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.Kaehler.JacobiZariski | {
"line": 166,
"column": 2
} | {
"line": 178,
"column": 98
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\n⊢ LinearMap.fst T Q.toExtension.Cota... | classical
apply (Q.comp P).cotangentSpaceBasis.ext
intro i
apply Q.cotangentSpaceBasis.repr.injective
ext j
simp only [compEquiv, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, ofComp_val,
LinearEquiv.trans_apply, Basis.repr_self, LinearMap.fst_apply, repr_CotangentSpaceMap]
obtain (i | i... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 166,
"column": 2
} | {
"line": 178,
"column": 98
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\n⊢ LinearMap.fst T Q.toExtension.Cota... | classical
apply (Q.comp P).cotangentSpaceBasis.ext
intro i
apply Q.cotangentSpaceBasis.repr.injective
ext j
simp only [compEquiv, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, ofComp_val,
LinearEquiv.trans_apply, Basis.repr_self, LinearMap.fst_apply, repr_CotangentSpaceMap]
obtain (i | i... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 424,
"column": 4
} | {
"line": 424,
"column": 93
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nι' : Type w₃\nσ : Type w₂\nσ' : Type w₄\nQ : Generators S T ι\nP : Generators R S σ\nQ' : Gen... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 398,
"column": 6
} | {
"line": 399,
"column": 45
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nx : ↥P.ker\nhx : P.toInfinitesimal.toRingHom ↑x ∈ ⊥\n⊢ ↑x ∈ P.ker ^ 2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 417,
"column": 4
} | {
"line": 418,
"column": 89
} | [
{
"pp": "case right\nR : Type u_1\nS : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Extension R S\nx : P.Cotangent\nhx : (Cotangent.map P.toInfinitesimal) x ∈ P.infinitesimal.cotangentComplex.ker\n⊢ x ∈ P.cotangentComplex.ker",
"usedConstants": [
"Eq.mpr",
"Submod... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Locus | {
"line": 155,
"column": 16
} | {
"line": 155,
"column": 27
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : FinitePresentation R A\np : Ideal A\ninst✝¹ : p.IsPrime\ninst✝ : IsSmoothAt R p\nf : A\nhxf : { asIdeal := p, isPrime := inst✝¹ } ∈ ↑(basicOpen f)\nhf : ↑(basicOpen f) ⊆ smoothLocus R A\n⊢ f ∉ p",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RingHom.Unramified | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 15
} | [
{
"pp": "R S : Type u_3\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\ns : Set S\nhs : Ideal.span s = ⊤\nH :\n ∀ (r : ↑s), (fun {R S} [CommRing R] [CommRing S] ↦ FormallyUnramified) ((algebraMap S (Localization.Away ↑r)).comp f)\nalgInst✝ : Algebra R S := f.toAlgebra\nx : PrimeSpectrum S\n⊢ ∃ r ∈ s, x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 552,
"column": 39
} | {
"line": 552,
"column": 73
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ≠ Q x * P z\n⊢ ?m.162 ≠ ?m.163",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"congrArg",
"Field.toDivisionRing",
"Divisi... | by rwa [ne_eq, ← X_eq_iff hPz hQz] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Etale.Kaehler | {
"line": 57,
"column": 2
} | {
"line": 58,
"column": 61
} | [
{
"pp": "case h.e'_3\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\ninst✝ : Algebra.FormallyEtale S T\n⊢ ⇑(lift (↑S ((Algebra.linearMap T (Module.End T (Ω[S⁄... | change _ = ((tensorKaehlerEquivOfFormallyEtale
R S T).toLinearMap.restrictScalars S : T ⊗[S] Ω[S⁄R] → _) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.Etale.Kaehler | {
"line": 284,
"column": 7
} | {
"line": 284,
"column": 18
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\nM : Submonoid S\ninst✝ : IsLocalization M T\nP : Extension R S := (Generators.self R S).toExtension\nM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Pi | {
"line": 59,
"column": 34
} | {
"line": 59,
"column": 45
} | [
{
"pp": "R : Type u_1\nI : Type u_2\nA : I → Type u_3\ninst✝³ : CommRing R\ninst✝² : (i : I) → CommRing (A i)\ninst✝¹ : (i : I) → Algebra R (A i)\ninst✝ : Finite I\nval✝ : Fintype I\nH : ∀ (i : I), FormallySmooth R (A i)\nB : Type (max u_1 u_2 u_3)\nx✝¹ : CommRing B\nx✝ : Algebra R B\nJ : Ideal B\nhJ : J ^ 2 = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.Kaehler | {
"line": 357,
"column": 7
} | {
"line": 357,
"column": 18
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\nM : Submonoid S\ninst✝ : IsLocalization M T\nx : H1Cotangent R S\nP : Extension R S := (Generators.sel... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Pi | {
"line": 104,
"column": 72
} | {
"line": 104,
"column": 83
} | [
{
"pp": "R : Type u_1\nI : Type u_2\nA : I → Type u_3\ninst✝³ : CommRing R\ninst✝² : (i : I) → CommRing (A i)\ninst✝¹ : (i : I) → Algebra R (A i)\ninst✝ : Finite I\nval✝ : Fintype I\nH : ∀ (i : I), FormallySmooth R (A i)\nB : Type (max u_1 u_2 u_3)\nx✝² : CommRing B\nx✝¹ : Algebra R B\nJ : Ideal B\nhJ : J ^ 2 =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Pi | {
"line": 51,
"column": 2
} | {
"line": 119,
"column": 37
} | [
{
"pp": "R : Type u_1\nI : Type u_2\nA : I → Type u_3\ninst✝³ : CommRing R\ninst✝² : (i : I) → CommRing (A i)\ninst✝¹ : (i : I) → Algebra R (A i)\ninst✝ : Finite I\n⊢ FormallySmooth R ((i : I) → A i) ↔ ∀ (i : I), FormallySmooth R (A i)",
"usedConstants": [
"Ideal.Quotient.isScalarTower",
"AddGro... | classical
cases nonempty_fintype I
constructor
· exact fun _ ↦ of_pi A
· refine fun H ↦ .of_comp_surjective fun B _ _ J hJ g ↦ ?_
have hJ' (x) (hx : x ∈ RingHom.ker (Ideal.Quotient.mk J)) : IsNilpotent x := by
refine ⟨2, show x ^ 2 ∈ (⊥ : Ideal B) from ?_⟩
rw [← hJ]
exact Ideal.pow_mem_pow... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.RingTheory.Smooth.Pi | {
"line": 51,
"column": 2
} | {
"line": 119,
"column": 37
} | [
{
"pp": "R : Type u_1\nI : Type u_2\nA : I → Type u_3\ninst✝³ : CommRing R\ninst✝² : (i : I) → CommRing (A i)\ninst✝¹ : (i : I) → Algebra R (A i)\ninst✝ : Finite I\n⊢ FormallySmooth R ((i : I) → A i) ↔ ∀ (i : I), FormallySmooth R (A i)",
"usedConstants": [
"Ideal.Quotient.isScalarTower",
"AddGro... | classical
cases nonempty_fintype I
constructor
· exact fun _ ↦ of_pi A
· refine fun H ↦ .of_comp_surjective fun B _ _ J hJ g ↦ ?_
have hJ' (x) (hx : x ∈ RingHom.ker (Ideal.Quotient.mk J)) : IsNilpotent x := by
refine ⟨2, show x ^ 2 ∈ (⊥ : Ideal B) from ?_⟩
rw [← hJ]
exact Ideal.pow_mem_pow... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Smooth.Pi | {
"line": 51,
"column": 2
} | {
"line": 119,
"column": 37
} | [
{
"pp": "R : Type u_1\nI : Type u_2\nA : I → Type u_3\ninst✝³ : CommRing R\ninst✝² : (i : I) → CommRing (A i)\ninst✝¹ : (i : I) → Algebra R (A i)\ninst✝ : Finite I\n⊢ FormallySmooth R ((i : I) → A i) ↔ ∀ (i : I), FormallySmooth R (A i)",
"usedConstants": [
"Ideal.Quotient.isScalarTower",
"AddGro... | classical
cases nonempty_fintype I
constructor
· exact fun _ ↦ of_pi A
· refine fun H ↦ .of_comp_surjective fun B _ _ J hJ g ↦ ?_
have hJ' (x) (hx : x ∈ RingHom.ker (Ideal.Quotient.mk J)) : IsNilpotent x := by
refine ⟨2, show x ^ 2 ∈ (⊥ : Ideal B) from ?_⟩
rw [← hJ]
exact Ideal.pow_mem_pow... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 747,
"column": 13
} | {
"line": 747,
"column": 64
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ≠ Q x * P z\n⊢ W.negY ![W.addX P Q, W.negAddY P Q, W.addZ P Q] / W.addZ P Q =\n W.toAffine.addY (P x / P z) (Q x / Q z) (P y / P ... | negY_of_Z_ne_zero <| addZ_ne_zero_of_X_ne hP hQ hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Unramified.Field | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 53
} | [
{
"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\nI : Ideal B\nhI : I ^ 2 = ⊥\nf₁ f₂ : L →ₐ[K] B\ne : (Ideal.Quotient.mkₐ K I).comp f₁ = (Ideal.Quotient.mkₐ K I).comp f₂\nx : L\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.Field | {
"line": 64,
"column": 2
} | {
"line": 109,
"column": 40
} | [
{
"pp": "K : Type u_1\nA : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra K A\ninst✝³ : FormallyUnramified K A\ninst✝² : EssFiniteType K A\ninst✝¹ : IsAlgClosed K\ninst✝ : IsLocalRing A\n⊢ Function.Bijective ⇑(algebraMap K A)",
"usedConstants": [
"Ideal.mem_bot",
"Ideal.Quotie... | have := finite_of_free (R := K) (S := A)
have : IsArtinianRing A := isArtinian_of_tower K inferInstance
have hA : IsNilpotent (IsLocalRing.maximalIdeal A) := by
rw [← IsLocalRing.jacobson_eq_maximalIdeal ⊥]
· exact IsArtinianRing.isNilpotent_jacobson_bot
· exact bot_ne_top
let e : K ≃ₐ[K] A ⧸ IsLocalR... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Unramified.Field | {
"line": 64,
"column": 2
} | {
"line": 109,
"column": 40
} | [
{
"pp": "K : Type u_1\nA : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra K A\ninst✝³ : FormallyUnramified K A\ninst✝² : EssFiniteType K A\ninst✝¹ : IsAlgClosed K\ninst✝ : IsLocalRing A\n⊢ Function.Bijective ⇑(algebraMap K A)",
"usedConstants": [
"Ideal.mem_bot",
"Ideal.Quotie... | have := finite_of_free (R := K) (S := A)
have : IsArtinianRing A := isArtinian_of_tower K inferInstance
have hA : IsNilpotent (IsLocalRing.maximalIdeal A) := by
rw [← IsLocalRing.jacobson_eq_maximalIdeal ⊥]
· exact IsArtinianRing.isNilpotent_jacobson_bot
· exact bot_ne_top
let e : K ≃ₐ[K] A ⧸ IsLocalR... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Unramified.Finite | {
"line": 76,
"column": 8
} | {
"line": 77,
"column": 57
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : EssFiniteType R S\nthis✝ : ∀ (t : S ⊗[R] S), (TensorProduct.lmul' R) t = 1 ↔ 1 - t ∈ KaehlerDifferential.ideal R S\nt e : S ⊗[R] S\nht₁ : ∀ (s : S), (1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) * (1 - e) = 0\nht₂ : e ∈ Ideal... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.Finite | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 34
} | [
{
"pp": "case h_add\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nI : Type u_4\ninst✝ : DecidableEq I\nb : Basis I R S\nf : I →₀ S\nx : S\na : I → I →₀ R := fun i ↦ b.repr (b i * x)\ni : I\n⊢ ∀ (a : I) (b₁ b₂ : R), (b₁ + b₂) • b a ⊗ₜ[R] b i = b₁ • b a ⊗ₜ[R] b i + b... | · intros; simp only [add_smul] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Unramified.Finite | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 67
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nI : Type u_4\ninst✝ : DecidableEq I\nb : Basis I R S\nf : I →₀ S\nx : S\na : I → I →₀ R := fun i ↦ b.repr (b i * x)\nh₁ :\n ∀ (k : I),\n ((f.sum fun i y ↦ (a i) k • b.repr y).sum fun j z ↦ z • b j ⊗ₜ[R] b k)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.Field | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 15
} | [
{
"pp": "case refine_6.H\nK : 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 ↦ (Idea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Functoriality | {
"line": 388,
"column": 4
} | {
"line": 388,
"column": 62
} | [
{
"pp": "case h.right\nR : 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\ny : N\nn : ℕ\nhxy : f x ≡ y [SMOD I ^ n • ⊤]\nz : M\nhz... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Exactness | {
"line": 105,
"column": 12
} | {
"line": 105,
"column": 23
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nI : Ideal R\nM : Type u\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nN : Type u\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R N\nf : M →ₗ[R] N\nhf : Function.Injective ⇑f\nk : ℕ\nhk : ∀ n ≥ k, I ^ n • ⊤ ⊓ f.range =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 15
} | [
{
"pp": "case tmul.tmul.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\ni... | | tmul x z => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Smooth.AdicCompletion | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 13
} | [
{
"pp": "case H\nR : Type u_1\nA : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nS : Type u_3\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FormallySmooth R A\nI : Ideal S\ninst✝ : IsAdicComplete I S\nf : A →ₐ[R] S ⧸ I\ng : A →ₐ[R] AdicCompletion I S\nhg : (AlgHom.restrictScal... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.AdicCompletion | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 13
} | [
{
"pp": "case h.H\nR : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nS : Type u_3\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : FormallySmooth R A\nf : S →ₐ[R] A\nhf : Function.Surjective ⇑f\ng : A →ₐ[R] AdicCompletion (ker f) S\nhg :\n (AlgHom.restrictScalars R ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Exactness | {
"line": 191,
"column": 6
} | {
"line": 191,
"column": 17
} | [
{
"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] ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Presentation.Submersive | {
"line": 456,
"column": 4
} | {
"line": 456,
"column": 29
} | [
{
"pp": "case intro.intro\nR : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : PreSubmersivePresentation R S ι σ\nι' : Type u_1\nσ' : Type u_2\ne : ι' ≃ ι\nf : σ' ≃ σ\ninst✝¹ : Finite σ\ninst✝ : Finite σ'\nval✝¹ : Fintype σ\nval✝ : Fintype σ'\n⊢ (a... | ← AlgHom.mapMatrix_apply, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 15
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Flat R S\ninst✝⁹ : CommRing P\ninst✝⁸ : Algebra R P\ninst✝⁷ : Algebra P S\ninst✝⁶ : IsScalarTower R P S\ninst✝⁵ : IsLocalRing R\ninst✝⁴ : IsLocalRing S\ninst✝³ : IsLocalHom (alg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 177,
"column": 6
} | {
"line": 177,
"column": 81
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Flat R S\ninst✝⁹ : CommRing P\ninst✝⁸ : Algebra R P\ninst✝⁷ : Algebra P S\ninst✝⁶ : IsScalarTower R P S\ninst✝⁵ : IsLocalRing R\ninst✝⁴ : IsLocalRing S\ninst✝³ : IsLocalHom (alg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 205,
"column": 6
} | {
"line": 205,
"column": 17
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : Module.Flat R S\ninst✝⁴ : FinitePresentation R S\np : Ideal R\nq : Ideal S\ninst✝³ : p.IsPrime\ninst✝² : q.IsPrime\ninst✝¹ : q.LiesOver p\ninst✝ : FormallySmooth p.ResidueField (p.Fiber S)\nRp : Type u_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 26
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : Module.Flat R S\ninst✝⁴ : FinitePresentation R S\np : Ideal R\nq : Ideal S\ninst✝³ : p.IsPrime\ninst✝² : q.IsPrime\ninst✝¹ : q.LiesOver p\ninst✝ : FormallySmooth p.ResidueField (p.Fiber S)\nRp : Type u_... | algebraize [f.toRingHom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.Smooth.StandardSmoothCotangent | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 15
} | [
{
"pp": "case a.hr\nR : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Finite σ\nP : SubmersivePresentation R S ι σ\nx : ↥P.ker\nhx : ∀ (i : σ), (aeval P.val) ((pderiv (P.map i)) ↑x) = 0\nthis✝ : ↑x ∈ Ideal.span (Set.range P.relation)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Presentation.Core | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 53
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Presentation R S ι σ\nR₀ : Type u_5\ninst✝¹⁰ : CommRing R₀\ninst✝⁹ : Algebra R₀ R\ninst✝⁸ : Algebra R₀ S\ninst✝⁷ : IsScalarTower R₀ R S\ninst✝⁶ : P.HasCoeffs R₀\nR₁ : Type u_6\... | refine ⟨subset_trans (P.coeffs_subset_range R₀) ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Extension.Presentation.Core | {
"line": 142,
"column": 58
} | {
"line": 145,
"column": 10
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nP : Presentation R S ι σ\nR₀ : Type u_5\ninst✝⁴ : CommRing R₀\ninst✝³ : Algebra R₀ R\ninst✝² : Algebra R₀ S\ninst✝¹ : IsScalarTower R₀ R S\ninst✝ : P.HasCoeffs R₀\n⊢ ∀ a ∈ Ideal.span ... | by
simp_rw [← RingHom.mem_ker, ← SetLike.le_def, Ideal.span_le]
rintro a ⟨i, rfl⟩
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Extension.Presentation.Core | {
"line": 317,
"column": 2
} | {
"line": 317,
"column": 78
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : Finite σ\nP : SubmersivePresentation R S ι σ\nR₀ : Type u_5\ninst✝⁴ : CommRing R₀\ninst✝³ : Algebra R₀ R\ninst✝² : Algebra R₀ S\ninst✝¹ : IsScalarTower R₀ R S\ninst✝ : P.HasC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.LocalizationAway | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 15
} | [
{
"pp": "case h\nR : Type u_1\nS : Type u_2\nT : Type u_3\nι : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\ng : S\ninst✝ : IsLocalization.Away g T\nP : Generators R S ι\nQ : Generators S T... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.LocalizationAway | {
"line": 153,
"column": 2
} | {
"line": 156,
"column": 45
} | [
{
"pp": "case a\nR : Type u_1\nS : Type u_2\nT : Type u_3\nι : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\ng : S\ninst✝ : IsLocalization.Away g T\nP : Generators R S ι\nQ : Generators S T... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.LocalizationAway | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 87
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nι : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\ng : S\ninst✝ : IsLocalization.Away g T\nP : Generators R S ι\nx : ((localizationAway T g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RingHom.StandardSmooth | {
"line": 243,
"column": 8
} | {
"line": 244,
"column": 48
} | [
{
"pp": "n : ℕ\nR : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Algebra.IsStandardSmoothOfRelativeDimension n R S\nthis✝¹ : (α : Type) → [_root_.Finite α] → Fintype α := Fintype.ofFinite\nι σ : Type\nw✝¹ : _root_.Finite σ\nw✝ : _root_.Finite ι\nP : Algebra.Submers... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 92,
"column": 6
} | {
"line": 92,
"column": 27
} | [
{
"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\nq₁ q₂ : MvPolynomial ι k\ne : F = q₁ * q₂\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 13
} | [
{
"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\nq₁ q₂ : MvPolynomial ι k\ne : F = q₁ * q₂\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 21
} | [
{
"pp": "case refine_3\nk : 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\nσ : ι →₀ ℕ\nhσ : σ ∈ F.support\ni : ι\nhσi : σ i ≠ 0\nH✝ : (F.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.NoetherianDescent | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 62
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nA : Type u\nB : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : Smooth A B\nP : Presentation A B (Fin (Presentation.ofFinitePresentationVars A B))\n (Fin (Presentation.ofFinitePresentationRels A B)) :=\n Presen... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Smooth | {
"line": 238,
"column": 78
} | {
"line": 241,
"column": 62
} | [
{
"pp": "n m : ℕ\nX Y : Scheme\nf : X ⟶ Y\nhf : Smooth f\n⊢ LocallyOfFinitePresentation f",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"RingHom.FinitePresentation",
"CommRing",
"AlgebraicGeometry.Scheme",
"congrArg",
"CommSemiring.toSemiring",
... | by
rw [HasRingHomProperty.eq_affineLocally @LocallyOfFinitePresentation]
rw [HasRingHomProperty.eq_affineLocally @Smooth] at hf
exact affineLocally_le (fun hf ↦ hf.finitePresentation) f hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 150,
"column": 4
} | {
"line": 151,
"column": 25
} | [
{
"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\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι✝ → K\nF : MvPolynomial ι✝ k\nHF : ∀ (F' : MvPolynomial ι✝ k), F' ≠ 0 → (aeval a) F' = 0 → ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 151,
"column": 54
} | {
"line": 151,
"column": 65
} | [
{
"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\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι✝ → K\nF : MvPolynomial ι✝ k\nHF : ∀ (F' : MvPolynomial ι✝ k), F' ≠ 0 → (aeval a) F' = 0 → ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 154,
"column": 39
} | {
"line": 154,
"column": 77
} | [
{
"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), ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 208,
"column": 43
} | {
"line": 208,
"column": 94
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\nn : ι\ninst✝ : ExpChar k p\nha' : IsTranscendenceBasis k fun i ↦ a ↑i\nS : Set (MvPol... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 237,
"column": 8
} | {
"line": 237,
"column": 19
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\nn : ι\ninst✝ : ExpChar k p\nha' : IsTranscendenceBasis k fun i ↦ a ↑i\nS : Set (MvPol... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 67,
"column": 2
} | {
"line": 68,
"column": 65
} | [
{
"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\ni : σ\n⊢ (Subtype.val ∘ D.f ∘ ⇑b) i ∈ ↑(RingHom.ker (aeval P.val))",
"usedConstants"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 91,
"column": 65
} | {
"line": 91,
"column": 76
} | [
{
"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⊢ D.g ^ 1 • RingHom.ker (algebraMap P.Ring S) ≤ RingHom.ker (algebraMap P.Ring D.T)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 131,
"column": 2
} | {
"line": 133,
"column": 19
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\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\nx : D.presLeft.Ring\nhx : x ∈ D.presLeft.ker\n⊢ x ∈ P.ker",
"usedConstants": [
"Algebra.Gener... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 172,
"column": 4
} | {
"line": 173,
"column": 36
} | [
{
"pp": "case refine_1\nR : 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\ni : σ\n⊢ (D.tensorCotangentHom ∘ₗ D.tensorCotangentInv) (b i) = LinearMap... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalRing | {
"line": 108,
"column": 69
} | {
"line": 108,
"column": 84
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsLocalRing R\ninst✝³ : IsLocalRing S\ninst✝² : IsLocalHom (algebraMap R S)\ninst✝¹ : EssFiniteType R S\ninst✝ : Algebra.IsSeparable (ResidueField R) (ResidueField S)\nH : Ideal.map (algebraMap R S) (ma... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 21
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\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\ninst✝ : Nontrivial S\n⊢ D.basis (Sum.inl ()) = D.cotangentEquivProd.symm (cMulXSubOneCotangent S D.gba... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalRing | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 79
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\np : Ideal R\ninst✝³ : p.IsPrime\nq : Ideal S\ninst✝² : q.IsPrime\nhq : p.primesOver S = {q}\ninst✝¹ : Algebra.IsIntegral R S\ninst✝ : FaithfulSMul R S\nx : R\ns : ↥p.primeCompl\nhx : (localRingHom p q (algebraMa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 290,
"column": 41
} | {
"line": 290,
"column": 52
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FinitePresentation R S\nα : Type u_4\nP : Generators R S α\ninst✝ : Finite α\nσ : Type u_5\nb₀ : Module.Basis σ S P.toExtension.Cotangent\nh✝ : Subsingleton S\n⊢ Ideal.span (Set.range fun x ↦ 1) = (ofSu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 309,
"column": 65
} | {
"line": 309,
"column": 76
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FinitePresentation R S\nα : Type u_4\nP : Generators R S α\ninst✝ : Finite α\nσ : Type u_5\nb₀ : Module.Basis σ S P.toExtension.Cotangent\nh✝ : Nontrivial S\nf : P.toExtension.Cotangent → ↥P.toExtension... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 343,
"column": 41
} | {
"line": 343,
"column": 52
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FinitePresentation R S\nα : Type u_4\nP : Generators R S α\ninst✝¹ : Finite α\ninst✝ : Module.Free S P.toExtension.Cotangent\nh✝ : Subsingleton S\n⊢ Ideal.span (Set.range fun x ↦ 1) = (ofSurjective (fun... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.FormallyUnramified | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 76
} | [
{
"pp": "X Y Z' Z : Scheme\ni : Z' ⟶ Z\nhi : IsNilpotent (Scheme.Hom.ker i)\ninst✝¹ : IsClosedImmersion i\nf : X ⟶ Y\ninst✝ : FormallyUnramified f\ng₁ g₂ : Z ⟶ X\nhig : i ≫ g₁ = i ≫ g₂\nhgf : g₁ ≫ f = g₂ ≫ f\nthis✝² : IsDominant i\nx : ↥Z\nU : Opens ↥Y\nhU : U ∈ Y.affineOpens\nhxU : f (g₁ x) ∈ ↑U\nV : Opens ↥X\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.LocalClosure | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 40
} | [
{
"pp": "W P Q : MorphismProperty Scheme\nX✝ Y✝ : Scheme\ninst✝³ : W.IsStableUnderBaseChange\ninst✝² : Scheme.IsJointlySurjectivePreserving W\ninst✝¹ : P.RespectsLeft Q\ninst✝ : Q.IsStableUnderBaseChange\nX Y Z : Scheme\nf : X ⟶ Y\nhf : Q f\ng : Y ⟶ Z\nx✝ : sourceLocalClosure W P g\n𝒰 : Scheme.Cover (Scheme.pr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.LocalClosure | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 42
} | [
{
"pp": "case refine_1\nW P Q : MorphismProperty Scheme\nX Y : Scheme\ninst✝³ : W.IsStableUnderBaseChange\ninst✝² : Scheme.IsJointlySurjectivePreserving W\ninst✝¹ : P.RespectsIso\ninst✝ : P.RespectsLeft IsOpenImmersion\nX✝ Y✝ : Scheme\nf : X✝ ⟶ Y✝\n𝒰 : Scheme.zariskiPrecoverage.ZeroHypercover X✝\nx✝ : sourceLo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.LocalClosure | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 82
} | [
{
"pp": "W P Q : MorphismProperty Scheme\nX✝ Y✝ : Scheme\ninst✝² : W.IsStableUnderBaseChange\ninst✝¹ : Scheme.IsJointlySurjectivePreserving W\ninst✝ : P.IsStableUnderBaseChange\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nx✝¹ : HasPullback f g\nx✝ : sourceLocalClosure W P g\n𝒰 : Scheme.Cover (Scheme.precoverage W) Y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.LocalClosure | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 89
} | [
{
"pp": "W P Q : MorphismProperty Scheme\nX✝ Y✝ : Scheme\ninst✝⁴ : W.IsStableUnderBaseChange\ninst✝³ : Scheme.IsJointlySurjectivePreserving W\ninst✝² : W.IsStableUnderComposition\ninst✝¹ : P.IsStableUnderBaseChange\ninst✝ : P.IsStableUnderComposition\nX Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\nx✝² : sourceLocalClosu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalRing | {
"line": 299,
"column": 60
} | {
"line": 299,
"column": 71
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\np : Ideal R\ninst✝⁷ : p.IsPrime\nq : Ideal S\ninst✝⁶ : q.IsPrime\nhq : p.primesOver S = {q}\ninst✝⁵ : Module.Finite R S\ninst✝⁴ : FaithfulSMul R S\ninst✝³ : q.LiesOver p\ninst✝² : Algebra.IsUnramifiedAt R q\nin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.FormallyUnramified | {
"line": 221,
"column": 17
} | {
"line": 221,
"column": 28
} | [
{
"pp": "X Y Z' Z : Scheme\ni : Z' ⟶ Z\ninst✝¹ : IsClosedImmersion i\nf : X ⟶ Y\ninst✝ : FormallyUnramified f\ng₁ g₂ : Z ⟶ X\nhig : i ≫ g₁ = i ≫ g₂\nhgf : g₁ ≫ f = g₂ ≫ f\nthis✝¹ : IsDominant i\nx : ↥Z\nU : Opens ↥Y\nhU : U ∈ Y.affineOpens\nhxU : f (g₁ x) ∈ ↑U\nV : Opens ↥X\nhV : V ∈ X.affineOpens\nhxV : g₁ x ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Descent | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 20
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nP : MorphismProperty C\ninst✝³ : (isomorphisms C).DescendsAlong P\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : HasEqualizers C\ninst✝ : HasPullbacks C\nX Y S T : C\nf g : X ⟶ Y\ns : X ⟶ S\nt : Y ⟶ S\nhf : f ≫ t = s\nhg : g ≫ t = s\nv : T ⟶ S\nhv : P v\nH :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Descent | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 20
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nT : Type u_3\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\ninst✝ : FaithfullyFlat R S\nH : Function.Injective ⇑(algebraMap S (S ⊗[R] T))\nthis :\n LinearMap.lTensor S (Algebra.linearMap R T) = ↑R (Algebra.linearM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Flat.FaithfullyFlat.Descent | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 20
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nT : Type u_3\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\ninst✝ : FaithfullyFlat R S\nH : Function.Surjective ⇑(algebraMap S (S ⊗[R] T))\nthis :\n LinearMap.lTensor S (Algebra.linearMap R T) = ↑R (Algebra.linear... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Descent | {
"line": 58,
"column": 21
} | {
"line": 58,
"column": 36
} | [
{
"pp": "P : MorphismProperty Scheme\nX : Scheme\ninst✝² : CompactSpace ↥X\ninst✝¹ : IsZariskiLocalAtSource P\ninst✝ : P.ContainsIdentities\n𝒰 : X.OpenCover := X.affineCover.finiteSubcover\np : (∐ fun i ↦ 𝒰.X i) ⟶ X := Sigma.desc fun i ↦ 𝒰.f i\ni : (sigmaOpenCover fun i ↦ 𝒰.X i).I₀\n⊢ P ((sigmaOpenCover fun... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Jacobson.Artinian | {
"line": 52,
"column": 6
} | {
"line": 52,
"column": 32
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Algebra.FiniteType R A\ninst✝ : IsArtinianRing R\nthis : IsNoetherianRing A\n⊢ Module.Finite R A ↔ Ring.KrullDimLE 0 A",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAss... | finite_iff_isArtinianRing, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Jacobson.Artinian | {
"line": 52,
"column": 33
} | {
"line": 52,
"column": 85
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Algebra.FiniteType R A\ninst✝ : IsArtinianRing R\nthis : IsNoetherianRing A\n⊢ IsArtinianRing A ↔ Ring.KrullDimLE 0 A",
"usedConstants": [
"Eq.mpr",
"IsArtinianRing",
"congrArg",
... | isArtinianRing_iff_isNoetherianRing_krullDimLE_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Descent | {
"line": 140,
"column": 34
} | {
"line": 140,
"column": 81
} | [
{
"pp": "P P' : MorphismProperty Scheme\ninst✝³ : P'.IsStableUnderBaseChange\ninst✝² : P'.IsStableUnderComposition\ninst✝¹ : P.IsStableUnderBaseChange\nH₁ : @IsLocalIso ⊓ @Surjective ≤ P'\ninst✝ : IsZariskiLocalAtTarget P\nH :\n ∀ {R S : CommRingCat} {Y : Scheme} (φ : R ⟶ S) (g : Y ⟶ Spec R),\n P' (Spec.map... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.NilpotentKer | {
"line": 47,
"column": 6
} | {
"line": 47,
"column": 93
} | [
{
"pp": "case succ.refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : Module.Finite R T\nf : S →ₐ[R] T\nhf₁ : Function.Surjective ⇑f\nI : Ideal S\nhI : RingHom.ker f = I\nhf₂ : I ≤ nilradical S\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.NilpotentKer | {
"line": 57,
"column": 8
} | {
"line": 57,
"column": 31
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : Module.Finite R T\nf : S →ₐ[R] T\nhf₁ : Function.Surjective ⇑f\nI : Ideal S\nhI : RingHom.ker f = I\nhf₂ : I ≤ nilradical S\nhf₃ : I.FG\nthis✝ : M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.NilpotentKer | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 45
} | [
{
"pp": "case h.e'_6.h.a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : CommRing T\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : Module.Finite R T\nf : S →ₐ[R] T\nhf₁ : Function.Surjective ⇑f\nI : Ideal S\nhI : RingHom.ker f = I\nhf₂ : I ≤ nilradical S\nhf₃ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.FlatDescent | {
"line": 138,
"column": 2
} | {
"line": 139,
"column": 72
} | [
{
"pp": "case H\nX Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : HasPullback f g\nhf : (@Surjective ⊓ @Flat ⊓ @QuasiCompact) f\nhg : IsOpenImmersion (pullback.fst f g)\nthis : UniversallyOpen g\nU : Z.Opens := { carrier := Set.range ⇑g, is_open' := ⋯ }\nf' : pullback f U.ι ⟶ ↑U := pullback.snd f U.ι\ng' : Y ⟶ ↑U ... | have : Surjective g' := ⟨fun ⟨x, ⟨y, hy⟩⟩ ↦
⟨y, by apply U.ι.injective; simp [← Scheme.Hom.comp_apply, g', hy]⟩⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Morphisms.FlatDescent | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 30
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : HasPullback f g\nhf : (@Surjective ⊓ @Flat ⊓ @QuasiCompact) f\nhg : IsOpenImmersion (pullback.fst f g)\nthis✝ : UniversallyOpen g\nU : Z.Opens := { carrier := Set.range ⇑g, is_open' := ⋯ }\nf' : pullback f U.ι ⟶ ↑U := pullback.snd f U.ι\ng' : Y ⟶ ↑U := IsOp... | refine ⟨?_, inferInstance⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 43,
"column": 7
} | {
"line": 43,
"column": 61
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nx : S\nh : IsStronglyTranscendental R x\ninst✝ : FaithfulSMul R S\np : R[X]\nhp : (aeval x) p = 0\n⊢ ∀ (n : ℕ), p.coeff n = coeff 0 n",
"usedConstants": [
"CommSemiring.toSemiring",
"id",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 48,
"column": 2
} | {
"line": 49,
"column": 90
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : FaithfulSMul R K\nx : K\nh : Transcendental R x\n⊢ IsStronglyTranscendental R x",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"RingHom.instRing... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 67,
"column": 4
} | {
"line": 68,
"column": 43
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\nM : Submonoid S\ninst✝¹ : IsLocalization M T\ninst✝ : IsScalarTower R S T\nx : S\nh : IsStronglyTranscendental R x\np : R[X]\nu : S\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Conductor | {
"line": 34,
"column": 25
} | {
"line": 34,
"column": 51
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx a✝ b✝ : S\nha : a✝ ∈ {a | ∀ (b : S), a * b ∈ R[x]}\nhb : b✝ ∈ {a | ∀ (b : S), a * b ∈ R[x]}\nc : S\n⊢ (a✝ + b✝) * c ∈ R[x]",
"usedConstants": [
"Subalgebra.instSetLike",
"add_mul",
"Eq.mpr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 74
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\nM : Submonoid S\ninst✝¹ : IsLocalization M T\ninst✝ : IsScalarTower R S T\nx : S\nh : IsStronglyTranscendental R x\np : R[X]\nu : S\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Conductor | {
"line": 35,
"column": 27
} | {
"line": 35,
"column": 83
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx c a : S\nha : a ∈ {a | ∀ (b : S), a * b ∈ R[x]}\nb : S\n⊢ c • a * b ∈ R[x]",
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"Semigroup.toMul",
"instHSMul",
"Semiring.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Conductor | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 28
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx y : S\nhy : y ∈ ↑(conductor R x)\n⊢ y ∈ ↑R[x]",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 42
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\nM : Submonoid R\ninst✝¹ : IsLocalization M S\ninst✝ : IsScalarTower R S T\nx : T\nh : IsStronglyTranscendental R x\nt : T\np : S[X]\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 43
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\nM : Submonoid R\ninst✝¹ : IsLocalization M S\ninst✝ : IsScalarTower R S T\nx : T\nh : IsStronglyTranscendental R x\nt : T\np : S[X]\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 54
} | [
{
"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\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nx : T\nh : IsStronglyTranscendental S x\nt : T\np : R[X]\nhp : (aeval x) p * t = 0\n⊢ map (algebraMap R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 54
} | [
{
"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\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nx : T\nh : IsStronglyTranscendental R x\nH : Function.Surjective ⇑(algebraMap R S)\nt : T\np : R[X]\nhp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 99,
"column": 65
} | {
"line": 99,
"column": 76
} | [
{
"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\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nx : T\nh : IsStronglyTranscendental R x\nH : Function.Surjective ⇑(algebraMap R S)\nt : T\np : R[X]\nhp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 29
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsReduced S\nx : S\nhx : IsStronglyTranscendental R x\nq : Ideal S\nhq : q ∈ minimalPrimes S\nu : S\np : R[X]\ne : (aeval x) p * u ∈ q\nthis✝¹ : q.IsPrime\nthis✝ : Ring.KrullDimLE 0 (Localization.AtPrime... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 138,
"column": 2
} | {
"line": 139,
"column": 89
} | [
{
"pp": "case a\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsReduced S\nx : S\nhx : IsStronglyTranscendental R x\nq : Ideal S\nhq : q ∈ minimalPrimes S\nu : S\np : R[X]\ne : (aeval x) p * u ∈ q\nthis✝¹ : q.IsPrime\nthis✝ : Ring.KrullDimLE 0 (Localization... | have : algebraMap R S (p.coeff i) * u * m = 0 := by
simpa [← mul_assoc] using congr(($(hx (u * m) p (by linear_combination hm))).coeff i) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.IntegralClosure.Algebra.Ideal | {
"line": 44,
"column": 35
} | {
"line": 44,
"column": 46
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\nI : Ideal R\nP x y : R[X]\nhx✝ : x ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nhy✝ : y ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nhx : ∀ (i : ℕ), x.coeff i ∈ I ^ i\nhy : ∀ (i : ℕ), y.coeff i ∈ I ^ i\ni : ℕ\nx✝ : ℕ × ℕ\nj₁ j₂ : ℕ\nhj : (j₁, j₂) ∈ Finset.antidiago... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.Algebra.Ideal | {
"line": 59,
"column": 64
} | {
"line": 59,
"column": 93
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal R\nx : S\nh✝ : Nontrivial R\np : (↥(Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}))[X]\nhp : p.Monic\ne : eval₂ (algebraMap (↥(Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x})) S[X]) (C x * X) p = 0\nq : R[X]... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.Algebra.Ideal | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 22
} | [
{
"pp": "case inr.refine_1\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nI : Ideal R\nx : S\nh✝ : Nontrivial R\np : (↥(Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}))[X]\nhp : p.Monic\ne : eval₂ (algebraMap (↥(Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x})) S[X]) (C x *... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.Algebra.Ideal | {
"line": 82,
"column": 25
} | {
"line": 82,
"column": 46
} | [
{
"pp": "case add\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Algebra.IsIntegral R S\nI : Ideal R\nx✝ : S\nA : Subalgebra R R[X] := Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nthis : Algebra R[X] S[X] := algebra R S\nx y : S\nhx✝ : x ∈ Submodule.span S (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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