal.{u}
hao : a < o
H : IsNormal (op a)
ho : Principal op o
ho' : o.IsLimit
⊢ op a o = o | op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u}
a o : Ordinal.{u}
hao : a < o
H : IsNormal (op a)
ho : Principal op o
ho' : o.IsLimit
⊢ op a o ≤ o |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 4 | Ordinal.op_eq_self_of_principal | [
[
78,
83
],
[
81,
37
]
] | 1 | 3 | rw [← IsNormal.bsup_eq.{u, u} H ho', bsup_le_iff] | op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u}
a o : Ordinal.{u}
hao : a < o
H : IsNormal (op a)
ho : Principal op o
ho' : o.IsLimit
⊢ op a o ≤ o | op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u}
a o : Ordinal.{u}
hao : a < o
H : IsNormal (op a)
ho : Principal op o
ho' : o.IsLimit
⊢ ∀ i < o, op a i ≤ o |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 4 | Ordinal.op_eq_self_of_principal | [
[
78,
83
],
[
81,
37
]
] | 2 | 3 | exact fun b hbo => (ho hao hbo).le | op : Ordinal.{u} → Ordinal.{u} → Ordinal.{u}
a o : Ordinal.{u}
hao : a < o
H : IsNormal (op a)
ho : Principal op o
ho' : o.IsLimit
⊢ ∀ i < o, op a i ≤ o | no goals |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 9 | Ordinal.principal_add_of_le_one | [
[
125,
85
],
[
128,
28
]
] | 0 | 5 | rcases le_one_iff.1 ho with (rfl | rfl) | o : Ordinal.{u_1}
ho : o ≤ 1
⊢ Principal (fun x x_1 => x + x_1) o | case inl
ho : 0 ≤ 1
⊢ Principal (fun x x_1 => x + x_1) 0
case inr
ho : 1 ≤ 1
⊢ Principal (fun x x_1 => x + x_1) 1 |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 9 | Ordinal.principal_add_of_le_one | [
[
125,
85
],
[
128,
28
]
] | 1 | 5 | · exact principal_zero | case inl
ho : 0 ≤ 1
⊢ Principal (fun x x_1 => x + x_1) 0
case inr
ho : 1 ≤ 1
⊢ Principal (fun x x_1 => x + x_1) 1 | case inr
ho : 1 ≤ 1
⊢ Principal (fun x x_1 => x + x_1) 1 |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 9 | Ordinal.principal_add_of_le_one | [
[
125,
85
],
[
128,
28
]
] | 2 | 5 | · exact principal_add_one | case inr
ho : 1 ≤ 1
⊢ Principal (fun x x_1 => x + x_1) 1 | no goals |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 9 | Ordinal.principal_add_of_le_one | [
[
125,
85
],
[
128,
28
]
] | 3 | 5 | exact principal_zero | case inl
ho : 0 ≤ 1
⊢ Principal (fun x x_1 => x + x_1) 0 | no goals |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 9 | Ordinal.principal_add_of_le_one | [
[
125,
85
],
[
128,
28
]
] | 4 | 5 | exact principal_add_one | case inr
ho : 1 ≤ 1
⊢ Principal (fun x x_1 => x + x_1) 1 | no goals |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 10 | Ordinal.principal_add_isLimit | [
[
132,
18
],
[
140,
71
]
] | 0 | 12 | refine ⟨fun ho₀ => ?_, fun a hao => ?_⟩ | o : Ordinal.{u_1}
ho₁ : 1 < o
ho : Principal (fun x x_1 => x + x_1) o
⊢ o.IsLimit | case refine_1
o : Ordinal.{u_1}
ho₁ : 1 < o
ho : Principal (fun x x_1 => x + x_1) o
ho₀ : o = 0
⊢ False
case refine_2
o : Ordinal.{u_1}
ho₁ : 1 < o
ho : Principal (fun x x_1 => x + x_1) o
a : Ordinal.{u_1}
hao : a < o
⊢ succ a < o |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 10 | Ordinal.principal_add_isLimit | [
[
132,
18
],
[
140,
71
]
] | 1 | 12 | · rw [ho₀] at ho₁
exact not_lt_of_gt zero_lt_one ho₁ | case refine_1
o : Ordinal.{u_1}
ho₁ : 1 < o
ho : Principal (fun x x_1 => x + x_1) o
ho₀ : o = 0
⊢ False
case refine_2
o : Ordinal.{u_1}
ho₁ : 1 < o
ho : Principal (fun x x_1 => x + x_1) o
a : Ordinal.{u_1}
hao : a < o
⊢ succ a < o | case refine_2
o : Ordinal.{u_1}
ho₁ : 1 < o
ho : Principal (fun x x_1 => x + x_1) o
a : Ordinal.{u_1}
hao : a < o
⊢ succ a < o |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 10 | Ordinal.principal_add_isLimit | [
[
132,
18
],
[
140,
71
]
] | 2 | 12 | · rcases eq_or_ne a 0 with ha | ha
· rw [ha, succ_zero]
exact ho₁
· refine lt_of_le_of_lt ?_ (ho hao hao)
rwa [← add_one_eq_succ, add_le_add_iff_left, one_le_iff_ne_zero] | case refine_2
o : Ordinal.{u_1}
ho₁ : 1 < o
ho : Principal (fun x x_1 => x + x_1) o
a : Ordinal.{u_1}
hao : a < o
⊢ succ a < o | no goals |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 10 | Ordinal.principal_add_isLimit | [
[
132,
18
],
[
140,
71
]
] | 3 | 12 | rw [ho₀] at ho₁ | case refine_1
o : Ordinal.{u_1}
ho₁ : 1 < o
ho : Principal (fun x x_1 => x + x_1) o
ho₀ : o = 0
⊢ False | case refine_1
o : Ordinal.{u_1}
ho₁ : 1 < 0
ho : Principal (fun x x_1 => x + x_1) o
ho₀ : o = 0
⊢ False |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 10 | Ordinal.principal_add_isLimit | [
[
132,
18
],
[
140,
71
]
] | 4 | 12 | exact not_lt_of_gt zero_lt_one ho₁ | case refine_1
o : Ordinal.{u_1}
ho₁ : 1 < 0
ho : Principal (fun x x_1 => x + x_1) o
ho₀ : o = 0
⊢ False | no goals |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 10 | Ordinal.principal_add_isLimit | [
[
132,
18
],
[
140,
71
]
] | 5 | 12 | rcases eq_or_ne a 0 with ha | ha | case refine_2
o : Ordinal.{u_1}
ho₁ : 1 < o
ho : Principal (fun x x_1 => x + x_1) o
a : Ordinal.{u_1}
hao : a < o
⊢ succ a < o | case refine_2.inl
o : Ordinal.{u_1}
ho₁ : 1 < o
ho : Principal (fun x x_1 => x + x_1) o
a : Ordinal.{u_1}
hao : a < o
ha : a = 0
⊢ succ a < o
case refine_2.inr
o : Ordinal.{u_1}
ho₁ : 1 < o
ho : Principal (fun x x_1 => x + x_1) o
a : Ordinal.{u_1}
hao : a < o
ha : a ≠ 0
⊢ succ a < o |
Mathlib/SetTheory/Ordinal/Principal.lean | [
[
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib/SetTheory/Ordinal/FixedPoint.lean"
],
[
"Init",
".lake/packages/lean4/src/lean/Init.lean"
]
] | [
{
"code": "def Principal (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) : Prop :=\n ∀ ⦃a b⦄, a < o → b < o → op a b < o",
"end": [
49,
38
],
"full_name": "Ordinal.Principal",
"kind": "commanddeclaration",
"start": [
43,
1
]
},
{
"code": "theorem principal_iff_principal_swap {op : Ordinal → Ordinal → Ordinal} {o : Ordinal} :\n Principal op o ↔ Principal (Function.swap op) o",
"end": [
54,
51
],
"full_name": "Ordinal.principal_iff_principal_swap",
"kind": "commanddeclaration",
"start": [
52,
1
]
},
{
"code": "theorem principal_zero {op : Ordinal → Ordinal → Ordinal} : Principal op 0",
"end": [
58,
33
],
"full_name": "Ordinal.principal_zero",
"kind": "commanddeclaration",
"start": [
57,
1
]
},
{
"code": "@[simp]\ntheorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0",
"end": [
66,
39
],
"full_name": "Ordinal.principal_one_iff",
"kind": "commanddeclaration",
"start": [
61,
1
]
},
{
"code": "theorem Principal.iterate_lt {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) (n : ℕ) : (op a)^[n] a < o",
"end": [
74,
20
],
"full_name": "Ordinal.Principal.iterate_lt",
"kind": "commanddeclaration",
"start": [
69,
1
]
},
{
"code": "theorem op_eq_self_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal.{u}} (hao : a < o)\n (H : IsNormal (op a)) (ho : Principal op o) (ho' : IsLimit o) : op a o = o",
"end": [
81,
37
],
"full_name": "Ordinal.op_eq_self_of_principal",
"kind": "commanddeclaration",
"start": [
77,
1
]
},
{
"code": "theorem nfp_le_of_principal {op : Ordinal → Ordinal → Ordinal} {a o : Ordinal} (hao : a < o)\n (ho : Principal op o) : nfp (op a) a ≤ o",
"end": [
86,
43
],
"full_name": "Ordinal.nfp_le_of_principal",
"kind": "commanddeclaration",
"start": [
84,
1
]
},
{
"code": "theorem principal_nfp_blsub₂ (op : Ordinal → Ordinal → Ordinal) (o : Ordinal) :\n Principal op (nfp (fun o' => blsub₂.{u, u, u} o' o' (@fun a _ b _ => op a b)) o)",
"end": [
110,
68
],
"full_name": "Ordinal.principal_nfp_blsub₂",
"kind": "commanddeclaration",
"start": [
96,
1
]
},
{
"code": "theorem unbounded_principal (op : Ordinal → Ordinal → Ordinal) :\n Set.Unbounded (· < ·) { o | Principal op o }",
"end": [
115,
54
],
"full_name": "Ordinal.unbounded_principal",
"kind": "commanddeclaration",
"start": [
113,
1
]
},
{
"code": "theorem principal_add_one : Principal (· + ·) 1",
"end": [
122,
36
],
"full_name": "Ordinal.principal_add_one",
"kind": "commanddeclaration",
"start": [
121,
1
]
},
{
"code": "theorem principal_add_of_le_one {o : Ordinal} (ho : o ≤ 1) : Principal (· + ·) o",
"end": [
128,
28
],
"full_name": "Ordinal.principal_add_of_le_one",
"kind": "commanddeclaration",
"start": [
125,
1
]
},
{
"code": "theorem principal_add_isLimit {o : Ordinal} (ho₁ : 1 < o) (ho : Principal (· + ·) o) :\n o.IsLimit",
"end": [
140,
71
],
"full_name": "Ordinal.principal_add_isLimit",
"kind": "commanddeclaration",
"start": [
131,
1
]
},
{
"code": "theorem principal_add_iff_add_left_eq_self {o : Ordinal} :\n Principal (· + ·) o ↔ ∀ a < o, a + o = o",
"end": [
153,
42
],
"full_name": "Ordinal.principal_add_iff_add_left_eq_self",
"kind": "commanddeclaration",
"start": [
143,
1
]
},
{
"code": "theorem exists_lt_add_of_not_principal_add {a} (ha : ¬Principal (· + ·) a) :\n ∃ b c, b < a ∧ c < a ∧ b + c = a",
"end": [
164,
20
],
"full_name": "Ordinal.exists_lt_add_of_not_principal_add",
"kind": "commanddeclaration",
"start": [
156,
1
]
},
{
"code": "theorem principal_add_iff_add_lt_ne_self {a} :\n Principal (· + ·) a ↔ ∀ ⦃b c⦄, b < a → c < a → b + c ≠ a",
"end": [
172,
28
],
"full_name": "Ordinal.principal_add_iff_add_lt_ne_self",
"kind": "commanddeclaration",
"start": [
167,
1
]
},
{
"code": "theorem add_omega {a : Ordinal} (h : a < omega) : a + omega = omega",
"end": [
179,
68
],
"full_name": "Ordinal.add_omega",
"kind": "commanddeclaration",
"start": [
175,
1
]
},
{
"code": "theorem principal_add_omega : Principal (· + ·) omega",
"end": [
183,
58
],
"full_name": "Ordinal.principal_add_omega",
"kind": "commanddeclaration",
"start": [
182,
1
]
},
{
"code": "theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b)",
"end": [
201,
72
],
"full_name": "Ordinal.add_omega_opow",
"kind": "commanddeclaration",
"start": [
186,
1
]
},
{
"code": "theorem principal_add_omega_opow (o : Ordinal) : Principal (· + ·) (omega^o)",
"end": [
205,
63
],
"full_name": "Ordinal.principal_add_omega_opow",
"kind": "commanddeclaration",
"start": [
204,
1
]
},
{
"code": "theorem principal_add_iff_zero_or_omega_opow {o : Ordinal} :\n Principal (· + ·) o ↔ o = 0 ∨ ∃ a : Ordinal, o = (omega^a)",
"end": [
230,
64
],
"full_name": "Ordinal.principal_add_iff_zero_or_omega_opow",
"kind": "commanddeclaration",
"start": [
208,
1
]
},
{
"code": "theorem opow_principal_add_of_principal_add {a} (ha : Principal (· + ·) a) (b : Ordinal) :\n Principal (· + ·) (a^b)",
"end": [
241,
37
],
"full_name": "Ordinal.opow_principal_add_of_principal_add",
"kind": "commanddeclaration",
"start": [
233,
1
]
},
{
"code": "theorem add_absorp {a b c : Ordinal} (h₁ : a < (omega^b)) (h₂ : (omega^b) ≤ c) : a + c = c",
"end": [
245,
73
],
"full_name": "Ordinal.add_absorp",
"kind": "commanddeclaration",
"start": [
244,
1
]
},
{
"code": "theorem mul_principal_add_is_principal_add (a : Ordinal.{u}) {b : Ordinal.{u}} (hb₁ : b ≠ 1)\n (hb : Principal (· + ·) b) : Principal (· + ·) (a * b)",
"end": [
263,
36
],
"full_name": "Ordinal.mul_principal_add_is_principal_add",
"kind": "commanddeclaration",
"start": [
248,
1
]
},
{
"code": "theorem principal_mul_one : Principal (· * ·) 1",
"end": [
271,
19
],
"full_name": "Ordinal.principal_mul_one",
"kind": "commanddeclaration",
"start": [
269,
1
]
},
{
"code": "theorem principal_mul_two : Principal (· * ·) 2",
"end": [
279,
25
],
"full_name": "Ordinal.principal_mul_two",
"kind": "commanddeclaration",
"start": [
274,
1
]
},
{
"code": "theorem principal_mul_of_le_two {o : Ordinal} (ho : o ≤ 2) : Principal (· * ·) o",
"end": [
290,
28
],
"full_name": "Ordinal.principal_mul_of_le_two",
"kind": "commanddeclaration",
"start": [
282,
1
]
},
{
"code": "theorem principal_add_of_principal_mul {o : Ordinal} (ho : Principal (· * ·) o) (ho₂ : o ≠ 2) :\n Principal (· + ·) o",
"end": [
302,
58
],
"full_name": "Ordinal.principal_add_of_principal_mul",
"kind": "commanddeclaration",
"start": [
293,
1
]
},
{
"code": "theorem principal_mul_isLimit {o : Ordinal.{u}} (ho₂ : 2 < o) (ho : Principal (· * ·) o) :\n o.IsLimit",
"end": [
308,
55
],
"full_name": "Ordinal.principal_mul_isLimit",
"kind": "commanddeclaration",
"start": [
305,
1
]
},
{
"code": "theorem principal_mul_iff_mul_left_eq {o : Ordinal} :\n Principal (· * ·) o ↔ ∀ a, 0 < a → a < o → a * o = o",
"end": [
325,
43
],
"full_name": "Ordinal.principal_mul_iff_mul_left_eq",
"kind": "commanddeclaration",
"start": [
311,
1
]
},
{
"code": "theorem principal_mul_omega : Principal (· * ·) omega",
"end": [
332,
23
],
"full_name": "Ordinal.principal_mul_omega",
"kind": "commanddeclaration",
"start": [
328,
1
]
},
{
"code": "theorem mul_omega {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : a * omega = omega",
"end": [
336,
62
],
"full_name": "Ordinal.mul_omega",
"kind": "commanddeclaration",
"start": [
335,
1
]
},
{
"code": "theorem mul_lt_omega_opow {a b c : Ordinal} (c0 : 0 < c) (ha : a < (omega^c)) (hb : b < omega) :\n a * b < (omega^c)",
"end": [
351,
19
],
"full_name": "Ordinal.mul_lt_omega_opow",
"kind": "commanddeclaration",
"start": [
339,
1
]
},
{
"code": "theorem mul_omega_opow_opow {a b : Ordinal} (a0 : 0 < a) (h : a < (omega^omega^b)) :\n a * (omega^omega^b) = (omega^omega^b)",
"end": [
364,
37
],
"full_name": "Ordinal.mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
354,
1
]
},
{
"code": "theorem principal_mul_omega_opow_opow (o : Ordinal) : Principal (· * ·) (omega^omega^o)",
"end": [
368,
63
],
"full_name": "Ordinal.principal_mul_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
367,
1
]
},
{
"code": "theorem principal_add_of_principal_mul_opow {o b : Ordinal} (hb : 1 < b)\n (ho : Principal (· * ·) (b^o)) : Principal (· + ·) o",
"end": [
374,
71
],
"full_name": "Ordinal.principal_add_of_principal_mul_opow",
"kind": "commanddeclaration",
"start": [
371,
1
]
},
{
"code": "theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :\n Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a)",
"end": [
393,
44
],
"full_name": "Ordinal.principal_mul_iff_le_two_or_omega_opow_opow",
"kind": "commanddeclaration",
"start": [
377,
1
]
},
{
"code": "theorem mul_omega_dvd {a : Ordinal} (a0 : 0 < a) (ha : a < omega) : ∀ {b}, omega ∣ b → a * b = b",
"end": [
397,
56
],
"full_name": "Ordinal.mul_omega_dvd",
"kind": "commanddeclaration",
"start": [
396,
1
]
},
{
"code": "theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)\n (hb₂ : 2 < b) : a * b = (b^succ (log b a))",
"end": [
414,
54
],
"full_name": "Ordinal.mul_eq_opow_log_succ",
"kind": "commanddeclaration",
"start": [
400,
1
]
},
{
"code": "theorem principal_opow_omega : Principal (·^·) omega",
"end": [
424,
23
],
"full_name": "Ordinal.principal_opow_omega",
"kind": "commanddeclaration",
"start": [
420,
1
]
},
{
"code": "theorem opow_omega {a : Ordinal} (a1 : 1 < a) (h : a < omega) : (a^omega) = omega",
"end": [
431,
25
],
"full_name": "Ordinal.opow_omega",
"kind": "commanddeclaration",
"start": [
427,
1
]
}
] | 10 | Ordinal.principal_add_isLimit | [
[
132,
18
],
[
140,
71
]
] | 6 | 12 | · rw [ha, succ_zero]
exact ho₁ | case refine_2.inl
o : Ordinal.{u_1}
ho₁ : 1 < o
ho : Principal (fun x x_1 => x + x_1) o
a : Ordinal.{u_1}
hao : a < o
ha : a = 0
⊢ succ a < o
case refine_2.inr
o : Ordinal.{u_1}
ho₁ : 1 < o
ho : Principal (fun x x_1 => x + x_1) o
a : Ordinal.{u_1}
hao : a < o
ha : a ≠ 0
⊢ succ a < o | case refine_2.inr
o : Ordinal.{u_1}
ho₁ : 1 < o
ho : Principal (fun x x_1 => x + x_1) o
a : Ordinal.{u_1}
hao : a < o
ha : a ≠ 0
⊢ succ a < o |
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