module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.SetTheory.ZFC.Basic | {
"line": 495,
"column": 15
} | {
"line": 495,
"column": 17
} | {
"line": 495,
"column": 18
} | [
{
"pp": "x y : ZFSet.{u}\n⊢ y ∈ ⋃₀ {x} ↔ y ∈ x",
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"usedConstants": [
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Mathlib.SetTheory.ZFC.Basic | {
"line": 499,
"column": 15
} | {
"line": 499,
"column": 17
} | {
"line": 499,
"column": 18
} | [
{
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Mathlib.SetTheory.ZFC.Basic | {
"line": 502,
"column": 95
} | {
"line": 502,
"column": 97
} | {
"line": 503,
"column": 2
} | [
{
"pp": "x : ZFSet.{u}\n⊢ ↑(⋃₀ x) = ⋃₀ (SetLike.coe '' ↑x)",
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... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 507,
"column": 96
} | {
"line": 507,
"column": 98
} | {
"line": 508,
"column": 2
} | [
{
"pp": "x : ZFSet.{u}\nh : x.Nonempty\n⊢ ↑(⋂₀ x) = ⋂₀ (SetLike.coe '' ↑x)",
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Mathlib.SetTheory.ZFC.Basic | {
"line": 511,
"column": 92
} | {
"line": 511,
"column": 94
} | {
"line": 512,
"column": 2
} | [
{
"pp": "x y : ZFSet.{u_1}\nH : {x} = {y}\n⊢ x = y",
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... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 194,
"column": 81
} | {
"line": 194,
"column": 83
} | {
"line": 195,
"column": 2
} | [
{
"pp": "f : Ordinal.{u} → Ordinal.{u}\nhf : IsNormal f\nhp : 0 < f 0\n⊢ IsNormal fun x ↦ veblenWith f x 0",
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Mathlib.SetTheory.ZFC.Basic | {
"line": 545,
"column": 55
} | {
"line": 545,
"column": 57
} | {
"line": 545,
"column": 58
} | [
{
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Mathlib.SetTheory.ZFC.Basic | {
"line": 546,
"column": 55
} | {
"line": 546,
"column": 57
} | {
"line": 546,
"column": 58
} | [
{
"pp": "x y z : ZFSet.{u}\n⊢ z ∈ x ∩ y ↔ z ∈ x ∧ z ∈ y",
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Mathlib.SetTheory.Ordinal.Veblen | {
"line": 214,
"column": 87
} | {
"line": 214,
"column": 89
} | {
"line": 215,
"column": 2
} | [
{
"pp": "f : Ordinal.{u} → Ordinal.{u}\no₁ o₂ a : Ordinal.{u}\nhf : IsNormal f\nh : o₂ ≤ o₁\n⊢ veblenWith f o₁ (veblenWith f o₂ a) = veblenWith f o₂ a ↔ veblenWith f o₁ a = a",
"ppTerm": "?m.9",
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"usedConstants": [
"_private.Mathlib.SetTheory.Ordinal.Veblen.0.Ordinal.veblenWit... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 547,
"column": 55
} | {
"line": 547,
"column": 57
} | {
"line": 547,
"column": 58
} | [
{
"pp": "x y z : ZFSet.{u}\n⊢ z ∈ x \\ y ↔ z ∈ x ∧ z ∉ y",
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Mathlib.SetTheory.ZFC.Basic | {
"line": 550,
"column": 72
} | {
"line": 550,
"column": 74
} | {
"line": 550,
"column": 75
} | [
{
"pp": "x y : ZFSet.{u}\n⊢ ↑(x ∪ y) = ↑x ∪ ↑y",
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"ZFSet.in... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 553,
"column": 72
} | {
"line": 553,
"column": 74
} | {
"line": 553,
"column": 75
} | [
{
"pp": "x y : ZFSet.{u}\n⊢ ↑(x ∩ y) = ↑x ∩ ↑y",
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"Inter.inter",
"iff_s... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 218,
"column": 87
} | {
"line": 218,
"column": 89
} | {
"line": 219,
"column": 2
} | [
{
"pp": "f : Ordinal.{u} → Ordinal.{u}\no₁ o₂ a : Ordinal.{u}\nhf : IsNormal f\nh : o₂ ≤ o₁\n⊢ veblenWith f o₂ a < veblenWith f o₁ (veblenWith f o₂ a) ↔ a < veblenWith f o₁ a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
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"Ordinal.partialOrder"... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 556,
"column": 72
} | {
"line": 556,
"column": 74
} | {
"line": 556,
"column": 75
} | [
{
"pp": "x y : ZFSet.{u}\n⊢ ↑(x \\ y) = ↑x \\ ↑y",
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... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 222,
"column": 93
} | {
"line": 222,
"column": 95
} | {
"line": 223,
"column": 2
} | [
{
"pp": "f : Ordinal.{u} → Ordinal.{u}\no a : Ordinal.{u}\nhf : IsNormal f\n⊢ veblenWith f o (f a) = f a ↔ veblenWith f o a = a",
"ppTerm": "?m.7",
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"usedConstants": [
"zero_le",
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"congrArg",
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"AddMonoid.toAdd... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 558,
"column": 67
} | {
"line": 558,
"column": 69
} | {
"line": 558,
"column": 70
} | [
{
"pp": "x y : ZFSet.{u}\nhxy : x ⊆ y\n⊢ x ∩ y = x",
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"usedConstants": [
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"a... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 225,
"column": 93
} | {
"line": 225,
"column": 95
} | {
"line": 226,
"column": 2
} | [
{
"pp": "f : Ordinal.{u} → Ordinal.{u}\no a : Ordinal.{u}\nhf : IsNormal f\n⊢ f a < veblenWith f o (f a) ↔ a < veblenWith f o a",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"zero_le",
"Preorder.toLT",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 559,
"column": 68
} | {
"line": 559,
"column": 70
} | {
"line": 559,
"column": 71
} | [
{
"pp": "x y : ZFSet.{u}\nhyx : y ⊆ x\n⊢ x ∩ y = y",
"ppTerm": "?m.6",
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"congrFun'",
"Z... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 565,
"column": 14
} | {
"line": 565,
"column": 16
} | {
"line": 565,
"column": 17
} | [
{
"pp": "x✝ y z x : ZFSet.{u}\n⊢ Function.LeftInverse (fun s ↦ ⟨ZFSet.sep (fun x_1 ↦ x_1 ∈ ↑s) x, ⋯⟩) fun y ↦ ⟨↑↑y, ⋯⟩",
"ppTerm": "?m.35",
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... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 566,
"column": 15
} | {
"line": 566,
"column": 17
} | {
"line": 566,
"column": 18
} | [
{
"pp": "x✝ y z x : ZFSet.{u}\n⊢ Function.RightInverse (fun s ↦ ⟨ZFSet.sep (fun x_1 ↦ x_1 ∈ ↑s) x, ⋯⟩) fun y ↦ ⟨↑↑y, ⋯⟩",
"ppTerm": "?m.36",
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... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 568,
"column": 58
} | {
"line": 568,
"column": 60
} | {
"line": 569,
"column": 2
} | [
{
"pp": "x y : ZFSet.{u_1}\n⊢ insert x y = {x} ∪ y",
"ppTerm": "?m.11",
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"usedConstants": [
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"ZFSet.instUnion... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 619,
"column": 52
} | {
"line": 619,
"column": 54
} | {
"line": 620,
"column": 4
} | [
{
"pp": "f : ZFSet.{u} → ZFSet.{u}\ninst✝ : Definable₁ f\nx y : ZFSet.{u}\nx✝² x✝¹ : PSet.{u}\nα✝ : Type u\nA✝ : α✝ → PSet.{u}\nx✝ : ⟦x✝¹⟧ ∈ ⟦PSet.mk α✝ A✝⟧\na : (PSet.mk α✝ A✝).Type\nya : x✝¹.Equiv ((PSet.mk α✝ A✝).Func a)\n⊢ f ⟦x✝¹⟧ ∈ image f ⟦PSet.mk α✝ A✝⟧",
"ppTerm": "?m.28",
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... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 632,
"column": 40
} | {
"line": 632,
"column": 42
} | {
"line": 632,
"column": 43
} | [
{
"pp": "f : ZFSet.{u_1} → ZFSet.{u_1}\ninst✝ : Definable₁ f\nx : ZFSet.{u_1}\n⊢ ↑(image f x) = f '' ↑x",
"ppTerm": "?m.5",
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... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 645,
"column": 34
} | {
"line": 645,
"column": 36
} | {
"line": 646,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : Small.{u, u_1} α\nf : α → ZFSet.{u}\nx : ZFSet.{u}\ny : PSet.{u}\n⊢ ⟦y⟧ ∈ range f ↔ ∃ i, f i = ⟦y⟧",
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"usedConstants": [
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... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 654,
"column": 74
} | {
"line": 654,
"column": 76
} | {
"line": 654,
"column": 77
} | [
{
"pp": "α : Type u_1\ninst✝ : Small.{u, u_1} α\nf : α → ZFSet.{u}\n⊢ ↑(range f) = Set.range f",
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"E... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 656,
"column": 70
} | {
"line": 656,
"column": 72
} | {
"line": 656,
"column": 73
} | [
{
"pp": "α : Type u_1\ninst✝ : Small.{u, u_1} α\nf : α → ZFSet.{u}\na : α\n⊢ f a ∈ range f",
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Mathlib.SetTheory.ZFC.Basic | {
"line": 665,
"column": 88
} | {
"line": 665,
"column": 90
} | {
"line": 666,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : Small.{u, u_1} α\nf : α → ZFSet.{u}\nx : ZFSet.{u}\n⊢ x ∈ ⋃ (i : α), f i ↔ ∃ i, x ∈ f i",
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"usedConstants": [
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"ZFSet",
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"Membership.m... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 669,
"column": 79
} | {
"line": 669,
"column": 81
} | {
"line": 670,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : Small.{u, u_1} α\nf : α → ZFSet.{u}\n⊢ ↑(⋃ (i : α), f i) = ⋃ i, ↑(f i)",
"ppTerm": "?m.10",
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... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 673,
"column": 70
} | {
"line": 673,
"column": 72
} | {
"line": 674,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : Small.{u, u_1} α\nf : α → ZFSet.{u}\ni : α\n⊢ f i ⊆ ⋃ (i : α), f i",
"ppTerm": "?m.6",
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"usedConstants": [
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"ZFSet",
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"Exists",
"id",
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... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 684,
"column": 77
} | {
"line": 684,
"column": 79
} | {
"line": 684,
"column": 80
} | [
{
"pp": "x y : ZFSet.{u}\n⊢ ↑(x.pair y) = {{x}, {x, y}}",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
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"Set.instInsert",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 692,
"column": 66
} | {
"line": 692,
"column": 68
} | {
"line": 693,
"column": 2
} | [
{
"pp": "p : ZFSet.{u} → ZFSet.{u} → Prop\nx y z : ZFSet.{u}\n⊢ z ∈ pairSep p x y ↔ ∃ a ∈ x, ∃ b ∈ y, z = a.pair b ∧ p a b",
"ppTerm": "?m.16",
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"usedConstants": [
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... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 233,
"column": 42
} | {
"line": 233,
"column": 44
} | {
"line": 234,
"column": 2
} | [
{
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"ppTerm": "?m.35",
"... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Rank | {
"line": 40,
"column": 32
} | {
"line": 40,
"column": 34
} | {
"line": 41,
"column": 4
} | [
{
"pp": "α✝¹ : Type u_1\nA✝¹ : α✝¹ → PSet.{u_1}\nα✝ : Type u_1\nA✝ : α✝ → PSet.{u_1}\nαβ : ∀ (a : α✝¹), ∃ b, (A✝¹ a).Equiv (A✝ b)\nβα : ∀ (b : α✝), ∃ a, (A✝¹ a).Equiv (A✝ b)\n⊢ (mk α✝¹ A✝¹).rank = (mk α✝ A✝).rank",
"ppTerm": "?m.17",
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"usedConstants": [
"Set.ext",
"Eq.mp... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Rank | {
"line": 52,
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} | {
"line": 52,
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} | {
"line": 53,
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} | [
{
"pp": "α✝ : Type u_1\nA✝ : α✝ → PSet.{u_1}\nx✝ : PSet.{u_1}\nw✝ : (mk α✝ A✝).Type\nh : x✝.Equiv ((mk α✝ A✝).Func w✝)\n⊢ x✝.rank < (mk α✝ A✝).rank",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Order.succ",
"Ordinal.partialOrder",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Rank | {
"line": 57,
"column": 14
} | {
"line": 57,
"column": 16
} | {
"line": 58,
"column": 4
} | [
{
"pp": "o : Ordinal.{u_1}\nα✝ : Type u_1\nA : α✝ → PSet.{u_1}\n⊢ (mk α✝ A).rank ≤ o ↔ ∀ ⦃y : PSet.{u_1}⦄, y ∈ mk α✝ A → y.rank < o",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Order.succ",
"PSet.instMembership",
"Ordinal.partial... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 246,
"column": 93
} | {
"line": 246,
"column": 95
} | {
"line": 247,
"column": 2
} | [
{
"pp": "f : Ordinal.{u} → Ordinal.{u}\no₁ o₂ a b : Ordinal.{u}\nhf : IsNormal f\n⊢ veblenWith f o₁ a < veblenWith f o₂ b ↔\n o₁ = o₂ ∧ a < b ∨ o₁ < o₂ ∧ a < veblenWith f o₂ b ∨ o₂ < o₁ ∧ veblenWith f o₁ a < b",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Ordering.gt",
"E... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 699,
"column": 37
} | {
"line": 699,
"column": 39
} | {
"line": 699,
"column": 40
} | [
{
"pp": "x x' y y' : ZFSet.{u_1}\nH : ∀ (z : ZFSet.{u_1}), z = {x} ∨ z = {x, y} ↔ z = {x'} ∨ z = {x', y'}\n⊢ x = x' ∧ ?m.27",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"congrArg",
"ZFSet",
"Eq.mp",
"Insert.insert",
"_private.Mathlib.SetTheory.ZFC.Basic.0.Z... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Rank | {
"line": 62,
"column": 83
} | {
"line": 62,
"column": 85
} | {
"line": 63,
"column": 2
} | [
{
"pp": "o : Ordinal.{u_1}\nx : PSet.{u_1}\n⊢ o < x.rank ↔ ∃ y ∈ x, o ≤ y.rank",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"PSet.rank_le_iff",
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
"Mathlib.Tactic.Push.not_and_eq",
"Ordinal.instLinearOrder",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Rank | {
"line": 71,
"column": 35
} | {
"line": 71,
"column": 37
} | {
"line": 71,
"column": 38
} | [
{
"pp": "⊢ ∅.rank = 0",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Ordinal.instLinearOrder",
"Lattice.toSemilatticeSup",
"Order.succ",
"Order.succ_eq_add_one",
"Ordinal.partialOrder",
"congrArg",
"iSup",
"Ordinal.instOrderBot",
"Orde... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 256,
"column": 93
} | {
"line": 256,
"column": 95
} | {
"line": 257,
"column": 2
} | [
{
"pp": "f : Ordinal.{u} → Ordinal.{u}\no₁ o₂ a b : Ordinal.{u}\nhf : IsNormal f\n⊢ veblenWith f o₁ a ≤ veblenWith f o₂ b ↔\n o₁ = o₂ ∧ a ≤ b ∨ o₁ < o₂ ∧ a ≤ veblenWith f o₂ b ∨ o₂ < o₁ ∧ veblenWith f o₁ a ≤ b",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Ordering.gt",
"_... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 427,
"column": 29
} | {
"line": 427,
"column": 31
} | {
"line": 428,
"column": 4
} | [
{
"pp": "e : ONote\nn : ℕ+\na o : ONote\nh₁ : (e.oadd n a).NF\nh₂ : o.NF\n⊢ (e.oadd n a + o).repr = (e.oadd n a).repr + o.repr",
"ppTerm": "?m.29",
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"usedConstants": [
"PNat.val",
"Iff.mpr",
"Ordering.gt",
"ONote.NF",
"Eq.mpr",
"lt_of_le_of_lt",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 266,
"column": 93
} | {
"line": 266,
"column": 95
} | {
"line": 267,
"column": 2
} | [
{
"pp": "f : Ordinal.{u} → Ordinal.{u}\no₁ o₂ a b : Ordinal.{u}\nhf : IsNormal f\n⊢ veblenWith f o₁ a = veblenWith f o₂ b ↔\n o₁ = o₂ ∧ a = b ∨ o₁ < o₂ ∧ a = veblenWith f o₂ b ∨ o₂ < o₁ ∧ veblenWith f o₁ a = b",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Ordering.gt",
"E... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 700,
"column": 30
} | {
"line": 700,
"column": 32
} | {
"line": 701,
"column": 4
} | [
{
"pp": "x y y' : ZFSet.{u_1}\nH : ∀ (z : ZFSet.{u_1}), z = {x} ∨ z = {x, y} ↔ z = {x} ∨ z = {x, y'}\n⊢ y = x → y = y'",
"ppTerm": "?m.60",
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"usedConstants": [
"_private.Mathlib.SetTheory.ZFC.Basic.0.ZFSet.pair_injective._simp_1_5",
"congrArg",
"ZFSet",
"iff_... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 286,
"column": 50
} | {
"line": 286,
"column": 52
} | {
"line": 287,
"column": 2
} | [
{
"pp": "⊢ veblen 0 = fun a ↦ ω ^ a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Ordinal.veblen.eq_1",
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"Eq.mpr",
"Ordinal.omega0",
"congrArg",
"id",
"Ordinal.veblenWith",
"HPow.hPow",
"Ordinal.zero",
"Ordinal... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 289,
"column": 64
} | {
"line": 289,
"column": 66
} | {
"line": 290,
"column": 2
} | [
{
"pp": "a : Ordinal.{u_1}\n⊢ veblen 0 a = ω ^ a",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Eq.mpr",
"Ordinal.omega0",
"congrArg",
"id",
"Ordinal.veblen_zero",
"HPow.hPow",
"Ordinal.zero",
"Zero.toOfNat0",
"... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 343,
"column": 48
} | {
"line": 343,
"column": 50
} | {
"line": 343,
"column": 51
} | [
{
"pp": "o a : Ordinal.{u}\n⊢ 0 < ω ^ 0",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.omega0",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",
"AddMonoid.toAddZeroClass",
"Partia... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 346,
"column": 60
} | {
"line": 346,
"column": 62
} | {
"line": 346,
"column": 63
} | [
{
"pp": "⊢ 0 < ω ^ 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.omega0",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreorder"... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 449,
"column": 22
} | {
"line": 449,
"column": 24
} | {
"line": 449,
"column": 25
} | [
{
"pp": "o : ONote\nb : Ordinal.{0}\nx✝ : NFBelow 0 b\nh₂ : o.NF\n⊢ (0 - o).NFBelow b",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"ONote.NF",
"ONote.oadd",
"ONote.instZero",
"HSub.hSub",
"ONote.zero",
"ONote.NFBelow.zero",
"ONote.NFBelow",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 361,
"column": 52
} | {
"line": 361,
"column": 54
} | {
"line": 361,
"column": 55
} | [
{
"pp": "o a : Ordinal.{u_1}\n⊢ 0 < ω ^ 0",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.omega0",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",
"AddMonoid.toAddZeroClass",
"Part... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 364,
"column": 58
} | {
"line": 364,
"column": 60
} | {
"line": 364,
"column": 61
} | [
{
"pp": "⊢ 0 < ω ^ 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.omega0",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreorder"... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 380,
"column": 52
} | {
"line": 380,
"column": 54
} | {
"line": 381,
"column": 2
} | [
{
"pp": "a : Ordinal.{u_1}\n⊢ a < veblen a a",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Eq.mpr",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.omega0",
"Ordinal.partialOrder",
"Ordinal.veblen_lt_veblen_iff_right._simp_... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 432,
"column": 78
} | {
"line": 432,
"column": 80
} | {
"line": 433,
"column": 2
} | [
{
"pp": "o x : Ordinal.{u}\nh : o < x.invVeblen₁\n⊢ veblen o x = x",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Ordinal.instLinearOrder",
"LinearOrder.toDecidableEq",
"Set.ofPred",
"Membership.mem",
"Eq.mp",
"Ne",
"Decida... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 441,
"column": 75
} | {
"line": 441,
"column": 77
} | {
"line": 442,
"column": 2
} | [
{
"pp": "o a : Ordinal.{u}\n⊢ a < veblen o a ↔ a.invVeblen₁ ≤ o",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Eq.mpr",
"False",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.partialOrder",
"congrArg",
"Ordinal.veble... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Rank | {
"line": 74,
"column": 87
} | {
"line": 74,
"column": 89
} | {
"line": 75,
"column": 2
} | [
{
"pp": "x y : PSet.{u_1}\n⊢ (insert x y).rank = max (succ x.rank) y.rank",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"Order.succ",
"Order.succ_eq_add_... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Rank | {
"line": 86,
"column": 27
} | {
"line": 86,
"column": 29
} | {
"line": 86,
"column": 30
} | [
{
"pp": "x : PSet.{u_1}\n⊢ max (succ x.rank) ∅.rank = succ x.rank",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Order.succ",
"Order.succ_eq_add_one",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",
"zero_le.... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Rank | {
"line": 88,
"column": 86
} | {
"line": 88,
"column": 88
} | {
"line": 89,
"column": 2
} | [
{
"pp": "x y : PSet.{u_1}\n⊢ {x, y}.rank = max (succ x.rank) (succ y.rank)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Order.succ",
"Order.succ_eq_add_one",
"Ordinal.partialOrder",
"congrArg",
"PartialOrder.toPreorder",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 447,
"column": 92
} | {
"line": 447,
"column": 94
} | {
"line": 448,
"column": 2
} | [
{
"pp": "o x : Ordinal.{u}\n⊢ ω ^ x ∈ range (veblen o) ↔ o ≤ x.invVeblen₁",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Ordinal.veblen",
"Eq.mpr",
"False",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"le_rfl",
"LinearOrder.toDec... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Rank | {
"line": 92,
"column": 72
} | {
"line": 92,
"column": 74
} | {
"line": 93,
"column": 2
} | [
{
"pp": "x : PSet.{u_1}\n⊢ x.powerset.rank = succ x.rank",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PSet.mem_powerset._simp_1",
"_private.Mathlib.SetTheory.ZFC.Rank.0.PSet.rank_powerset._simp_1_3",
"Ordinal.instLinearOrder",
"Preorder.toLT",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 461,
"column": 80
} | {
"line": 461,
"column": 82
} | {
"line": 462,
"column": 2
} | [
{
"pp": "o a : Ordinal.{u}\nh : a < veblen o a\n⊢ (veblen o a).invVeblen₁ = o",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Eq.mpr",
"Preorder.toLT",
"Ordinal.veblen_inj._simp_1",
"Ordinal.omega0",
"Ordinal.partialOrder",
"Ordin... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 470,
"column": 68
} | {
"line": 470,
"column": 70
} | {
"line": 471,
"column": 2
} | [
{
"pp": "a : Ordinal.{u}\nh : a < ω ^ a\n⊢ a.invVeblen₁ = 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Eq.mpr",
"Preorder.toLT",
"Ordinal.omega0",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",
"AddMonoid.to... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 475,
"column": 27
} | {
"line": 475,
"column": 29
} | {
"line": 475,
"column": 30
} | [
{
"pp": "⊢ 0 < ω ^ 0",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.omega0",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreorder",... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Rank | {
"line": 104,
"column": 60
} | {
"line": 104,
"column": 62
} | {
"line": 105,
"column": 2
} | [
{
"pp": "x : PSet.{u_1}\n⊢ (⋃₀ x).rank ≤ x.rank",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"PSet.instMembership",
"Ordinal.partialOrder",
"PartialOrder.toPreorder",
"PSet.rank_lt_of_mem",
"Preorder.toLE",
"_priv... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 485,
"column": 82
} | {
"line": 485,
"column": 84
} | {
"line": 486,
"column": 2
} | [
{
"pp": "a x : Ordinal.{u}\n⊢ x.invVeblen₂ = a ↔ ω ^ x = veblen x.invVeblen₁ a",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Eq.mpr",
"Ordinal.omega0",
"congrArg",
"Iff.rfl",
"id",
"Ordinal.veblen_inj",
"Ordinal.veblen_inv... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 488,
"column": 82
} | {
"line": 488,
"column": 84
} | {
"line": 489,
"column": 2
} | [
{
"pp": "a x : Ordinal.{u}\n⊢ x.invVeblen₂ < a ↔ ω ^ x < veblen x.invVeblen₁ a",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Eq.mpr",
"Preorder.toLT",
"Ordinal.omega0",
"Ordinal.partialOrder",
"congrArg",
"Iff.rfl",
"Parti... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 491,
"column": 82
} | {
"line": 491,
"column": 84
} | {
"line": 492,
"column": 2
} | [
{
"pp": "a x : Ordinal.{u}\n⊢ x.invVeblen₂ ≤ a ↔ ω ^ x ≤ veblen x.invVeblen₁ a",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Eq.mpr",
"Ordinal.omega0",
"Ordinal.partialOrder",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 696,
"column": 53
} | {
"line": 696,
"column": 55
} | {
"line": 697,
"column": 2
} | [
{
"pp": "⊢ Function.Injective2 pair",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"_private.Mathlib.SetTheory.ZFC.Basic.0.ZFSet.pair_injective._simp_1_5",
"congrArg",
"ZFSet",
"iff_true",
"true_iff",
"ZFSet.pair_eq_singleton",
"Membership.mem",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 494,
"column": 82
} | {
"line": 494,
"column": 84
} | {
"line": 495,
"column": 2
} | [
{
"pp": "a x : Ordinal.{u}\n⊢ a < x.invVeblen₂ ↔ veblen x.invVeblen₁ a < ω ^ x",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Eq.mpr",
"Preorder.toLT",
"Ordinal.omega0",
"Ordinal.partialOrder",
"congrArg",
"Iff.rfl",
"Parti... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 497,
"column": 82
} | {
"line": 497,
"column": 84
} | {
"line": 498,
"column": 2
} | [
{
"pp": "a x : Ordinal.{u}\n⊢ a ≤ x.invVeblen₂ ↔ veblen x.invVeblen₁ a ≤ ω ^ x",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Eq.mpr",
"Ordinal.omega0",
"Ordinal.partialOrder",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 500,
"column": 62
} | {
"line": 500,
"column": 64
} | {
"line": 501,
"column": 2
} | [
{
"pp": "x : Ordinal.{u_1}\n⊢ x.invVeblen₂ < ω ^ x",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Eq.mpr",
"Preorder.toLT",
"Ordinal.omega0",
"Ordinal.partialOrder",
"congrArg",
"PartialOrder.toPreorder",
"id",
"Ordin... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 717,
"column": 88
} | {
"line": 717,
"column": 90
} | {
"line": 718,
"column": 2
} | [
{
"pp": "x y z : ZFSet.{u}\n⊢ z ∈ x.prod y ↔ ∃ a ∈ x, ∃ b ∈ y, z = a.pair b",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"and_true",
"congrArg",
"ZFSet",
"Membership.mem",
"Exists",
"ZFSet.mem_pairSep._simp_1",
"ZFSet.prod",
"ZFSet.pairSep... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 720,
"column": 85
} | {
"line": 720,
"column": 87
} | {
"line": 721,
"column": 2
} | [
{
"pp": "x y a b : ZFSet.{u}\n⊢ a.pair b ∈ x.prod y ↔ a ∈ x ∧ b ∈ y",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"ZFSet.mem_prod._simp_1",
"congrArg",
"ZFSet",
"ZFSet.pair_inj._simp_1",
"Membership.mem",
"Exists",
"ZFSet.prod",
"exists_eq_... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Rank | {
"line": 109,
"column": 72
} | {
"line": 109,
"column": 74
} | {
"line": 110,
"column": 2
} | [
{
"pp": "x : PSet.{u_1}\n⊢ x.rank ≤ succ (⋃₀ x).rank",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Order.succ",
"PSet.instMembership",
"Ordinal.partialOrder",
"PSet.rank_mono",
"congrArg",
"PSet.powerset",
"PartialOrder.toPreorder... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Rank | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 41
} | {
"line": 121,
"column": 42
} | [
{
"pp": "x : PSet.{u}\n⊢ lift.{u + 1, u} x.rank = IsWellFounded.rank (fun x1 x2 ↦ x1 ∈ x2) x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"PSet.mem_wf",
"PSet.instIsWellFoundedMem",
"PSet.instMembership",
"Ordinal.lift",
"Membership.mem",
"PSet",
... | [
"case h\nx✝ x : PSet.{u}\nih : ∀ y ∈ x, lift.{u + 1, u} y.rank = IsWellFounded.rank (fun x1 x2 ↦ x1 ∈ x2) y\n⊢ lift.{u + 1, u} x.rank = IsWellFounded.rank (fun x1 x2 ↦ x1 ∈ x2) x"
] | induction x using mem_wf.induction with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.SetTheory.ZFC.Basic | {
"line": 733,
"column": 70
} | {
"line": 733,
"column": 72
} | {
"line": 733,
"column": 73
} | [
{
"pp": "x y f : ZFSet.{u}\n⊢ f ∈ x.funs y ↔ x.IsFunc y f",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"ZFSet.sep",
"congrArg",
"ZFSet",
"ZFSet.mem_sep._simp_1",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"LE.le",
"... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 504,
"column": 58
} | {
"line": 504,
"column": 60
} | {
"line": 505,
"column": 2
} | [
{
"pp": "x : Ordinal.{u_1}\n⊢ x.invVeblen₂ ≤ x",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Eq.mpr",
"le_refl",
"Preorder.toLT",
"Ordinal.omega0",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",
"Ordinal.v... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Rank | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 50
} | {
"line": 121,
"column": 50
} | [
{
"pp": "x : PSet.{u}\n⊢ lift.{u + 1, u} x.rank = IsWellFounded.rank (fun x1 x2 ↦ x1 ∈ x2) x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"PSet.mem_wf",
"PSet.instIsWellFoundedMem",
"PSet.instMembership",
"Ordinal.lift",
"Membership.mem",
"PSet",
... | [
"case h\nx✝ x : PSet.{u}\nih : ∀ y ∈ x, lift.{u + 1, u} y.rank = IsWellFounded.rank (fun x1 x2 ↦ x1 ∈ x2) y\n⊢ lift.{u + 1, u} x.rank = IsWellFounded.rank (fun x1 x2 ↦ x1 ∈ x2) x"
] | induction x using mem_wf.induction with | _ x ih
=> _ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 510,
"column": 68
} | {
"line": 510,
"column": 70
} | {
"line": 511,
"column": 2
} | [
{
"pp": "a : Ordinal.{u}\nh : a < ω ^ a\n⊢ a.invVeblen₂ = a",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Eq.mpr",
"Ordinal.invVeblen₂_eq_iff",
"Ordinal.omega0",
"congrArg",
"Ordinal.invVeblen₁_of_lt_opow",
"id",
"Ordinal.... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 514,
"column": 46
} | {
"line": 514,
"column": 48
} | {
"line": 515,
"column": 2
} | [
{
"pp": "⊢ invVeblen₂ 0 = 0",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.omega0",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPre... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 518,
"column": 93
} | {
"line": 518,
"column": 95
} | {
"line": 519,
"column": 2
} | [
{
"pp": "o a : Ordinal.{u}\nho : o ≠ 0\nh : a < veblen o a\n⊢ (veblen o a).invVeblen₂ = a",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Eq.mpr",
"Ordinal.invVeblen₂_eq_iff",
"Ordinal.omega0",
"Ordinal.partialOrder",
"congrArg",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 750,
"column": 37
} | {
"line": 750,
"column": 39
} | {
"line": 751,
"column": 4
} | [
{
"pp": "f : ZFSet.{u} → ZFSet.{u}\ninst✝ : Definable₁ f\nx z : ZFSet.{u}\nzx : z ∈ x\ny : ZFSet.{u}\nyx : (fun w ↦ z.pair w ∈ map f x) y\n⊢ y = f z",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"ZFSet",
"ZFSet.pair_injective",
"ZFSet.D... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 777,
"column": 81
} | {
"line": 777,
"column": 83
} | {
"line": 778,
"column": 2
} | [
{
"pp": "p : ZFSet.{u} → Prop\nx : ZFSet.{u}\n⊢ Hereditarily p x ↔ p x ∧ ∀ y ∈ x, Hereditarily p y",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"ZFSet",
"Iff.rfl",
"Membership.mem",
"id",
"ZFSet.Hereditarily",
"And",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 523,
"column": 64
} | {
"line": 523,
"column": 66
} | {
"line": 524,
"column": 2
} | [
{
"pp": "o a x : Ordinal.{u}\nh : a < veblen o a\n⊢ veblen o a = ω ^ x ↔ x.invVeblen₁ = o ∧ x.invVeblen₂ = a",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Eq.mpr",
"Preorder.toLT",
"Ordinal.omega0",
"Ordinal.partialOrder",
"Ordinal.o... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 548,
"column": 75
} | {
"line": 548,
"column": 77
} | {
"line": 549,
"column": 2
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ ε_ o = deriv (fun a ↦ ω ^ a) o",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Ordinal.omega0",
"congrArg",
"AddMonoid.toAddZeroClass",
"Eq.mp",
"id",
"Ordinal.veblen_zero",
"zero_add",
"O... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 551,
"column": 60
} | {
"line": 551,
"column": 62
} | {
"line": 552,
"column": 2
} | [
{
"pp": "⊢ ε_ 0 = nfp (fun a ↦ ω ^ a) 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.omega0",
"Ordinal.epsilon_eq_deriv",
"congrArg",
"id",
"Ordinal.nfp",
"HPow.hPow",
"Ordinal.deriv",
"Ordinal.zero",
"Ordina... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 557,
"column": 95
} | {
"line": 557,
"column": 97
} | {
"line": 558,
"column": 2
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ ε_ (succ o) = nfp (fun a ↦ ω ^ a) (succ (ε_ o))",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Order.succ",
"Ordinal.omega0",
"Ordinal.partialOrder",
"Ordinal.epsilon_eq_deriv",
"congrArg",
"Ordinal.deriv_s... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Rank | {
"line": 120,
"column": 96
} | {
"line": 120,
"column": 98
} | {
"line": 121,
"column": 2
} | [
{
"pp": "x : PSet.{u}\n⊢ lift.{u + 1, u} x.rank = IsWellFounded.rank (fun x1 x2 ↦ x1 ∈ x2) x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PSet.mem_wf",
"PSet.instIsWellFoundedMem",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Order.succ",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 560,
"column": 70
} | {
"line": 560,
"column": 72
} | {
"line": 561,
"column": 2
} | [
{
"pp": "o : Ordinal.{u}\nh : ω ^ o ≤ o\n⊢ ε_ 0 ≤ o",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"zero_le",
"Eq.mpr",
"Ordinal.nfp_le_fp",
"Ordinal.omega0",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",
"AddMon... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Rank | {
"line": 154,
"column": 72
} | {
"line": 154,
"column": 74
} | {
"line": 155,
"column": 2
} | [
{
"pp": "x : ZFSet.{u}\no : Ordinal.{u}\n⊢ o < x.rank ↔ ∃ y ∈ x, o ≤ y.rank",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
"Mathlib.Tactic.Push.not_and_eq",
"Ordinal.instLinearOrder",
"Preorder.toLT",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Basic | {
"line": 788,
"column": 55
} | {
"line": 788,
"column": 57
} | {
"line": 789,
"column": 2
} | [
{
"pp": "p : ZFSet.{u} → Prop\nx : ZFSet.{u}\n⊢ Hereditarily p x → p ∅",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"ZFSet.instEmptyCollection",
"ZFSet.Hereditarily.mem",
"ZFSet",
"ZFSet.inductionOn",
"Membership.mem",
"ZFSet.eq_empty_or_nonempty",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Rank | {
"line": 169,
"column": 27
} | {
"line": 169,
"column": 29
} | {
"line": 169,
"column": 30
} | [
{
"pp": "x : ZFSet.{u_1}\n⊢ max (succ x.rank) ∅.rank = succ x.rank",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"ZFSet.instEmptyCollection",
"ZFSet.rank_empty",
"Lattice.toSemilatticeSup",
"Order.succ",
"Order.succ_eq_add_one",
"Ordinal.partialOrder",... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 568,
"column": 63
} | {
"line": 568,
"column": 65
} | {
"line": 569,
"column": 2
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ ω ^ ε_ o = ε_ o",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.omega0",
"Ordinal.epsilon_eq_deriv",
"congrArg",
"Ordinal.deriv_fp",
"id",
"Ordinal.isNormal_opow",
"HPow.hPow",
"Ordinal.de... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 451,
"column": 47
} | {
"line": 451,
"column": 49
} | {
"line": 452,
"column": 4
} | [
{
"pp": "e₁ : ONote\nn₁ : ℕ+\na₁ e₂ : ONote\nn₂ : ℕ+\na₂ : ONote\nb : Ordinal.{0}\nh₁ : (e₁.oadd n₁ a₁).NFBelow b\nh₂ : (e₂.oadd n₂ a₂).NF\n⊢ (e₁.oadd n₁ a₁ - e₂.oadd n₂ a₂).NFBelow b",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"PNat.val",
"Ordering.gt",
"ONote.NF",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.ZFC.Rank | {
"line": 171,
"column": 87
} | {
"line": 171,
"column": 89
} | {
"line": 172,
"column": 2
} | [
{
"pp": "x y : ZFSet.{u_1}\n⊢ {x, y}.rank = max (succ x.rank) (succ y.rank)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Order.succ",
"Order.succ_eq_add_one",
"Ordinal.partialOrder",
"congrArg",
"ZFSet",
"PartialOrde... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 572,
"column": 73
} | {
"line": 572,
"column": 75
} | {
"line": 573,
"column": 2
} | [
{
"pp": "o : Ordinal.{u}\n⊢ o < ε_ 0 ↔ ∃ n, o < (fun a ↦ ω ^ a)^[n] 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Ordinal.omega0",
"Ordinal.partialOrder",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"Exis... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 472,
"column": 19
} | {
"line": 472,
"column": 21
} | {
"line": 472,
"column": 22
} | [
{
"pp": "o : ONote\nx✝ : NF 0\nh₂ : o.NF\n⊢ (0 - o).repr = repr 0 - o.repr",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"ONote.NF",
"ONote.oadd",
"Ordinal.zero_sub",
"ONote.instZero",
"HSub.hSub",
"ONote.zero",
"ONote.instSub",
"instHSub",... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 579,
"column": 84
} | {
"line": 579,
"column": 86
} | {
"line": 580,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ (fun a ↦ ω ^ a)^[n] 0 < ε_ 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.omega0",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",
"AddMonoid.toAddZ... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 587,
"column": 54
} | {
"line": 587,
"column": 56
} | {
"line": 588,
"column": 2
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ ω < ε_ o",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"zero_le",
"Ordinal.opow_one",
"Ordinal.veblen",
"Preorder.toLT",
"Ordinal.omega0",
"StrictMono.monotone",
"Ordinal.partialOrder",
"congrArg",
"instIs... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 656,
"column": 68
} | {
"line": 656,
"column": 70
} | {
"line": 657,
"column": 2
} | [
{
"pp": "o : Ordinal.{u}\nh : veblen o 0 ≤ o\n⊢ Γ_ 0 ≤ o",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"zero_le",
"Ordinal.veblen",
"Eq.mpr",
"Ordinal.gamma_zero_eq_nfp",
"Ordinal.nfp_le_fp",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZer... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 664,
"column": 76
} | {
"line": 664,
"column": 78
} | {
"line": 665,
"column": 2
} | [
{
"pp": "o : Ordinal.{u}\n⊢ o < Γ_ 0 ↔ ∃ n, o < (fun a ↦ veblen a 0)^[n] 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Eq.mpr",
"Preorder.toLT",
"Ordinal.gamma_zero_eq_nfp",
"Ordinal.partialOrder",
"congrArg",
"Iff.rfl",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 671,
"column": 82
} | {
"line": 671,
"column": 84
} | {
"line": 672,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ (fun a ↦ veblen a 0)^[n] 0 < Γ_ 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Ordinal.veblen",
"Eq.mpr",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.gamma_zero_eq_nfp",
"Ordinal.omega0",
"Ordinal.partialOrder",
... | [] | by | [anonymous] | by |
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