module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.FieldTheory.Finite.Basic | {
"line": 313,
"column": 6
} | {
"line": 314,
"column": 31
} | [
{
"pp": "case h\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype K\ni : ℕ\nh : i < q - 1\nhi : ¬i = 0\nhiq : ¬q - 1 ∣ i\nφ : Kˣ ↪ K := { toFun := fun x ↦ ↑x, inj' := ⋯ }\nx : K\n⊢ x ∈ map φ univ ↔ x ∈ univ \\ {0}",
"usedConstants": [
"Units.val",
"Eq.mpr",
"Finset.univ",
"_private.Ma... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 191,
"column": 38
} | {
"line": 191,
"column": 54
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Invertible R M\nN : Type (max v u) := Dual R M\ne : M ⊗[R] N ≃ₗ[R] R := TensorProduct.comm R M N ≪≫ₗ linearEquiv R M\nS : Finset (M × N)\nhS : e.symm 1 = ∑ i ∈ S, i.1 ⊗ₜ[R] i.2\nf : (↥S →₀ N) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.Finite.Quotient | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 15
} | [
{
"pp": "case refine_1\nM : Type u_3\ninst✝² : AddCommGroup M\ninst✝¹ : Free ℤ M\ninst✝ : Module.Finite ℤ M\nN : Submodule ℤ M\nh : Finite (M ⧸ N)\nx : M\n⊢ ((LinearMap.lsmul ℤ M) ↑(Nat.card (M ⧸ N))) x ∈ N",
"usedConstants": [
"Eq.mpr",
"Submodule",
"instHSMul",
"Semiring.toModule",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 13
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\nP : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : AddCommMonoid P\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module.Invertible R M\nf : N →ₗ[R] P\nh : Function.Injective ⇑(LinearMap.lTen... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 13
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\nP : Type u_2\ninst✝⁷ : CommSemiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : AddCommMonoid P\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module.Invertible R M\nf : N →ₗ[R] P\nh : Function.Surjective ⇑(LinearMap.lTe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 246,
"column": 52
} | {
"line": 246,
"column": 63
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Invertible R M\nx✝ : Free R M\na✝ : Nontrivial R\ne : M ≃ₗ[R] Free.ChooseBasisIndex R M →₀ R\nthis : Fintype.card (Free.ChooseBasisIndex R M × Free.ChooseBasisIndex R M) = Fintype.card Unit\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 266,
"column": 2
} | {
"line": 266,
"column": 20
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Invertible R M\nthis :\n ⇑(toModuleEnd R M) =\n ⇑(lid R M).conj ∘\n ⇑(rTensorEquiv R R (TensorProduct.comm R M (Dual R M) ≪≫ₗ linearEquiv R M)) ∘\n ⇑(RingEquiv.moduleEndSelf R)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 285,
"column": 2
} | {
"line": 285,
"column": 37
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module.Invertible R M\ninst✝ : Module.Invertible R N\nf : M →ₗ[R] N\nhf : Function.Surjective ⇑f\n⊢ Function.Bijective ⇑f",
"usedCons... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 286,
"column": 60
} | {
"line": 286,
"column": 96
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module.Invertible R M\ninst✝ : Module.Invertible R N\nf : M →ₗ[R] N\nhf : Function.Surjective ⇑f\n⊢ Function.Surjective ⇑(LinearMap.lTens... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 381,
"column": 6
} | {
"line": 381,
"column": 58
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : Fintype K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Finite L\nthis✝ : Fintype L\nm : ℕ\nlt : m < Module.finrank K L\npos : 0 < m\neq : frobeniusAlgHom K L ^ m = 1\nx : L\nx✝ : x ∈ univ.val\nthis : (fun x ↦ x ^ q ^ m) x - x = 0\nh : X ^ q ^ m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 388,
"column": 2
} | {
"line": 388,
"column": 67
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : Fintype K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Finite L\n⊢ orderOf (frobeniusAlgEquivOfAlgebraic K L) = Module.finrank K L",
"usedConstants": [
"Nontrivial",
"Monoid",
"Eq.mpr",
"IsDomain",
"MulOne.toOn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 400,
"column": 4
} | {
"line": 400,
"column": 49
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : Fintype K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Finite L\n⊢ Function.Bijective (⇑(Algebra.IsAlgebraic.algEquivEquivAlgHom K L) ∘ fun n ↦ frobeniusAlgEquivOfAlgebraic K L ^ ↑n)",
"usedConstants": [
"Eq.mpr",
"MulEquiv.inst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 413,
"column": 30
} | {
"line": 413,
"column": 100
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : Fintype K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Finite L\nx : L\n⊢ ((aeval (frobeniusAlgHom K L).toLinearMap) (X ^ Module.finrank K L - 1)) x = 0 x",
"usedConstants": [
"Module.End.instRing",
"AlgHom.toLinearMap",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 416,
"column": 6
} | {
"line": 416,
"column": 51
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : Fintype K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Finite L\n⊢ LinearIndependent K fun i ↦ (frobeniusAlgHom K L).toLinearMap ^ ↑i",
"usedConstants": [
"AlgHom.toEnd",
"Module.End.instRing",
"AlgHom.toLinearMap",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.Finite.CardQuotient | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 70
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : AddCommGroup M\ninst✝³ : Free ℤ M\ninst✝² : Module.Finite ℤ M\nN : Submodule ℤ M\nE : Type u_2\ninst✝¹ : EquivLike E M ↥N\ninst✝ : AddEquivClass E M ↥N\ne : E\nb : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := Free.chooseBasis ℤ M\nh : finrank ℤ ↥N = finrank ℤ M\na : Free.ChooseBasisI... | let f_apply : ∀ x, f x = b'.equiv ab (Equiv.refl _) x := fun x ↦ rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.FieldTheory.Finite.Basic | {
"line": 504,
"column": 2
} | {
"line": 504,
"column": 78
} | [
{
"pp": "case inr\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ZMod p\nhp_odd : p % 2 = 1\nf : (ZMod p)[X] := X ^ 2\ng : (ZMod p)[X] := X ^ 2 - C x\na b : ZMod p\nhab : eval a f + eval b g = 0\n⊢ a ^ 2 + b ^ 2 - x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 526,
"column": 2
} | {
"line": 526,
"column": 44
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Prime p)\nx : ℕ\n⊢ ∃ a b, a ≤ p / 2 ∧ b ≤ p / 2 ∧ a ^ 2 + b ^ 2 ≡ x [MOD p]",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"congrArg",
"Nat.instMonoid",
"Exists",
"id",
"HDiv.hDiv",
"instOfNatNat",
"Int",
"LE.le",
"i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 535,
"column": 2
} | {
"line": 535,
"column": 13
} | [
{
"pp": "R : Type u_3\ninst✝³ : Ring R\ninst✝² : IsDomain R\np : ℕ\ninst✝¹ : NeZero p\ninst✝ : CharP R p\nx : ℤ\nthis : Fact (Nat.Prime p)\na b : ZMod p\nhab : a ^ 2 + b ^ 2 = ↑x\n⊢ ↑a.val ^ 2 + ↑b.val ^ 2 = ↑x",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"ZMod.cast",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 639,
"column": 2
} | {
"line": 639,
"column": 45
} | [
{
"pp": "p : ℕ\nhp : Nat.Prime p\nn : ℤ\nhpn : IsCoprime n ↑p\nthis✝ : Fact (Nat.Prime p)\nthis : ¬↑n = 0\n⊢ n ^ (p - 1) ≡ 1 [ZMOD ↑p]",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"ZMod.commRing",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"Int.cast_pow",
"HSub... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 771,
"column": 2
} | {
"line": 771,
"column": 87
} | [
{
"pp": "F : Type u_3\ninst✝¹ : Field F\ninst✝ : Finite F\nhF : ringChar F ≠ 2\nh : ¬Function.Injective fun x ↦ x * x\n⊢ ∃ a, ¬IsSquare a",
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
"HMul.hMul",
"congrArg",
"Exists",
"id",
"Field.toSemifield",
"instDi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 804,
"column": 4
} | {
"line": 804,
"column": 81
} | [
{
"pp": "F : Type u_3\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na g : Fˣ\nhg : ∀ (x : Fˣ), x ∈ Subgroup.zpowers g\nn : ℕ\nhn : (fun x ↦ g ^ x) n = a\n⊢ IsSquare a ↔ a ^ (Fintype.card F / 2) = 1",
"usedConstants": [
"instHDiv",
"HMul.hMul",
"HSub.hSub",
"FiniteField.o... | have hodd := Nat.two_mul_odd_div_two (FiniteField.odd_card_of_char_ne_two hF) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.Finite.Basic | {
"line": 827,
"column": 21
} | {
"line": 827,
"column": 41
} | [
{
"pp": "F : Type u_3\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ Units.mk0 a ha ^ (Fintype.card F / 2) = 1 ↔ a ^ (Fintype.card F / 2) = 1",
"usedConstants": [
"MulOne.toOne",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"AddGroupWithOne.t... | simp [Units.ext_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.Finite.Basic | {
"line": 827,
"column": 21
} | {
"line": 827,
"column": 41
} | [
{
"pp": "F : Type u_3\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ Units.mk0 a ha ^ (Fintype.card F / 2) = 1 ↔ a ^ (Fintype.card F / 2) = 1",
"usedConstants": [
"MulOne.toOne",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"AddGroupWithOne.t... | simp [Units.ext_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Finite.Basic | {
"line": 827,
"column": 21
} | {
"line": 827,
"column": 41
} | [
{
"pp": "F : Type u_3\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ Units.mk0 a ha ^ (Fintype.card F / 2) = 1 ↔ a ^ (Fintype.card F / 2) = 1",
"usedConstants": [
"MulOne.toOne",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"AddGroupWithOne.t... | simp [Units.ext_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Finite.Basic | {
"line": 831,
"column": 2
} | {
"line": 833,
"column": 37
} | [
{
"pp": "case mpr\nF : Type u_3\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ (∃ r, a = r * r) → ∃ r, a = ↑r * ↑r",
"usedConstants": [
"Units.val",
"GroupWithZero.toMonoidWithZero",
"False",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg"... | · rintro ⟨y, rfl⟩
have hy : y ≠ 0 := by rintro rfl; simp at ha
refine ⟨Units.mk0 y hy, ?_⟩; simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.PicardGroup | {
"line": 725,
"column": 47
} | {
"line": 725,
"column": 74
} | [
{
"pp": "R : Type u\nA : Type u_4\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : FaithfulSMul R A\nI : (Submodule R A)ˣ\nh✝ : I ∈ (unitsToPic R A).ker\ne : R ≃ₗ[R] ↥↑I\ne' : R ≃ₗ[R] ↥↑I⁻¹\nh : ↑I = R ∙ ↑(↑e 1)\nh' : ↑I⁻¹ = R ∙ ↑(↑e' 1)\n⊢ R ∙ ↑(e 1) * ↑(e' 1) = 1",
"usedConstan... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 728,
"column": 47
} | {
"line": 728,
"column": 74
} | [
{
"pp": "R : Type u\nA : Type u_4\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : FaithfulSMul R A\nI : (Submodule R A)ˣ\nh✝ : I ∈ (unitsToPic R A).ker\ne : R ≃ₗ[R] ↥↑I\ne' : R ≃ₗ[R] ↥↑I⁻¹\nh : ↑I = R ∙ ↑(↑e 1)\nh' : ↑I⁻¹ = R ∙ ↑(↑e' 1)\n⊢ R ∙ ↑(e' 1) * ↑(e 1) = 1",
"usedConstan... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative | {
"line": 63,
"column": 13
} | {
"line": 63,
"column": 24
} | [
{
"pp": "case empty\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\nP : α → Prop\ni : α → ℕ\nh1 : ∀ {x : α}, IsUnit x → P x\nhpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)\nhcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)\nthis : DecidableEq α := Classical.decEq α\nis_p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 733,
"column": 61
} | {
"line": 733,
"column": 72
} | [
{
"pp": "R : Type u\nA : Type u_4\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : FaithfulSMul R A\nx : Aˣ\nx✝¹ x✝ : R\neq : (LinearMap.toSpanSingleton R A ↑x) x✝¹ = (LinearMap.toSpanSingleton R A ↑x) x✝\n⊢ (fun r ↦ r • 1) x✝¹ = (fun r ↦ r • 1) x✝",
"usedConstants": [
"Mul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 758,
"column": 2
} | {
"line": 758,
"column": 25
} | [
{
"pp": "R : Type u\nM : Type v\nA : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ne : A ⊗[R] M ≃ₗ[A] A\ninst✝¹ : Flat R M\ninst✝ : FaithfulSMul R A\n⊢ Function.Injective ⇑(toAlgebra e)",
"usedConstants": [
"AlgHom.toLinear... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative | {
"line": 103,
"column": 13
} | {
"line": 103,
"column": 24
} | [
{
"pp": "case empty\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\nβ : Type u_3\ninst✝ : CommMonoidWithZero β\nf : α → β\ni j : α → ℕ\nh1 : ∀ {x y : α}, IsUnit y → f (x * y) = f x * f y\nhpr : ∀ {p : α} (i : ℕ), Prime p → f (p ^ i) = f p ^ i\nhcp : ∀ {x y : α}, IsRelPrime x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PicardGroup | {
"line": 783,
"column": 50
} | {
"line": 783,
"column": 61
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\nP : Type u_2\nQ : Type u_3\nA : Type u_4\ninst✝¹² : CommSemiring R\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid N\ninst✝⁹ : AddCommMonoid P\ninst✝⁸ : AddCommMonoid Q\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : Module R P\ninst✝⁴ : Module R Q\ninst✝³ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Norm.Basic | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 48
} | [
{
"pp": "case mp.refine_1\nR : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : Ring S\ninst✝⁴ : Algebra R S\ninst✝³ : IsDomain R\ninst✝² : IsDomain S\ninst✝¹ : Free R S\ninst✝ : Module.Finite R S\nx : S\nb : Basis (Free.ChooseBasisIndex R S) R S := Free.chooseBasis R S\ndecEq : DecidableEq (Free.ChooseBas... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 92
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsSeparable F E\nhα : IsIntegral F α\nhβ : IsIntegral F β\nf : F[X] := minpoly F α\ng : F[X] := minpoly F β\nιFE : F →+* E := algebraMap F E\nιEE' : E →+* (Polynomial.map ... | obtain ⟨c, hc⟩ := primitive_element_inf_aux_exists_c (ιEE'.comp ιFE) (ιEE' α) (ιEE' β) f g | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 152,
"column": 8
} | {
"line": 152,
"column": 31
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Infinite F\nE : Type u_2\ninst✝² : Field E\nα β : E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsSeparable F E\nhα : IsIntegral F α\nhβ : IsIntegral F β\nf : F[X] := minpoly F α\ng : F[X] := minpoly F β\nιFE : F →+* E := algebraMap F E\nιEE' : E →+* (Polynomial.map ... | mem_roots_map h_ne_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.RamificationInertia.Inertia | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 55
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommRing R\nS : Type u_1\ninst✝⁷ : CommRing S\ninst✝⁶ : IsDedekindDomain S\ninst✝⁵ : Free ℤ S\ninst✝⁴ : IsDedekindDomain R\ninst✝³ : Free ℤ R\ninst✝² : Algebra S R\ninst✝¹ : Module.Finite S R\nP : Ideal R\np : Ideal S\ninst✝ : P.LiesOver p\nhp : p.IsPrime\nhp_ne_bot : p ≠ ⊥\nthis :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RamificationInertia.Inertia | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 13
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Free ℤ R\ninst✝¹ : Module.Finite ℤ R\np : ℤ\nP : Ideal R\ninst✝ : P.LiesOver (span {p})\nhp : Prime p\n⊢ absNorm P = p.natAbs ^ (span {p}).inertiaDeg P",
"usedConstants": [
"Ideal.Quotient.commSemiring",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RamificationInertia.Inertia | {
"line": 155,
"column": 51
} | {
"line": 155,
"column": 62
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Free ℤ R\ninst✝¹ : Module.Finite ℤ R\np : ℤ\nP : Ideal R\ninst✝ : P.LiesOver (span {p})\nhp : Prime p\n⊢ span {p} ≠ ⊥",
"usedConstants": [
"Submodule.span_eq_bot._simp_1",
"Eq.mpr",
"Submodule",
"Semiring... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 295,
"column": 17
} | {
"line": 295,
"column": 64
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Finite (IntermediateField F E)\nK : IntermediateField F E\nthis : FiniteDimensional F E\nh✝ : Finite F\nα : ↥K\nh : F⟮α⟯ = ⊤\n⊢ F⟮↑α⟯ = K",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 303,
"column": 2
} | {
"line": 303,
"column": 24
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsAlgebraic F E\nh : ∃ α, F⟮α⟯ = ⊤\n⊢ FiniteDimensional F E",
"usedConstants": []
}
] | obtain ⟨α, hprim⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 312,
"column": 2
} | {
"line": 312,
"column": 24
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsAlgebraic F E\nh : ∃ α, F⟮α⟯ = ⊤\nthis : FiniteDimensional F E\n⊢ Finite (IntermediateField F E)",
"usedConstants": []
}
] | obtain ⟨α, hprim⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 172,
"column": 4
} | {
"line": 173,
"column": 34
} | [
{
"pp": "case succ.refine_1\nS : Type u_1\ninst✝¹ : CommRing S\nP : Ideal S\nP_prime : P.IsPrime\ninst✝ : IsDedekindDomain S\nhP : P ≠ ⊥\ni : ℕ\nih : cardQuot (P ^ i) = cardQuot P ^ i\nthis : P ^ (i + 1) < P ^ i\na : S\na_mem : a ∈ P ^ i\na_notMem : a ∉ P ^ (i + 1)\nf g : (c : S) → c ∈ P ^ i → S\nhg : ∀ (c : S)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 329,
"column": 4
} | {
"line": 329,
"column": 79
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.IsAlgebraic F E\nthis : FiniteDimensional F E\nα : E\nhprim : F⟮α⟯ = ⊤\nf : F[X] := minpoly F α\nG : Type (max 0 u_2) := { g // g.Monic ∧ g ∣ Polynomial.map (algebraMap F E) f }\nhfin : Finite G\ng : I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 110,
"column": 37
} | {
"line": 110,
"column": 48
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal R\nP : Ideal S\nh : ¬map f p ≤ P\n⊢ ¬map f p ≤ P ^ (0 + 1)",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"IsScalarTower.right",
"Algebra.algebraMap",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 239,
"column": 24
} | {
"line": 239,
"column": 95
} | [
{
"pp": "S : Type u_1\ninst✝³ : CommRing S\ninst✝² : Nontrivial S\ninst✝¹ : IsDedekindDomain S\ninst✝ : Free ℤ S\nI : Ideal S\nhI : Irreducible (absNorm I)\nh : IsUnit I\n⊢ IsUnit (absNorm I)",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroOneClass",
"Semiring.toModule",
"Ideal.absNor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 242,
"column": 6
} | {
"line": 242,
"column": 77
} | [
{
"pp": "S : Type u_1\ninst✝³ : CommRing S\ninst✝² : Nontrivial S\ninst✝¹ : IsDedekindDomain S\ninst✝ : Free ℤ S\na b : Ideal S\nhI : Irreducible (absNorm (a * b))\n⊢ IsUnit a ∨ IsUnit b",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.RingTheory.Ideal.Norm.AbsNorm.0.Ideal.irreducible_of_irreduc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 388,
"column": 67
} | {
"line": 388,
"column": 78
} | [
{
"pp": "S : Type u_1\ninst✝⁴ : CommRing S\ninst✝³ : Nontrivial S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\np : ℤ\nhp : Prime p\nI : Ideal S\nhI : p ∣ ↑(absNorm I)\nthis✝ : IsAddTorsionFree S\nthis : CharZero S\n⊢ span {p} ≠ ⊥",
"usedConstants": [
"Submodule.span_eq_b... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 395,
"column": 29
} | {
"line": 395,
"column": 40
} | [
{
"pp": "S : Type u_1\ninst✝⁴ : CommRing S\ninst✝³ : Nontrivial S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\np : ℤ\nhp : Prime p\nthis✝ : IsAddTorsionFree S\nthis : CharZero S\nhpMax : (span {p}).IsMaximal\nI : Ideal S\nhI' : IsUnit I\nhI : p ∣ ↑(absNorm I)\n⊢ I = ⊤",
"usedC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 396,
"column": 29
} | {
"line": 396,
"column": 40
} | [
{
"pp": "S : Type u_1\ninst✝⁴ : CommRing S\ninst✝³ : Nontrivial S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\np : ℤ\nhp : Prime p\nthis✝ : IsAddTorsionFree S\nthis : CharZero S\nhpMax : (span {p}).IsMaximal\nhI' : IsUnit ⊤\nhI : p ∣ ↑(absNorm ⊤)\n⊢ p ∣ 1",
"usedConstants": []... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 405,
"column": 43
} | {
"line": 405,
"column": 54
} | [
{
"pp": "S : Type u_1\ninst✝⁴ : CommRing S\ninst✝³ : Nontrivial S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\np : ℤ\nhp : Prime p\nthis✝¹ : IsAddTorsionFree S\nthis✝ : CharZero S\nhpMax : (span {p}).IsMaximal\nI P : Ideal S\nhI' : I ≠ 0\nhP : Prime P\nIH : p ∣ ↑(absNorm I) → ∃ P,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 410,
"column": 80
} | {
"line": 410,
"column": 95
} | [
{
"pp": "S : Type u_1\ninst✝⁴ : CommRing S\ninst✝³ : Nontrivial S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\np : ℤ\nhp✝ : Prime p\nthis✝¹ : IsAddTorsionFree S\nthis✝ : CharZero S\nhpMax : (span {p}).IsMaximal\nI P : Ideal S\nhI' : I ≠ 0\nhP : Prime P\nIH : p ∣ ↑(absNorm I) → ∃ P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 431,
"column": 6
} | {
"line": 431,
"column": 91
} | [
{
"pp": "case inr.refine_1\nS : Type u_1\ninst✝⁵ : CommRing S\ninst✝⁴ : Nontrivial S\ninst✝³ : IsDedekindDomain S\ninst✝² : Free ℤ S\ninst✝¹ : Module.Finite ℤ S\ninst✝ : CharZero S\nn : ℕ\nhn : n > 0\nf : Ideal S → Ideal (S ⧸ span {↑n}) := fun I ↦ map (Quotient.mk (span {↑n})) I\n⊢ ((Algebra.norm ℤ) ↑n).natAbs ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 440,
"column": 6
} | {
"line": 441,
"column": 73
} | [
{
"pp": "S : Type u_1\ninst✝⁵ : CommRing S\ninst✝⁴ : Nontrivial S\ninst✝³ : IsDedekindDomain S\ninst✝² : Free ℤ S\ninst✝¹ : Module.Finite ℤ S\ninst✝ : CharZero S\nn : ℕ\n⊢ {I | absNorm I ≤ n}.Finite",
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Nat.instMulZeroOneClass",
"Nat.instMulZero... | show {I : Ideal S | Ideal.absNorm I ≤ n} =
(⋃ i ∈ Set.Icc 0 n, {I : Ideal S | Ideal.absNorm I = i}) by ext; simp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.RamificationInertia.Ramification | {
"line": 318,
"column": 29
} | {
"line": 318,
"column": 40
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nS : Type v\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsDedekindDomain S\ninst✝² : IsDedekindDomain R\ninst✝¹ : FaithfulSMul R S\nv : Ideal R\nw : Ideal S\nhv : Irreducible v\nhw : Irreducible w\nhw_bot : w ≠ ⊥\ninst✝ : w.LiesOver v\nI : Ideal R\nhI : IsUnit I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 58,
"column": 6
} | {
"line": 58,
"column": 17
} | [
{
"pp": "case mp\nA : Type u_1\nM : Type u_2\ninst✝² : CommRing A\nI : Ideal A\ninst✝¹ : AddCommMonoid M\ninst✝ : Module A M\nx : M\na : ∃ i, x ∈ torsionBySet A M ↑(I ^ i)\n⊢ ∃ n, ∀ a ∈ I ^ n, a • x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 78,
"column": 7
} | {
"line": 78,
"column": 53
} | [
{
"pp": "A : Type u_1\nM : Type u_2\nM₁ : Type u_3\nM₂ : Type u_4\ninst✝⁶ : CommRing A\nI : Ideal A\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module A M\ninst✝¹ : Module A M₁\ninst✝ : Module A M₂\nφ : M₁ →ₗ[A] M₂\nc : ↥(primaryComponent M₁ I)\n⊢ (φ.domRestrict (pr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Prod.TProd | {
"line": 150,
"column": 8
} | {
"line": 150,
"column": 18
} | [
{
"pp": "case h\nι : Type u\nα : ι → Type v\ni : ι\nl : List ι\nt : (i : ι) → Set (α i)\nf : (i : ι) → α i\nh : TProd.mk l f ∈ Set.tprod l t ↔ ∀ (i : ι), i ∈ l → f i ∈ t i\n⊢ f ∈ TProd.mk (i :: l) ⁻¹' Set.tprod (i :: l) t ↔ f ∈ {i_1 | i_1 ∈ i :: l}.pi t",
"usedConstants": [
"Set.instSProd",
"Eq.... | Set.tprod, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 147,
"column": 32
} | {
"line": 147,
"column": 43
} | [
{
"pp": "A : Type u_1\nM : Type u_2\ninst✝³ : CommRing A\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : IsDedekindDomain A\nh : IsTorsion A M\nx : M\na : A\nha : a ∈ A⁰\nhmem : x ∈ torsionBySet A M ↑(span {a})\n⊢ span {a} ≠ ⊥",
"usedConstants": [
"Submodule.span_eq_bot._simp_1",
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 117,
"column": 33
} | {
"line": 117,
"column": 44
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nf : β → Set α\ns : Set β\nhs : s.Countable\nh : ∀ b ∈ s, MeasurableSet (f b)\nthis : Countable ↑s\n⊢ ∀ (b : ↑s), MeasurableSet (f ↑b)",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"Subtype.forall._simp_1",
"Membership.mem"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 155,
"column": 2
} | {
"line": 156,
"column": 34
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns : Set (Set α)\nhs : s.Countable\nh : ∀ t ∈ s, MeasurableSet t\n⊢ MeasurableSet (⋂₀ s)",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"congrArg",
"Set.iInter",
"Membership.mem",
"id",
"Set.sInter_eq_biInter",
"Me... | rw [sInter_eq_biInter]
exact MeasurableSet.biInter hs h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 155,
"column": 2
} | {
"line": 156,
"column": 34
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns : Set (Set α)\nhs : s.Countable\nh : ∀ t ∈ s, MeasurableSet t\n⊢ MeasurableSet (⋂₀ s)",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"congrArg",
"Set.iInter",
"Membership.mem",
"id",
"Set.sInter_eq_biInter",
"Me... | rw [sInter_eq_biInter]
exact MeasurableSet.biInter hs h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 296,
"column": 28
} | {
"line": 296,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort u_6\ns t u : Set α\nm : MeasurableSpace α\np : Set α → Prop\nh : ∀ (s : Set α), p s ↔ MeasurableSet s\n⊢ p ∅",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"id",
"propext",
"Set.instEmptyC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 297,
"column": 28
} | {
"line": 297,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort u_6\ns t u : Set α\nm : MeasurableSpace α\np : Set α → Prop\nh : ∀ (s : Set α), p s ↔ MeasurableSet s\n⊢ ∀ (s : Set α), p s → p sᶜ",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"Compl.compl",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 298,
"column": 29
} | {
"line": 298,
"column": 49
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort u_6\ns t u : Set α\nm : MeasurableSpace α\np : Set α → Prop\nh : ∀ (s : Set α), p s ↔ MeasurableSet s\n⊢ ∀ (f : ℕ → Set α), (∀ (i : ℕ), p (f i)) → p (⋃ i, f i)",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 208,
"column": 6
} | {
"line": 208,
"column": 58
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDedekindDomain A\nM₁ : Type u_5\nM₂ : Type u_6\ninst✝³ : AddCommGroup M₁\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module A M₁\ninst✝ : Module A M₂\nhM₁ : IsTorsion A M₁\nP : HeightOneSpectrum A\nφ : M₁ →ₗ[A] M₂\nhf✝ : Surjective ⇑φ\nb : M₁\nhy : φ b ∈ primaryCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Basic | {
"line": 67,
"column": 34
} | {
"line": 67,
"column": 68
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort uι\ns : Set α\nm✝ m₁ m₂ : MeasurableSpace α\nm' : MeasurableSpace β\nf✝¹ : α → β\ng : β → α\nf✝ : α → β\nm : MeasurableSpace α\nf : ℕ → Set β\nhf : ∀ (i : ℕ), MeasurableSet (f✝ ⁻¹' f i)\n⊢ MeasurableSet (f✝ ⁻¹' ⋃ i, f i)",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Basic | {
"line": 298,
"column": 17
} | {
"line": 298,
"column": 49
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nf g : α → β\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nx✝ : DecidablePred fun x ↦ x ∈ s\nhs : MeasurableSet s\nhf : Measurable f\nhg : Measurable g\nt : Set β\nht : MeasurableSet t\n⊢ MeasurableSet (s.piecewise f g ⁻¹' t)",
"usedConstants": [
"Eq.mp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 93,
"column": 57
} | {
"line": 93,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI : Ideal R\nhI : I ≠ 0\nh_fin : Fintype { x // x ∣ I }\nv w : { x // x.asIdeal ∣ I }\nhvw : (fun v ↦ ⟨(↑v).asIdeal, ⋯⟩) v = (fun v ↦ ⟨(↑v).asIdeal, ⋯⟩) w\n⊢ ((fun a ↦ ↑a) v).asIdeal = ((fun a ↦ ↑a) w).asIdeal",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 12
} | [
{
"pp": "case coe\nα : Type u_6\ninst✝ : MeasurableSpace α\nf : α → ℕ∞\nh : ∀ (n : ℕ), MeasurableSet (f ⁻¹' {↑n})\nn : ℕ\n⊢ MeasurableSet (f ⁻¹' {↑n})",
"usedConstants": []
}
] | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 20
} | [
{
"pp": "case pos\nα : Type u_1\nmα : MeasurableSpace α\nf : α → Prop\nh : MeasurableSet (f ⁻¹' {True})\nx : Prop\nhx : x\n⊢ MeasurableSet (f ⁻¹' {x})",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"congrArg",
"setOf",
"Set.instSingletonSet",
"id",
"Set.preimage",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 97,
"column": 4
} | {
"line": 98,
"column": 11
} | [
{
"pp": "case neg\nα : Type u_1\nmα : MeasurableSpace α\nf : α → Prop\nh : MeasurableSet (f ⁻¹' {True})\nx : Prop\nhx : ¬x\n⊢ MeasurableSet (f ⁻¹' {x})",
"usedConstants": [
"Eq.mpr",
"False",
"MeasurableSet",
"eq_false",
"congrArg",
"setOf",
"Set.instSingletonSet",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 35
} | [
{
"pp": "case h.e'_3.h\nα : Type u_1\nm : MeasurableSpace α\ns t : Set α\nh : s ⊆ t\nhs : MeasurableSet (Subtype.val ⁻¹' s)\nu : Set α\nhu : MeasurableSet u\nx : α\nhx : x ∈ t\n⊢ ⟨x, hx⟩ ∈ inclusion h '' Subtype.val ⁻¹' u ↔ ⟨x, hx⟩ ∈ Subtype.val ⁻¹' u ∩ Subtype.val ⁻¹' s",
"usedConstants": [
"Eq.mpr",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 317,
"column": 18
} | {
"line": 317,
"column": 46
} | [
{
"pp": "β : Type u_2\ninst✝¹ : MeasurableSpace β\ninst✝ : Countable β\nx y : β\nhy : y ∉ measurableAtom x\n⊢ ∃ s, x ∈ s ∧ MeasurableSet s ∧ y ∉ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 320,
"column": 4
} | {
"line": 320,
"column": 25
} | [
{
"pp": "β : Type u_2\ninst✝¹ : MeasurableSpace β\ninst✝ : Countable β\nx : β\ns : β → Set β\nhs : ∀ y ∉ measurableAtom x, x ∈ s y ∧ MeasurableSet (s y) ∧ y ∉ s y\n⊢ measurableAtom x = ⋂ y ∈ (measurableAtom x)ᶜ, s y",
"usedConstants": [
"Set.Subset.antisymm",
"Set.iInter",
"Compl.compl",
... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 344,
"column": 2
} | {
"line": 345,
"column": 65
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI : FractionalIdeal R⁰ K\nhI : I ≠ 0\na : R\nJ : Ideal R\nh_aJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J\na₁ : R := choose ... | have h_a₁J₁ : I = spanSingleton R⁰ ((algebraMap R K) a₁)⁻¹ * ↑J₁ :=
(choose_spec (choose_spec (exists_eq_spanSingleton_mul I))).2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 361,
"column": 56
} | {
"line": 361,
"column": 82
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI : FractionalIdeal R⁰ K\nhI : I ≠ 0\na : R\nJ : Ideal R\na₁ : R := choose ⋯\nJ₁ : Ideal R := choose ⋯\nh_aJ : ↑J₁ / spanSingleton R... | div_eq_div_iff h_a₁' h_a', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 578,
"column": 2
} | {
"line": 578,
"column": 35
} | [
{
"pp": "β : Type u_2\nδ : Type u_4\nX : δ → Type u_6\ninst✝ : (a : δ) → MeasurableSpace (X a)\ng : (a : δ) → β → X a\na : δ\n⊢ MeasurableSpace.comap (g a) inferInstance ≤ MeasurableSpace.comap (fun b c ↦ g c b) pi",
"usedConstants": [
"Eq.mpr",
"MeasurableSpace.instLE",
"MeasurableSpace.c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 76,
"column": 15
} | {
"line": 76,
"column": 61
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\nmα : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nhf : MeasurableEmbedding f\nh : MeasurableSet (f '' s)\n⊢ MeasurableSet s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 37
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\nhf : MeasurableEmbedding f\ng : α → γ\ng' : β → γ\nhg : Measurable g\nhg' : Measurable g'\n⊢ Measurable fun x ↦ g (Exists.choose ⋯)",
"usedConstants": [... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 775,
"column": 6
} | {
"line": 775,
"column": 17
} | [
{
"pp": "case pos\nδ : Type u_4\nX : δ → Type u_6\ninst✝¹ : (i : δ) → MeasurableSpace (X i)\ninst✝ : DecidableEq δ\nis : List δ\nj : δ\nhj : j ∈ j :: is\n⊢ Measurable fun v ↦ v.elim hj",
"usedConstants": [
"Eq.mpr",
"List.TProd.elim_self",
"congrArg",
"Measurable",
"List.TProd.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 332,
"column": 15
} | {
"line": 332,
"column": 56
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\ne : α ≃ᵐ β\ns : Set β\nh : MeasurableSet (⇑e ⁻¹' s)\n⊢ MeasurableSet s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 860,
"column": 4
} | {
"line": 860,
"column": 15
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\np : α → Prop\nh : Measurable p\n⊢ MeasurableSet {a | p a}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 33
} | [
{
"pp": "case h\nR : Type u\ninst✝¹⁵ : CommRing R\nS : Type v\ninst✝¹⁴ : CommRing S\ninst✝¹³ : Algebra R S\nK : Type u_1\ninst✝¹² : Field K\ninst✝¹¹ : Algebra R K\nV : Type u_3\nV' : Type u_4\nV'' : Type u_5\ninst✝¹⁰ : AddCommGroup V\ninst✝⁹ : Module R V\ninst✝⁸ : Module K V\ninst✝⁷ : IsScalarTower R K V\ninst✝... | letI := Classical.propDecidable | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 567,
"column": 43
} | {
"line": 567,
"column": 54
} | [
{
"pp": "α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α✝\ninst✝⁵ : MeasurableSpace α✝\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nπ : δ' → Type u_6\nπ' : δ' → Type u_7\ninst✝² : (x : δ') → MeasurableSpace (π x)\ninst✝¹ : (x : δ') → MeasurableSpace... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 666,
"column": 8
} | {
"line": 666,
"column": 56
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nhf : Injective f ∧ MeasurableSpace.comap f inst✝ = inst✝¹ ∧ MeasurableSet (range f)\ns : Set β\nhs : MeasurableSet s\n⊢ MeasurableSet (f '' f ⁻¹' s)",
"usedConstants": [
"Eq.mpr",
"MeasurableSe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 675,
"column": 4
} | {
"line": 676,
"column": 11
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns✝ t u✝ : Set α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → α\ns : Set α\nhf : MeasurableEmbedding f\nu : Set α\nhu : MeasurableSet u\n⊢ MeasurableSet (⇑(Equiv.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 35
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\n⊢ MeasurableSet (limsup s atTop)",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"congrArg",
"CompleteLattice.toConditionallyCompleteLattice",
"Filter.blimsup",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 35
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\n⊢ MeasurableSet (liminf s atTop)",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"_private.Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated.0.MeasurableSet.measurableSet_liminf._s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Completion | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 69
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\nβ : Type u_2\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx y : β\nh : ofRat x = ofRat y\n⊢ x = y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Completion | {
"line": 200,
"column": 8
} | {
"line": 200,
"column": 73
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\nβ : Type u_2\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx : Cauchy abv\nf g : CauSeq β abv\nfg : f ≈ g\nthis : f.LimZero ↔ g.LimZero\nhf : ¬f.LimZero\nhg : ¬g.LimZero\n⊢ mk (f.inv ... | have If : mk (inv f hf) * mk f = 1 := mk_eq.2 (inv_mul_cancel hf) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 80
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε : α\nε0 : 0 < ε\na₁ a₂ b₁ b₂ : β\nh₁ : abv (a₁ - b₁) < ε / 2\nh₂ : abv (a₂ - b₂) < ε / 2\n⊢ abv (a₁ + a₂ - (b₁ + b₂)) < ε",
"usedConstan... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 63
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε K₁ K₂ : α\nε0 : 0 < ε\na₁ a₂ b₁ b₂ : β\nM : α := max 1 (max K₁ K₂)\nK0 : 0 < M\nεK : 0 < ε / 2 / M\nh₁ : abv (a₁ - b₁) < ε / 2 / M\nh₂ : abv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 15
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\nhf : IsCauSeq abv f\ni : ℕ\nh : ∀ j ≥ i, abv (f j - f i) < 1\nR : ℕ → α := Nat.rec (abv (f 0)) fun i c ↦ max c (abv (f i.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 130,
"column": 31
} | {
"line": 130,
"column": 53
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx : β\nε : α\nε0 : ε > 0\nj : ℕ\nx✝ : j ≥ 0\n⊢ abv ((fun x_1 ↦ x) j - (fun x_1 ↦ x) 0) < ε",
"usedConstants": [
"Eq.mpr",
"Pre... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 651,
"column": 27
} | {
"line": 651,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nJ I : Ideal R\nhIJ : J * I ≤ J\nhJ : ¬J = 0\nhI : ¬I = 0\ns : Finset (HeightOneSpectrum R) := ⋯.toFinset\nthis : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \\ {p}, q.asIdeal\na : HeightOneSpectrum R → R\nha : ∀ p ∈ s, a p ∈ J * ∏ q ∈ s \... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 373,
"column": 6
} | {
"line": 373,
"column": 32
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhf : f.LimZero\nhg : g.LimZero\nε : α\nε0 : ε > 0\nx✝ : ℕ\nH : ∀ j ≥ x✝, abv (↑f j) < ε / 2 ∧ abv (↑g j) < ε / 2\nj : ℕ\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 658,
"column": 63
} | {
"line": 658,
"column": 81
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nJ I : Ideal R\nhIJ : J * I ≤ J\nhJ : ¬J = 0\nhI : ¬I = 0\ns : Finset (HeightOneSpectrum R) := ⋯.toFinset\nthis : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \\ {p}, q.asIdeal\na : HeightOneSpectrum R → R\nha : ∀ p ∈ s, a p ∈ J * ∏ q ∈ s \... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 35
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhf : f.LimZero\nhg : g.LimZero\n⊢ (f - g).LimZero",
"usedConstants": [
"CauSeq.addGroup",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 397,
"column": 2
} | {
"line": 397,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhfg : (f - g).LimZero\n⊢ (g - f).LimZero",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 400,
"column": 31
} | {
"line": 400,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nε : α\nε0 : ε > 0\nj : ℕ\nx✝ : j ≥ 0\n⊢ abv (↑0 j) < ε",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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