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.RingTheory.Unramified.Basic | {
"line": 219,
"column": 14
} | {
"line": 219,
"column": 15
} | [
{
"pp": "R : Type u_1\ninst✝¹⁰ : CommRing R\nA : Type u_2\ninst✝⁹ : CommRing A\ninst✝⁸ : Algebra R A\nB : Type u_3\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra A B\ninst✝⁴ : IsScalarTower R A B\ninst✝³ : FormallyUnramified R A\ninst✝² : FormallyUnramified A B\nC : Type u_3\ninst✝¹ : CommRing C\n... | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Unramified.Basic | {
"line": 233,
"column": 14
} | {
"line": 233,
"column": 15
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : CommRing R\nA : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nB : Type u_3\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : Algebra A B\ninst✝³ : IsScalarTower R A B\ninst✝² : FormallyUnramified R B\nQ : Type u_3\ninst✝¹ : CommRing Q\ninst✝ : Algebra A Q\n⊢ ∀ (I : Idea... | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Unramified.Basic | {
"line": 252,
"column": 14
} | {
"line": 252,
"column": 15
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : Algebra R A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : FormallyUnramified R A\nf : A →ₐ[R] B\nH : Function.Surjective ⇑f\nQ : Type u_3\ninst✝¹ : CommRing Q\ninst✝ : Algebra R Q\n⊢ ∀ (I : Ideal Q), I ^... | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Unramified.Basic | {
"line": 280,
"column": 14
} | {
"line": 280,
"column": 15
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\nA : Type u_2\ninst✝⁶ : CommRing A\ninst✝⁵ : Algebra R A\nB : Type u_3\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : FormallyUnramified R A\nC : Type (max u_2 u_3)\ninst✝¹ : CommRing C\ninst✝ : Algebra B C\n⊢ ∀ (I : Ideal C), I ^ 2 = ⊥ → Function.Injective (Idea... | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Unramified.Basic | {
"line": 304,
"column": 14
} | {
"line": 304,
"column": 15
} | [
{
"pp": "R : Type u_1\nRₘ : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing Rₘ\nM : Submonoid R\ninst✝³ : Algebra R Rₘ\ninst✝² : IsLocalization M Rₘ\nQ : Type u_3\ninst✝¹ : CommRing Q\ninst✝ : Algebra R Q\n⊢ ∀ (I : Ideal Q), I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp",
"usedConstants": [
... | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Smooth.Basic | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 57
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : FormallySmooth R A\nI : Ideal B\nhI : I ^ 2 = ⊥\nf : A →ₐ[R] B ⧸ I\n⊢ ∃ a, (Ideal.Quotient.mkₐ R I).comp a = f",
"usedConstants": [
"Algebra... | let P : Algebra.Generators R A A := Generators.self R A | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 106,
"column": 31
} | {
"line": 106,
"column": 60
} | [
{
"pp": "case a.a\nF : Type u\ninst✝ : Field F\nW : Projective F\nP : Fin 3 → F\nhPz : P z ≠ 0\n⊢ W.negY P / P z + -W.a₃ * 1 - (W.toAffine.negY (P x / P z) (P y / P z) + -W.a₃ * (P z / P z)) = 0",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
... | rw [negY, Affine.negY]; ring1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 106,
"column": 31
} | {
"line": 106,
"column": 60
} | [
{
"pp": "case a.a\nF : Type u\ninst✝ : Field F\nW : Projective F\nP : Fin 3 → F\nhPz : P z ≠ 0\n⊢ W.negY P / P z + -W.a₃ * 1 - (W.toAffine.negY (P x / P z) (P y / P z) + -W.a₃ * (P z / P z)) = 0",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
... | rw [negY, Affine.negY]; ring1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Smooth.Basic | {
"line": 131,
"column": 14
} | {
"line": 131,
"column": 15
} | [
{
"pp": "case h₁\nR : Type u\nA : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nB✝ : Type u_1\ninst✝² : CommRing B✝\ninst✝¹ : FormallySmooth R A\nI : Ideal B✝\nhI : IsNilpotent I\nB : Type u_1\ninst✝ : CommRing B\n⊢ ∀ (I : Ideal B), I ^ 2 = ⊥ → ∀ [inst : Algebra R B], Function.Surjecti... | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Smooth.Basic | {
"line": 132,
"column": 14
} | {
"line": 132,
"column": 15
} | [
{
"pp": "case h₂\nR : Type u\nA : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nB✝ : Type u_1\ninst✝² : CommRing B✝\ninst✝¹ : FormallySmooth R A\nI : Ideal B✝\nhI : IsNilpotent I\nB : Type u_1\ninst✝ : CommRing B\n⊢ ∀ (I J : Ideal B),\n I ≤ J →\n (∀ [inst : Algebra R B], Functi... | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 67
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP Q : Fin 3 → R\nhx : P x * Q z = Q x * P z\n⊢ P y * Q z - Q y * P z + (P y * Q z - W'.negY Q * P z) = (P y - W'.negY P) * Q z",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Weierst... | linear_combination (norm := (rw [negY, negY]; ring1)) -W'.a₁ * hx | Mathlib.Tactic.LinearCombination._aux_Mathlib_Tactic_LinearCombination___elabRules_Mathlib_Tactic_LinearCombination_linearCombination_1 | Mathlib.Tactic.LinearCombination.linearCombination |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 67
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP Q : Fin 3 → R\nhx : P x * Q z = Q x * P z\n⊢ P y * Q z - Q y * P z + (P y * Q z - W'.negY Q * P z) = (P y - W'.negY P) * Q z",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Weierst... | linear_combination (norm := (rw [negY, negY]; ring1)) -W'.a₁ * hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 67
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP Q : Fin 3 → R\nhx : P x * Q z = Q x * P z\n⊢ P y * Q z - Q y * P z + (P y * Q z - W'.negY Q * P z) = (P y - W'.negY P) * Q z",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Weierst... | linear_combination (norm := (rw [negY, negY]; ring1)) -W'.a₁ * hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 261,
"column": 4
} | {
"line": 261,
"column": 70
} | [
{
"pp": "case refine_1.refine_2\nR : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\nhf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥\nhf : Surjective ⇑(algebraMap P S)\n... | · intro x y hx hy; simp only [map_add, hx, hy, tmul_add, zero_add] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 250,
"column": 2
} | {
"line": 270,
"column": 46
} | [
{
"pp": "R : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\nhf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥\nhf : Surjective ⇑(algebraMap P S)\n⊢ Derivation R (P ⧸ Ring... | letI := Submodule.liftQ ((RingHom.ker (algebraMap P S) ^ 2).restrictScalars R)
(((mk P S _ 1).restrictScalars R).comp (KaehlerDifferential.D R P).toLinearMap)
refine ⟨this ?_, ?_, ?_⟩
· rintro x hx
simp only [Submodule.restrictScalars_mem, pow_two] at hx
simp only [LinearMap.mem_ker, LinearMap.coe_comp,... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 250,
"column": 2
} | {
"line": 270,
"column": 46
} | [
{
"pp": "R : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\nhf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥\nhf : Surjective ⇑(algebraMap P S)\n⊢ Derivation R (P ⧸ Ring... | letI := Submodule.liftQ ((RingHom.ker (algebraMap P S) ^ 2).restrictScalars R)
(((mk P S _ 1).restrictScalars R).comp (KaehlerDifferential.D R P).toLinearMap)
refine ⟨this ?_, ?_, ?_⟩
· rintro x hx
simp only [Submodule.restrictScalars_mem, pow_two] at hx
simp only [LinearMap.mem_ker, LinearMap.coe_comp,... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 373,
"column": 2
} | {
"line": 374,
"column": 7
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhP : W'.Equation P\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\nhy : P y * Q z = Q y * P z\nhy' : P y * Q z = W'.negY Q * P z\n⊢ W'.negDblY P * P z ^ 2 = -(eval P) W'.polynomialX ^ 3",
"usedConstants": [
... | rw [negDblY_eq' hP, Y_eq_negY_of_Y_eq hQz hx hy hy']
ring1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 373,
"column": 2
} | {
"line": 374,
"column": 7
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhP : W'.Equation P\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\nhy : P y * Q z = Q y * P z\nhy' : P y * Q z = W'.negY Q * P z\n⊢ W'.negDblY P * P z ^ 2 = -(eval P) W'.polynomialX ^ 3",
"usedConstants": [
... | rw [negDblY_eq' hP, Y_eq_negY_of_Y_eq hQz hx hy hy']
ring1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Smooth.Locus | {
"line": 66,
"column": 2
} | {
"line": 75,
"column": 69
} | [
{
"pp": "case h.a.h.e'_2.a\nR : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : EssFiniteType R A\np : PrimeSpectrum A\n⊢ Module.Projective (Localization.AtPrime p.asIdeal) Ω[Localization.AtPrime p.asIdeal⁄R] ↔\n Module.Free (Localization.AtPrime p.asIdeal) (Lo... | · trans Module.Free (Localization.AtPrime p.asIdeal) Ω[Localization.AtPrime p.asIdeal⁄R]
· have : EssFiniteType A (Localization.AtPrime p.asIdeal) :=
.of_isLocalization _ p.asIdeal.primeCompl
have : EssFiniteType R (Localization.AtPrime p.asIdeal) := .comp _ A _
exact ⟨fun _ ↦ Module.free_of_fla... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Unramified.Field | {
"line": 46,
"column": 14
} | {
"line": 46,
"column": 15
} | [
{
"pp": "K : Type u_1\nL : Type u_3\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : Algebra.IsSeparable K L\nB : Type u_3\ninst✝¹ : CommRing B\ninst✝ : Algebra K B\n⊢ ∀ (I : Ideal B), I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ K I).comp",
"usedConstants": [
"CommSemiring.to... | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Unramified.Field | {
"line": 171,
"column": 2
} | {
"line": 172,
"column": 82
} | [
{
"pp": "K : Type u_1\nL : Type u_3\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : FormallyUnramified K L\ninst✝¹ : EssFiniteType K L\ninst✝ : IsPurelyInseparable K L\nthis : Nontrivial (L ⊗[K] L)\nx : L\na✝ : x ∈ ⊤\nn : ℕ\nhn : x ^ ringExpChar K ^ n ∈ (algebraMap K L).range\n⊢ x ∈ (algebra... | have : ExpChar (L ⊗[K] L) (ringExpChar K) := by
refine expChar_of_injective_ringHom (algebraMap K _).injective (ringExpChar K) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Etale.Locus | {
"line": 62,
"column": 2
} | {
"line": 63,
"column": 68
} | [
{
"pp": "case h\nR : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\np : PrimeSpectrum A\n⊢ p ∈ etaleLocus R A ↔\n Subsingleton (LocalizedModule p.asIdeal.primeCompl Ω[A⁄R]) ∧\n Subsingleton (LocalizedModule p.asIdeal.primeCompl (H1Cotangent R A))",
"usedConsta... | have h₁ := IsLocalizedModule.iso p.asIdeal.primeCompl
(KaehlerDifferential.map R R A (Localization.AtPrime p.asIdeal)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Etale.Field | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 61
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsSeparable K L\nB : Type (max u_1 u_2)\nx✝¹ : CommRing B\nx✝ : Algebra K B\nI : Ideal B\nh✝ : I ^ 2 = ⊥\nf : L →ₐ[K] B ⧸ I\ng : (k : L) → ↥K⟮k⟯ →ₐ[K] B\nhg₁ : ∀ (k : L), (fun g ↦ (Ideal.Quotient.mkₐ K... | rw [← hg₂ _ ((g _).comp (IntermediateField.inclusion e))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 83,
"column": 41
} | {
"line": 83,
"column": 52
} | [
{
"pp": "case tmul\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : Algebra S B\ninst✝¹ : IsScalarTower R A B\ninst✝ : IsS... | smul_tmul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 83,
"column": 53
} | {
"line": 83,
"column": 64
} | [
{
"pp": "case tmul\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : Algebra S B\ninst✝¹ : IsScalarTower R A B\ninst✝ : IsS... | smul_tmul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 142,
"column": 14
} | {
"line": 142,
"column": 79
} | [
{
"pp": "case zero\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : Algebra S B\ninst✝¹ : IsScalarTower R A B\ninst✝ : IsS... | rw [map_zero, zero_smul, smul_zero, zero_add, zero_mul, map_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 142,
"column": 14
} | {
"line": 142,
"column": 79
} | [
{
"pp": "case zero\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : Algebra S B\ninst✝¹ : IsScalarTower R A B\ninst✝ : IsS... | rw [map_zero, zero_smul, smul_zero, zero_add, zero_mul, map_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 142,
"column": 14
} | {
"line": 142,
"column": 79
} | [
{
"pp": "case zero\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : Algebra S B\ninst✝¹ : IsScalarTower R A B\ninst✝ : IsS... | rw [map_zero, zero_smul, smul_zero, zero_add, zero_mul, map_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AdicCompletion.Functoriality | {
"line": 395,
"column": 2
} | {
"line": 402,
"column": 24
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nh : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)\nx : M\nn : ℕ\ny : N ⧸ I ^ n • ⊤\ny' : N ⧸ I ^ (n + 1) • ⊤\nhyy' : (factor ⋯)... | obtain ⟨y0, hy0⟩ := mkQ_surjective _ y'
have : f x ≡ y0 [SMOD (I ^ n • ⊤ : Submodule R N)] := by
rw [SModEq, ← mkQ_apply, ← mkQ_apply, ← factor_mk (pow_smul_top_le I N n.le_succ) y0,
hy0, hyy', hxy]
obtain ⟨x', hxx', hx'y0⟩ :=
exists_smodEq_pow_smul_top_and_smodEq_pow_add_one_smul_top h this
use x... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AdicCompletion.Functoriality | {
"line": 395,
"column": 2
} | {
"line": 402,
"column": 24
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nh : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)\nx : M\nn : ℕ\ny : N ⧸ I ^ n • ⊤\ny' : N ⧸ I ^ (n + 1) • ⊤\nhyy' : (factor ⋯)... | obtain ⟨y0, hy0⟩ := mkQ_surjective _ y'
have : f x ≡ y0 [SMOD (I ^ n • ⊤ : Submodule R N)] := by
rw [SModEq, ← mkQ_apply, ← mkQ_apply, ← factor_mk (pow_smul_top_le I N n.le_succ) y0,
hy0, hyy', hxy]
obtain ⟨x', hxx', hx'y0⟩ :=
exists_smodEq_pow_smul_top_and_smodEq_pow_add_one_smul_top h this
use x... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AdicCompletion.Exactness | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 71
} | [
{
"pp": "case refine_2\nR : Type u\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type w\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nhf : Function.Surjective ⇑f\ny : AdicCompletion I N\nb : AdicCauchySequence I N\na : (n : ℕ) → ↑(⇑f ⁻¹' {↑b n}) ... | exact _root_.AdicCompletion.ext fun n ↦ congrArg _ ((a n).property) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.AdicCompletion.Exactness | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 71
} | [
{
"pp": "case refine_2\nR : Type u\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type w\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nhf : Function.Surjective ⇑f\ny : AdicCompletion I N\nb : AdicCauchySequence I N\na : (n : ℕ) → ↑(⇑f ⁻¹' {↑b n}) ... | exact _root_.AdicCompletion.ext fun n ↦ congrArg _ ((a n).property) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AdicCompletion.Exactness | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 71
} | [
{
"pp": "case refine_2\nR : Type u\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type w\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nhf : Function.Surjective ⇑f\ny : AdicCompletion I N\nb : AdicCauchySequence I N\na : (n : ℕ) → ↑(⇑f ⁻¹' {↑b n}) ... | exact _root_.AdicCompletion.ext fun n ↦ congrArg _ ((a n).property) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AdicCompletion.Exactness | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 27
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommRing R\nI : Ideal R\nM : Type u\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : Type u\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R N\nf : M →ₗ[R] N\ng : N →ₗ[R] ... | simp only [map_sub, hd] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 304,
"column": 44
} | {
"line": 304,
"column": 55
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : S\na : A\n⊢ (D S (S ⊗[R] A)) (s ⊗ₜ[R] a) = (D S (S ⊗[R] A)) (s • 1 ⊗ₜ[R] a)",
"usedConstants": [
"Derivation",
"Eq.mpr",
"NonAsso... | smul_tmul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdicCompletion.Exactness | {
"line": 172,
"column": 8
} | {
"line": 172,
"column": 19
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommRing R\nI : Ideal R\nM : Type u\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : Type u\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R N\nf : M →ₗ[R] N\ng : N →ₗ[R] ... | map_smul'', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Basis.Exact | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 22
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nK : Type u_3\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup K\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R K\ninst✝ : Module R P\nf : K →ₗ[R] M\ng : M →ₗ[R] P\ns : M →ₗ[R] K\nhs : s ∘ₗ f = LinearMap.id\nhfg : Function.Exact... | generalize f y = x | Lean.Elab.Tactic.evalGeneralize | Lean.Parser.Tactic.generalize |
Mathlib.LinearAlgebra.Basis.Exact | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 99
} | [
{
"pp": "R✝ : Type u_1\nM✝ : Type u_2\nK✝ : Type u_3\nP✝ : Type u_4\ninst✝¹³ : Ring R✝\ninst✝¹² : AddCommGroup M✝\ninst✝¹¹ : AddCommGroup K✝\ninst✝¹⁰ : AddCommGroup P✝\ninst✝⁹ : Module R✝ M✝\ninst✝⁸ : Module R✝ K✝\ninst✝⁷ : Module R✝ P✝\ng✝ : M✝ →ₗ[R✝] P✝\ns✝ : M✝ →ₗ[R✝] K✝\nκ✝ : Type u_6\nσ✝ : Type u_7\nR : Ty... | simp only [Finset.mem_toRight, Finsupp.mem_support_iff, Function.comp_apply, not_imp_self] at hli | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.RingHom.Etale | {
"line": 51,
"column": 72
} | {
"line": 53,
"column": 12
} | [
{
"pp": "R : Type u_3\nS : Type u_4\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nhf : f.Etale\n⊢ f.FormallyUnramified",
"usedConstants": [
"RingHom.Etale",
"congrArg",
"Eq.mp",
"RingHom.Smooth",
"And",
"And.left",
"propext",
"RingHom.etale_iff_formal... | by
rw [etale_iff_formallyUnramified_and_smooth] at hf
exact hf.1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RingHom.Etale | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 39
} | [
{
"pp": "⊢ IsStableUnderBaseChange fun {R S} [CommRing R] [CommRing S] ↦ Etale",
"usedConstants": [
"RingHom.Etale.eq_formallyUnramified_and_smooth",
"Eq.mpr",
"CommRing",
"RingHom.Etale",
"congrArg",
"CommSemiring.toSemiring",
"RingHom",
"id",
"RingHom.... | rw [eq_formallyUnramified_and_smooth] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.RingHom.Etale | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 39
} | [
{
"pp": "⊢ PropertyIsLocal fun {R S} [CommRing R] [CommRing S] ↦ Etale",
"usedConstants": [
"RingHom.Etale.eq_formallyUnramified_and_smooth",
"Eq.mpr",
"CommRing",
"RingHom.Etale",
"congrArg",
"CommSemiring.toSemiring",
"RingHom",
"id",
"RingHom.Smooth",... | rw [eq_formallyUnramified_and_smooth] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.RingHom.Etale | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 39
} | [
{
"pp": "⊢ StableUnderComposition fun {R S} [CommRing R] [CommRing S] ↦ Etale",
"usedConstants": [
"RingHom.Etale.eq_formallyUnramified_and_smooth",
"Eq.mpr",
"CommRing",
"RingHom.Etale",
"congrArg",
"CommSemiring.toSemiring",
"RingHom",
"id",
"RingHom.S... | rw [eq_formallyUnramified_and_smooth] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Smooth.NoetherianDescent | {
"line": 101,
"column": 92
} | {
"line": 110,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nA : Type u\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nD : DescentAux A B\ni : D.vars\n⊢ ↑(D.h i).coeffs ⊆ Set.range ⇑(algebraMap (↥(subalgebra R D)) A)",
"usedConstants": [
"Subalgebra.instSetLike",
... | by
have : ((D.h i).coeffs : Set _) ⊆ ⋃ i, ((D.h i).coeffs : Set A) :=
Set.subset_iUnion_of_subset i subset_rfl
#adaptation_note /-- Before https://github.com/leanprover/lean4/pull/13166
(replacing grind's canonicalizer with a type-directed normalizer), `grind` closed this goal
without the `rw`. It is not ye... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 71,
"column": 2
} | {
"line": 79,
"column": 7
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\na : ι → K\nF : MvPolynomial ι k\nhF : Irreducible F\ni : ι\nH : AlgebraicIndependent k fun x ↦ a ↑x\n⊢ Irreducible (F.toPolynomialAdjoinImageCompl a i)",
"usedConstants": [
"Classical.typeDecida... | have : a '' {i}ᶜ = Set.range (fun x : {j | j ≠ i} ↦ a x) := by ext; simp
delta toPolynomialAdjoinImageCompl
convert!
hF.map (renameEquiv k (Equiv.optionSubtypeNe i).symm) |>.map (optionEquivLeft k _) |>.map
(Polynomial.mapAlgEquiv
(H.aevalEquiv.trans (Subalgebra.equivOfEq _ _ congr(Algebra.adjoin ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 71,
"column": 2
} | {
"line": 79,
"column": 7
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\na : ι → K\nF : MvPolynomial ι k\nhF : Irreducible F\ni : ι\nH : AlgebraicIndependent k fun x ↦ a ↑x\n⊢ Irreducible (F.toPolynomialAdjoinImageCompl a i)",
"usedConstants": [
"Classical.typeDecida... | have : a '' {i}ᶜ = Set.range (fun x : {j | j ≠ i} ↦ a x) := by ext; simp
delta toPolynomialAdjoinImageCompl
convert!
hF.map (renameEquiv k (Equiv.optionSubtypeNe i).symm) |>.map (optionEquivLeft k _) |>.map
(Polynomial.mapAlgEquiv
(H.aevalEquiv.trans (Subalgebra.equivOfEq _ _ congr(Algebra.adjoin ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 68
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\na : ι → K\nF : MvPolynomial ι k\nHF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree\nhF0 : F ≠ 0\nhFa : (aeval a) F = 0\n⊢ Irreducible F",
"usedConstants": [
... | refine ⟨fun h' ↦ (h'.map (aeval a)).ne_zero hFa, fun q₁ q₂ e ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 178,
"column": 18
} | {
"line": 178,
"column": 28
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\np : ℕ\nhp : Nat.Prime p\na : ι → K\nF : MvPolynomial ι k\nHF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree\nhF0 : F ≠ 0\nhFa : (aeval a) F = 0\nthis : ∀ (i : ι), ... | hσ' _ σ.2, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 77,
"column": 41
} | {
"line": 82,
"column": 5
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ✝ : Type u_3\nι : Type u_4\nP : Generators R S ι\nσ : Type u_5\nb : Module.Basis σ S P.toExtension.Cotangent\nD : Aux P b\n⊢ IsScalarTower P.Ring D.T S",
"usedConstants": [
"Ideal.Quotient.commSemiring... | by
refine ⟨fun x y z ↦ ?_⟩
obtain ⟨y, rfl⟩ := Ideal.Quotient.mk_surjective y
obtain ⟨z, rfl⟩ := P.algebraMap_surjective z
simp only [Algebra.smul_def, map_mul, Generators.algebraMap_apply, ← mul_assoc]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 51
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nK : Type u_4\ninst✝⁵ : Field K\ninst✝⁴ : Algebra R K\ninst✝³ : Algebra S K\ninst✝² : IsScalarTower R S K\ninst✝¹ : FaithfulSMul R S\ninst✝ : FaithfulSMul S K\nx : S\nH : Transcendental R ((algebraMap S K) x)\nth... | rw [← isStronglyTranscendental_iff_of_field] at H | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.RatFunc.Defs | {
"line": 205,
"column": 2
} | {
"line": 206,
"column": 47
} | [
{
"pp": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nP : Sort v\np q : K[X]\nf : K[X] → K[X] → P\nf0 : ∀ (p : K[X]), f p 0 = f 0 1\nH : ∀ {p q a : K[X]}, q ≠ 0 → a ≠ 0 → f (a * p) (a * q) = f p q\n⊢ (RatFunc.mk p q).liftOn' f H = f p q",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"... | rw [RatFunc.liftOn', RatFunc.liftOn_mk _ _ _ f0]
apply liftOn_condition_of_liftOn'_condition H | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.RatFunc.Defs | {
"line": 205,
"column": 2
} | {
"line": 206,
"column": 47
} | [
{
"pp": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nP : Sort v\np q : K[X]\nf : K[X] → K[X] → P\nf0 : ∀ (p : K[X]), f p 0 = f 0 1\nH : ∀ {p q a : K[X]}, q ≠ 0 → a ≠ 0 → f (a * p) (a * q) = f p q\n⊢ (RatFunc.mk p q).liftOn' f H = f p q",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"... | rw [RatFunc.liftOn', RatFunc.liftOn_mk _ _ _ f0]
apply liftOn_condition_of_liftOn'_condition H | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.IsTrivialOn | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 46
} | [
{
"pp": "case inr\nΓ : Type u_1\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nA : Type u_2\ninst✝² : CommRing A\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Algebra A K\nv : Valuation K Γ\nhv : IsTrivialOn A v\ny : K\nh0 : y ≠ 0\nhy : v y ≠ 1\nthis : ∀ (y : K), y ≠ 0 → v y ≠ 1 → 1 < v y → Transcendental A y\nhlt : v ... | rw [Transcendental, ← IsAlgebraic.inv_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.RatFunc.AsPolynomial | {
"line": 152,
"column": 68
} | {
"line": 152,
"column": 90
} | [
{
"pp": "K : Type u\ninst✝¹ : Field K\nL : Type u\ninst✝ : Field L\nf : K →+* L\na : L\nx : K⟮X⟯\nh : Polynomial.eval₂ f a x.denom = 0\n⊢ eval f a x = 0",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"instHDiv",
"RatFunc.denom",
"GroupWithZero.toDivInvMonoid",... | rw [eval, h, div_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.RatFunc.AsPolynomial | {
"line": 152,
"column": 68
} | {
"line": 152,
"column": 90
} | [
{
"pp": "K : Type u\ninst✝¹ : Field K\nL : Type u\ninst✝ : Field L\nf : K →+* L\na : L\nx : K⟮X⟯\nh : Polynomial.eval₂ f a x.denom = 0\n⊢ eval f a x = 0",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"instHDiv",
"RatFunc.denom",
"GroupWithZero.toDivInvMonoid",... | rw [eval, h, div_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.RatFunc.AsPolynomial | {
"line": 152,
"column": 68
} | {
"line": 152,
"column": 90
} | [
{
"pp": "K : Type u\ninst✝¹ : Field K\nL : Type u\ninst✝ : Field L\nf : K →+* L\na : L\nx : K⟮X⟯\nh : Polynomial.eval₂ f a x.denom = 0\n⊢ eval f a x = 0",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"instHDiv",
"RatFunc.denom",
"GroupWithZero.toDivInvMonoid",... | rw [eval, h, div_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 125,
"column": 36
} | {
"line": 125,
"column": 60
} | [
{
"pp": "case inl\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\ni : ℕ\na✝¹ : Nontrivial R\na✝ : Nontrivial S\nhp : IsIntegral R[X] 0\n⊢ IsIntegral R (coeff 0 i)",
"usedConstants": [
"IsIntegral",
"of_eq_true",
"Zero.toOfNat0",
"CommRing.t... | · simp [isIntegral_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 164,
"column": 57
} | {
"line": 168,
"column": 100
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nf : S[X]\n⊢ IsIntegral R[X] f ↔ ∀ (n : ℕ), IsIntegral R (f.coeff n)",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"NonUnitalCommRing.toNonUn... | by
refine ⟨IsIntegral.coeff, fun H ↦ ?_⟩
rw [← f.sum_monomial_eq, Polynomial.sum]
simp only [← C_mul_X_pow_eq_monomial, ← map_X (algebraMap R S)]
exact .sum _ fun i _ ↦ ((H i).map (CAlgHom (R := R))).tower_top.mul (.pow isIntegral_algebraMap _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 56,
"column": 68
} | {
"line": 73,
"column": 29
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nq : Ideal S\ninst✝ : q.IsPrime\n⊢ WeaklyQuasiFiniteAt R q ↔\n QuasiFinite R (Localization.AtPrime q ⧸ Ideal.map (algebraMap R (Localization.AtPrime q)) (Ideal.under R q))",
"usedConstants": [
"Algeb... | by
let q' := q.map (Ideal.Quotient.mk ((q.under R).map (algebraMap R S)))
have hq' : q = q'.comap (Ideal.Quotient.mk _) := .trans
(by simp [← RingHom.ker_eq_comap_bot, Ideal.map_comap_le])
(Ideal.comap_map_of_surjective _ Ideal.Quotient.mk_surjective _).symm
let φ₁ : Localization.AtPrime q →ₐ[R] Localizat... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 30
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : CommRing T\ninst✝² : Algebra R T\np : Ideal R\nq : Ideal S\ninst✝¹ : q.IsPrime\ninst✝ : QuasiFiniteAt R q\n⊢ WeaklyQuasiFiniteAt R q",
"usedConstants": [
"Eq.mpr",
"Algebra... | rw [weaklyQuasiFiniteAt_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 180,
"column": 37
} | {
"line": 180,
"column": 85
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal S\ninst✝⁴ : p.IsPrime\ninst✝³ : QuasiFiniteAt R (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R S) (Ideal.under R p))) p)\nA : Type u_4\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nq : Ideal (A ⊗... | ← RingHom.ker_coe_toRingHom (F := AlgHom _ _ _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 23
} | [
{
"pp": "case refine_2\nX : Scheme\nU : X.AffineZariskiSite\ns : Set ↑Γ(X, U.toOpens)\nx✝¹ : X.AffineZariskiSite\nf✝ : ↑Γ(X, U.toOpens)\nY : X.AffineZariskiSite\nf : ↑Γ(X, U.toOpens)\nhfs : f ∈ s\nhV : IsAffineOpen (X.basicOpen ↑⟨f, hfs⟩)\nx✝ : ⟨X.basicOpen ↑⟨f, hfs⟩, hV⟩ ⟶ U\nhf : X.basicOpen f✝ = toOpens ⟨X.b... | exact ⟨f, hfs, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 56
} | [
{
"pp": "case mp\nX : Scheme\nU : X.AffineZariskiSite\ns : Set ↑Γ(X, U.toOpens)\nV : X.Opens\nhV : IsAffineOpen V\nf : ⟨V, hV⟩ ⟶ U\n⊢ (Sieve.generate (U.presieveOfSections s)).arrows f → ∃ f ∈ s, ∃ g, X.basicOpen (f * g) = toOpens ⟨V, hV⟩",
"usedConstants": [
"AlgebraicGeometry.Scheme.AffineZariskiSit... | rintro ⟨⟨W, hW⟩, ⟨f₁, hf₁⟩, -, ⟨f₂, hf₂s, rfl⟩, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 795,
"column": 4
} | {
"line": 795,
"column": 74
} | [
{
"pp": "case neg\nk : Type u_3\nK : Type u_4\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : Algebra.IsAlgebraic k K\nhf : ¬Module.Finite k⟮X⟯ K⟮X⟯\nhf' : ¬Module.Finite k K\n⊢ Module.finrank k⟮X⟯ K⟮X⟯ = Module.finrank k K",
"usedConstants": [
"Eq.mpr",
"IsNoetherianRing.stro... | rw [Module.finrank_of_not_finite hf, Module.finrank_of_not_finite hf'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 241,
"column": 8
} | {
"line": 241,
"column": 76
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP : StandardEtalePair R\n⊢ Ideal.span {(Bivariate.equivMvPolynomial R) (C P.f), (Bivariate.equivMvPolynomial R) (Y * C P.g - 1)} =\n Ideal.map (↑(Bivari... | simp only [Ideal.map_span, Set.image_insert_eq, Set.image_singleton] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.DegreeLT | {
"line": 161,
"column": 7
} | {
"line": 161,
"column": 25
} | [
{
"pp": "case h\nR : Type u_1\ninst✝ : CommRing R\nr : R\nn : ℕ\nx✝ : R[X]\n⊢ x✝ ∈ Submodule.comap (↑↑(taylorEquiv r)) R[X]_n ↔ x✝ ∈ R[X]_n",
"usedConstants": [
"Polynomial.degreeLT",
"NegZeroClass.toNeg",
"Submodule",
"MulOne.toOne",
"Function.LeftInverse",
"Semiring.toM... | simp [taylorEquiv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 257,
"column": 4
} | {
"line": 257,
"column": 28
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nφ : S[X] →ₐ[R] B\nhφ : Function.Surjective ⇑φ\nf : S[X]\nhf : f.Monic\nhf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)\nhfx : RingHom.ker φ.toRingHom = Ideal.spa... | simpa [ψ] using hy.map ψ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 206,
"column": 6
} | {
"line": 208,
"column": 77
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np : R[X]\nht : φ.IsIntegralElem t\nhp : φ p * t ∈ φ.range\na : R := p.leadingCoeff\nR' : Type u_1 := Localization.Away a\nS' : Type u_2 := Localization.Away ((algebraMap R S) a)\nthis✝ : ... | obtain ⟨q, hqm, hq⟩ := ht
refine ⟨q.map (mapRingHom (algebraMap _ _)), hqm.map _, ?_⟩
rw [eval₂_map, H, ← hom_eval₂, ← AlgHom.toRingHom_eq_coe, hq, map_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 206,
"column": 6
} | {
"line": 208,
"column": 77
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np : R[X]\nht : φ.IsIntegralElem t\nhp : φ p * t ∈ φ.range\na : R := p.leadingCoeff\nR' : Type u_1 := Localization.Away a\nS' : Type u_2 := Localization.Away ((algebraMap R S) a)\nthis✝ : ... | obtain ⟨q, hqm, hq⟩ := ht
refine ⟨q.map (mapRingHom (algebraMap _ _)), hqm.map _, ?_⟩
rw [eval₂_map, H, ← hom_eval₂, ← AlgHom.toRingHom_eq_coe, hq, map_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 457,
"column": 8
} | {
"line": 457,
"column": 40
} | [
{
"pp": "case e_a\nn : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multis... | refine Multiset.map_congr rfl ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 586,
"column": 6
} | {
"line": 592,
"column": 56
} | [
{
"pp": "n : ℕ\nIH :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal S) [inst_3 : p.IsPrime]\n [WeaklyQuasiFiniteAt R p] (f : MvPolynomial (Fin n) R →ₐ[R] S), f.Finite → ZariskisMainProperty R p\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : ... | let e : Localization.AtPrime (p.under R') ≃ₐ[R] Localization.AtPrime p :=
.ofBijective (IsScalarTower.toAlgHom _ _ _) <| by
refine Localization.localRingHom_bijective_of_saturated_inf_eq_top _ ?_ _
rw [← top_le_iff, ← hs, Algebra.adjoin_le_iff]
intro x hx
refine ⟨r ^ (s.s... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 259,
"column": 30
} | {
"line": 259,
"column": 68
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\nU : Y.Opens\nhU : IsAffineOpen U\nthis : Algebra ↑Γ(Y, U) ↑Γ(X, f ⁻¹ᵁ U) := (CommRingCat.Hom.hom (app f U)).toAlgebra\n⊢ fromNormalization f ⁻¹ᵁ U ≤ opensRange ((normalizationOpenCover f).f ⟨U, hU⟩)",
"usedConstants": [
... | by simp [← fromNormalization_preimage] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 464,
"column": 14
} | {
"line": 464,
"column": 72
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nn m k : ℕ\nhn : n = m + k\np : MonicDegreeEq R n\nq : { q // ↑q.1 * ↑q.2 = map (algebraMap R S) ↑p }\nthis✝² : Algebra (MvPolynomial (Fin n) R) (MvPolynomi... | simp [MvPolynomial.mapEquivMonic_symm_map_algebraMap]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 464,
"column": 14
} | {
"line": 464,
"column": 72
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nn m k : ℕ\nhn : n = m + k\np : MonicDegreeEq R n\nq : { q // ↑q.1 * ↑q.2 = map (algebraMap R S) ↑p }\nthis✝² : Algebra (MvPolynomial (Fin n) R) (MvPolynomi... | simp [MvPolynomial.mapEquivMonic_symm_map_algebraMap]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 284,
"column": 2
} | {
"line": 284,
"column": 42
} | [
{
"pp": "case h\nX Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nV : { x // QuasiFiniteAt f x } → (normalization f).Opens\nhxV✝ : ∀ (x : { x // QuasiFiniteAt f x }), (toNormalization f) ↑x ∈ V x\nhV : ∀ (x : { x // QuasiFiniteAt f x }), IsIso (toNormaliza... | have := hr.isLocalization_stalk ⟨x, hxV⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.ConcreteCategory.WithAlgebraicStructures | {
"line": 79,
"column": 8
} | {
"line": 79,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Ring R\nJ : Type w\ninst✝³ : Category.{r, w} J\nF : J ⥤ ModuleCat R\ninst✝² : PreservesColimit F (forget (ModuleCat R))\ninst✝¹ : IsFiltered J\ninst✝ : HasColimit F\nr : R\nj : J\nx : ToType (F.obj j)\nhx : (ModuleCat.Hom.hom (colimit.ι F j)) (r • x) = 0\nj' : J\ni : j ⟶ j'\nh : ... | ModuleCat.comp_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.ConcreteCategory.WithAlgebraicStructures | {
"line": 79,
"column": 8
} | {
"line": 79,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Ring R\nJ : Type w\ninst✝³ : Category.{r, w} J\nF : J ⥤ ModuleCat R\ninst✝² : PreservesColimit F (forget (ModuleCat R))\ninst✝¹ : IsFiltered J\ninst✝ : HasColimit F\nr : R\nj : J\nx : ToType (F.obj j)\nhx : (ModuleCat.Hom.hom (colimit.ι F j)) (r • x) = 0\nj' : J\ni : j ⟶ j'\nh : ... | ModuleCat.comp_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.FlatRank | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 92
} | [
{
"pp": "X S Y T : Scheme\nf : X ⟶ S\ninst✝³ : IsAffine S\ninst✝² : IsAffine T\nf' : Y ⟶ T\ng' : Y ⟶ X\ng : T ⟶ S\nh : IsPullback g' f' f g\ninst✝¹ : Flat f\ninst✝ : IsFinite f\nt : ↥T\nthis : IsAffine X\n⊢ finrank f' t = finrank f (g t)",
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopolog... | have : IsPushout f.appTop g.appTop g'.appTop f'.appTop := isPushout_appTop_of_isPullback h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Morphisms.FlatRank | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 68
} | [
{
"pp": "X S T : Scheme\nf : X ⟶ S\ng : T ⟶ S\ninst✝² : IsAffine T\nt : ↥T\ninst✝¹ : Flat f\ninst✝ : IsFinite f\ni : Spec (S.affineOpenCover.X (S.affineOpenCover.idx (g t))) ⟶ S := S.affineOpenCover.f (S.affineOpenCover.idx (g t))\ny : ↥(pullback (S.affineOpenCover.f (S.affineOpenCover.idx (g t))) g)\nhyl : (pu... | obtain ⟨R, u, hu, z, rfl⟩ := (pullback i g).exists_Spec_apply_eq y | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Bicategory.Adjunction.Mate | {
"line": 465,
"column": 48
} | {
"line": 468,
"column": 12
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b : B\nl : a ⟶ b\nr : b ⟶ a\nadj : l ⊣ r\nl' : a ⟶ b\nr' : b ⟶ a\nadj' : l' ⊣ r'\nφ : l' ⟶ l\n⊢ (conjugateEquiv adj ((Adjunction.id a).comp adj')) ((λ_ l').hom ≫ φ) = (conjugateEquiv adj adj') φ ≫ (ρ_ r').inv",
"usedConstants": [
"Eq.mpr",
"CategoryTh... | by
simp only [conjugateEquiv_apply, mateEquiv_id_comp_right,
id_whiskerLeft, Category.assoc, Iso.inv_hom_id_assoc]
bicategory | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.GradedAlgebra.FiniteType | {
"line": 45,
"column": 2
} | {
"line": 53,
"column": 57
} | [
{
"pp": "S : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : CommRing S\ninst✝³ : SetLike σ S\ninst✝² : AddSubgroupClass σ S\n𝒜 : ι → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) S\n⊢ ∃ s, Algebra.adjoin ↥(𝒜 0) ↑s = ⊤ ∧ ∀ i ∈ s, ∃ n, n ≠ 0 ∧ ... | obtain ⟨s, h₁, h₂⟩ := exists_finset_adjoin_eq_top_and_homogeneous 𝒜
choose! n hn using h₂
refine ⟨s.filter (n · ≠ 0), ?_, by simpa using fun i hi hin ↦ ⟨n i, hin, hn i hi⟩⟩
rw [← top_le_iff, ← h₁, Algebra.adjoin_le_iff]
rintro i hi
by_cases hi0 : n i = 0
· exact Subalgebra.algebraMap_mem
(Algebra.adj... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.GradedAlgebra.FiniteType | {
"line": 45,
"column": 2
} | {
"line": 53,
"column": 57
} | [
{
"pp": "S : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : CommRing S\ninst✝³ : SetLike σ S\ninst✝² : AddSubgroupClass σ S\n𝒜 : ι → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) S\n⊢ ∃ s, Algebra.adjoin ↥(𝒜 0) ↑s = ⊤ ∧ ∀ i ∈ s, ∃ n, n ≠ 0 ∧ ... | obtain ⟨s, h₁, h₂⟩ := exists_finset_adjoin_eq_top_and_homogeneous 𝒜
choose! n hn using h₂
refine ⟨s.filter (n · ≠ 0), ?_, by simpa using fun i hi hin ↦ ⟨n i, hin, hn i hi⟩⟩
rw [← top_le_iff, ← h₁, Algebra.adjoin_le_iff]
rintro i hi
by_cases hi0 : n i = 0
· exact Subalgebra.algebraMap_mem
(Algebra.adj... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.LocalRing.LocalSubring | {
"line": 91,
"column": 25
} | {
"line": 91,
"column": 61
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\nK : Type u_3\ninst✝² : Field K\nA✝ : Subring K\nP✝ : Ideal ↥A✝\ninst✝¹ : P✝.IsPrime\nA : Subring K\nP : Ideal ↥A\ninst✝ : P.IsPrime\n⊢ ∀ (y : ↥P.primeCompl), IsUnit (A.subtype ↑y)",
"usedConstants": [
"GroupWithZero.toMonoi... | simp [Ideal.primeCompl, not_imp_not] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.LocalRing.LocalSubring | {
"line": 91,
"column": 25
} | {
"line": 91,
"column": 61
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\nK : Type u_3\ninst✝² : Field K\nA✝ : Subring K\nP✝ : Ideal ↥A✝\ninst✝¹ : P✝.IsPrime\nA : Subring K\nP : Ideal ↥A\ninst✝ : P.IsPrime\n⊢ ∀ (y : ↥P.primeCompl), IsUnit (A.subtype ↑y)",
"usedConstants": [
"GroupWithZero.toMonoi... | simp [Ideal.primeCompl, not_imp_not] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.LocalRing.LocalSubring | {
"line": 91,
"column": 25
} | {
"line": 91,
"column": 61
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\nK : Type u_3\ninst✝² : Field K\nA✝ : Subring K\nP✝ : Ideal ↥A✝\ninst✝¹ : P✝.IsPrime\nA : Subring K\nP : Ideal ↥A\ninst✝ : P.IsPrime\n⊢ ∀ (y : ↥P.primeCompl), IsUnit (A.subtype ↑y)",
"usedConstants": [
"GroupWithZero.toMonoi... | simp [Ideal.primeCompl, not_imp_not] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 32
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\ninst✝² : F.IsOneHypercoverDense J₀ J\ninst✝¹ : F.IsLocallyFull J\ninst✝ : F.IsLocallyFaithful J\nh : ∀ {X₀ : C₀} {S₀ : Sieve X₀}, Sieve.functorPushf... | rintro Y _ ⟨_, a, _, h, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Sites.Point.Basic | {
"line": 244,
"column": 94
} | {
"line": 250,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝⁷ : Category.{v, u} C\nJ : GrothendieckTopology C\nΦ : J.Point\nA : Type u'\ninst✝⁶ : Category.{v', u'} A\ninst✝⁵ : HasColimitsOfSize.{w, w, v', u'} A\nFC : A → A → Type u_1\nCC : A → Type w'\ninst✝⁴ : (X Y : A) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝³ : ConcreteCategory A FC\nP Q : Cᵒ... | by
rintro q₁ q₂ h
obtain ⟨X, x, p₁, p₂, rfl, rfl⟩ := Φ.toPresheafFiber_jointly_surjective₂ q₁ q₂
simp only [toPresheafFiber_naturality_apply, toPresheafFiber_eq_iff'] at h
obtain ⟨Y, g, y, rfl, h⟩ := h
simp only [← NatTrans.naturality_apply] at h
simpa using this _ y _ _ h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Sites.QuasiCompact | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 26
} | [
{
"pp": "case h₂\nP Q : MorphismProperty Scheme\nh : P ≤ Q\n⊢ precoverage P ≤ precoverage Q",
"usedConstants": [
"AlgebraicGeometry.Scheme.precoverage_mono"
]
}
] | exact precoverage_mono h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Sites.EffectiveEpimorphic | {
"line": 159,
"column": 2
} | {
"line": 162,
"column": 58
} | [
{
"pp": "case mp\nC : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf : Y ⟶ X\n⊢ (Presieve.singleton f).EffectiveEpimorphic → EffectiveEpi f",
"usedConstants": [
"CategoryTheory.Over",
"CategoryTheory.Sieve.generateSingleton_eq",
"congrArg",
"CategoryTheory.effectiveEpiStructOfIsColimi... | · intro (h : Nonempty _)
rw [Sieve.generateSingleton_eq] at h
constructor
apply Nonempty.map (effectiveEpiStructOfIsColimit _) h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Sites.ConstantSheaf | {
"line": 80,
"column": 6
} | {
"line": 80,
"column": 79
} | [
{
"pp": "case refine_2.refine_1\nT : Type v\ninst✝ : TopologicalSpace T\nR S : CommRingCat\nf : R ⟶ S\nhf₁ : Flat (Spec.map f)\nhf₂ : Surjective (Spec.map f)\nthis : Topology.IsQuotientMap ⇑(Spec.map f)\nx : C(↥(Spec S), T)\nh :\n ∀ {Z : Scheme} (p₁ p₂ : Z ⟶ Spec S),\n p₁ ≫ Spec.map f = p₂ ≫ Spec.map f →\n ... | obtain ⟨c, rfl, rfl⟩ := Scheme.Pullback.exists_preimage_pullback a b hfab | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 142,
"column": 6
} | {
"line": 143,
"column": 72
} | [
{
"pp": "σ : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : SetLike σ A\ninst✝² : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) A\nx : Finset A\nhx : Algebra.adjoin ↥(𝒜 0) ↑x = ⊤\nd : (i : A) → i ∈ x → ℕ\nhd : ∀ (i : A) (a : i ∈ x), d i a ≠ 0\nhxd : ∀ (i ... | ← MorphismProperty.cancel_left_of_respectsIso (P := @LocallyOfFiniteType)
(Proj.basicOpenIsoSpec 𝒜 (i : A) (hxd _ i.2) (hd _ i.2).bot_lt).inv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 213,
"column": 41
} | {
"line": 213,
"column": 90
} | [
{
"pp": "σ : Type u_1\nA : Type u_2\ninst✝¹⁰ : CommRing A\ninst✝⁹ : SetLike σ A\ninst✝⁸ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝⁷ : GradedRing 𝒜\nO : Type u_3\ninst✝⁶ : CommRing O\ninst✝⁵ : IsDomain O\ninst✝⁴ : ValuationRing O\nK : Type u_4\ninst✝³ : Field K\ninst✝² : Algebra O K\ninst✝¹ : IsFractionRing O K\... | simpa using Finset.max'_mem (Finset.univ.image ψ) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 213,
"column": 41
} | {
"line": 213,
"column": 90
} | [
{
"pp": "σ : Type u_1\nA : Type u_2\ninst✝¹⁰ : CommRing A\ninst✝⁹ : SetLike σ A\ninst✝⁸ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝⁷ : GradedRing 𝒜\nO : Type u_3\ninst✝⁶ : CommRing O\ninst✝⁵ : IsDomain O\ninst✝⁴ : ValuationRing O\nK : Type u_4\ninst✝³ : Field K\ninst✝² : Algebra O K\ninst✝¹ : IsFractionRing O K\... | simpa using Finset.max'_mem (Finset.univ.image ψ) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 213,
"column": 41
} | {
"line": 213,
"column": 90
} | [
{
"pp": "σ : Type u_1\nA : Type u_2\ninst✝¹⁰ : CommRing A\ninst✝⁹ : SetLike σ A\ninst✝⁸ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝⁷ : GradedRing 𝒜\nO : Type u_3\ninst✝⁶ : CommRing O\ninst✝⁵ : IsDomain O\ninst✝⁴ : ValuationRing O\nK : Type u_4\ninst✝³ : Field K\ninst✝² : Algebra O K\ninst✝¹ : IsFractionRing O K\... | simpa using Finset.max'_mem (Finset.univ.image ψ) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Colim | {
"line": 134,
"column": 4
} | {
"line": 137,
"column": 68
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝⁴ : Category.{v, u} C\nJ : Type u'\ninst✝³ : Category.{v', u'} J\ninst✝² : HasColimitsOfShape J C\ninst✝¹ : HasExactColimitsOfShape J C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex (J ⥤ C)\nhS : S.Exact\nc₁ : Cocone S.X₁\nhc₁ : IsColimit c₁\nc₂ : Cocone S.X₂\nhc₂ : IsCo... | dsimp
rw [IsColimit.comp_coconePointUniqueUpToIso_hom_assoc,
colimit.cocone_ι, ι_colimMap, reassoc_of% (hg j),
IsColimit.comp_coconePointUniqueUpToIso_hom, colimit.cocone_ι] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Colim | {
"line": 134,
"column": 4
} | {
"line": 137,
"column": 68
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝⁴ : Category.{v, u} C\nJ : Type u'\ninst✝³ : Category.{v', u'} J\ninst✝² : HasColimitsOfShape J C\ninst✝¹ : HasExactColimitsOfShape J C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex (J ⥤ C)\nhS : S.Exact\nc₁ : Cocone S.X₁\nhc₁ : IsColimit c₁\nc₂ : Cocone S.X₂\nhc₂ : IsCo... | dsimp
rw [IsColimit.comp_coconePointUniqueUpToIso_hom_assoc,
colimit.cocone_ι, ι_colimMap, reassoc_of% (hg j),
IsColimit.comp_coconePointUniqueUpToIso_hom, colimit.cocone_ι] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Subcanonical | {
"line": 282,
"column": 4
} | {
"line": 282,
"column": 28
} | [
{
"pp": "case refine_2.h\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝¹ : J.Subcanonical\nι : Type u_1\nX : ι → C\nc : Cofan X\nH : Sieve.ofArrows X c.inj ∈ J c.pt\ninst✝ : ∀ (i : ι), Mono (c.inj i)\nhempty : ∀ (Y : C) (a : IsInitial Y), ⊥ ∈ J Y\nhdisj : ∀ {i j : ι}, i ≠ j → ∀ {Y : ... | ext Z (g : Z.unop ⟶ X j) | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Subobject | {
"line": 52,
"column": 45
} | {
"line": 56,
"column": 77
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝¹ : SmallCategory J\nF : J ⥤ MonoOver X\ninst✝ : IsFiltered J\nc : Cocone (F ⋙ MonoOver.forget X ⋙ Over.forget X)\nhc : IsColimit c\nf : c.pt ⟶ X\nhf : ∀ (j : J), c.ι.app j ≫ ... | by
let α : F ⋙ MonoOver.forget _ ⋙ Over.forget _ ⟶ (Functor.const _).obj X :=
{ app j := (F.obj j).obj.hom }
have := NatTrans.mono_of_mono_app α
exact colim.map_mono' α hc (isColimitConstCocone J X) f (by simpa using hf) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ColimCoyoneda | {
"line": 125,
"column": 2
} | {
"line": 126,
"column": 55
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝¹ : SmallCategory J\nY : J ⥤ C\nc : Cocone Y\nhc : IsColimit c\nκ : Cardinal.{w}\nhκ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhXκ : HasCardinalLT (Subobject X) κ\nj... | simpa only [← cancel_epi ((kernel.ι (g y)).app j), comp_zero]
using NatTrans.congr_app (kernel.condition (g y)) j | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.Interval.Set.SuccOrder | {
"line": 50,
"column": 36
} | {
"line": 52,
"column": 28
} | [
{
"pp": "J : Type u_1\ninst✝¹ : PartialOrder J\ninst✝ : PredOrder J\nj : J\ni : ↑(Ici j)\nhi : ¬IsMin i\n⊢ Order.pred ↑i ∈ Ici j",
"usedConstants": [
"Eq.mpr",
"Subtype.coe_prop",
"Set.Ici",
"congrArg",
"Set.ordConnected_Ici",
"PartialOrder.toPreorder",
"Membership.... | by
rw [← coe_pred_of_not_isMin hi]
apply Subtype.coe_prop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Set.SuccOrder | {
"line": 53,
"column": 2
} | {
"line": 54,
"column": 38
} | [
{
"pp": "J : Type u_1\ninst✝¹ : PartialOrder J\ninst✝ : PredOrder J\nj : J\ni : ↑(Ici j)\nhi : ¬IsMin i\n⊢ Order.pred i = ⟨Order.pred ↑i, ⋯⟩",
"usedConstants": [
"Set.Ici",
"Set.Ici.pred_eq_of_not_isMin._proof_1",
"congrArg",
"Set.ordConnected_Ici",
"PartialOrder.toPreorder",
... | ext
simp only [coe_pred_of_not_isMin hi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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