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.IntegralClosure.Algebra.Ideal | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 15
} | [
{
"pp": "case mem\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 : R\nhx : x ∈ ↑I\n⊢ IsIntegral (↥(A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RingHom.QuasiFinite | {
"line": 98,
"column": 6
} | {
"line": 98,
"column": 39
} | [
{
"pp": "R S : Type u_6\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf✝ : R →+* S\ns : Finset S\nhs : Ideal.span ↑s = ⊤\nH : ∀ (r : ↥s), (fun {R S} [CommRing R] [CommRing S] ↦ QuasiFinite) ((algebraMap S (Localization.Away ↑r)).comp f✝)\nalgInst✝ : Algebra R S := f✝.toAlgebra\nP : Ideal R\nx✝ : P.IsPrime\nthis✝¹ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.GoingDown | {
"line": 36,
"column": 6
} | {
"line": 36,
"column": 57
} | [
{
"pp": "case a.inl\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhq : q.Monic\nH : q ∣ p\ni✝ : ℕ\nhi✝ : i✝ ≠ q.natDegree\nP : Ideal R\nhPJ : Ideal.span {x | ∃ i < p.natDegree, p.coeff i = x} ≤ P\nhP : P.IsPrime\ni : ℕ\nhi : i < p.natDegree\n⊢ (map (Ideal.Quotient.mk P) p).coeff i = (X ^ p.natDeg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.GoingDown | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhq : q.Monic\nH : q ∣ p\ni : ℕ\nhi : i ≠ q.natDegree\nP : Ideal R\nhPJ : Ideal.span {x | ∃ i < p.natDegree, p.coeff i = x} ≤ P\nhP : P.IsPrime\nthis : map (Ideal.Quotient.mk P) p = X ^ p.natDegree\nj : ℕ\nhj : j ≤ p.natDegree\na : (R ⧸ P)[X]ˣ\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.GoingDown | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 50
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhq : q.Monic\nH : q ∣ p\ni : ℕ\nhi : i ≠ q.natDegree\nP : Ideal R\nhPJ : Ideal.span {x | ∃ i < p.natDegree, p.coeff i = x} ≤ P\nhP : P.IsPrime\nthis : map (Ideal.Quotient.mk P) p = X ^ p.natDegree\na : (R ⧸ P)[X]ˣ\nr : R ⧸ P\nhr : IsUnit r\ne ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Conductor | {
"line": 204,
"column": 39
} | {
"line": 204,
"column": 55
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nx : S\nP : Ideal S\ninst✝² : P.IsPrime\nhx : ¬conductor R x ≤ P\ns : Subalgebra R S\nhs : s = R[x]\np : Ideal ↥s\ninst✝¹ : p.IsPrime\ninst✝ : P.LiesOver p\na : S\nha : a ∈ conductor R x\nhaP : a ∉ P\nb : S\n⊢ a ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Conductor | {
"line": 206,
"column": 42
} | {
"line": 206,
"column": 53
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nx : S\nP : Ideal S\ninst✝² : P.IsPrime\nhx : ¬conductor R x ≤ P\ns : Subalgebra R S\nhs : s = R[x]\np : Ideal ↥s\ninst✝¹ : p.IsPrime\ninst✝ : P.LiesOver p\na : S\nhaP : a ∉ P\nha : ∀ (b : S), a * b ∈ s\ny : S\nx... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.GoingDown | {
"line": 62,
"column": 4
} | {
"line": 63,
"column": 74
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : IsDomain S\ninst✝⁴ : FaithfulSMul R S\ninst✝³ : Algebra.IsIntegral R S\ninst✝² : IsIntegrallyClosed R\nthis : IsDomain R\np : Ideal R\ninst✝¹ : p.IsPrime\nQ : Ideal S\ninst✝ : Q.IsPrime\nhpQ : p < Ideal... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.GoingDown | {
"line": 71,
"column": 8
} | {
"line": 71,
"column": 59
} | [
{
"pp": "case a.inl\nR : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : IsDomain S\ninst✝⁴ : FaithfulSMul R S\ninst✝³ : Algebra.IsIntegral R S\ninst✝² : IsIntegrallyClosed R\nthis : IsDomain R\np : Ideal R\ninst✝¹ : p.IsPrime\nQ : Ideal S\ninst✝ : Q.IsPrime\nhpQ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.GoingDown | {
"line": 75,
"column": 6
} | {
"line": 75,
"column": 77
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : IsDomain S\ninst✝⁴ : FaithfulSMul R S\ninst✝³ : Algebra.IsIntegral R S\ninst✝² : IsIntegrallyClosed R\nthis✝ : IsDomain R\np : Ideal R\ninst✝¹ : p.IsPrime\nQ : Ideal S\ninst✝ : Q.IsPrime\nhpQ : p < Idea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.IsIntegral.AlmostIntegral | {
"line": 101,
"column": 6
} | {
"line": 102,
"column": 43
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ns : S\ninst✝ : IsNoetherianRing R\nH' : R⁰ ≤ Submonoid.comap (algebraMap R S) S⁰\nr : R\nhr : r ∈ R⁰\nhr' : ∀ (n : ℕ), r • s ^ n ∈ (algebraMap R S).range\nn : ℕ\na : R\nha : (algebraMap R S) a = r • s ^ n\nthis ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Defs | {
"line": 218,
"column": 6
} | {
"line": 218,
"column": 59
} | [
{
"pp": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nP : K⟮X⟯ → Prop\nx : FractionRing K[X]\nf : ∀ (p q : K[X]), q ≠ 0 → P (RatFunc.mk p q)\nx✝ : K[X] × ↥K[X]⁰\np : K[X]\nq : ↥K[X]⁰\n⊢ P { toFractionRing := Localization.mk (p, q).1 (p, q).2 }",
"usedConstants": [
"Eq.mpr",
"Localization... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 72,
"column": 19
} | {
"line": 72,
"column": 30
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R[X]\nq : S[X]\nhp : p.Monic\nhq : q.Monic\nH : q ∣ map (algebraMap R S) p\ni : ℕ\na✝ : Nontrivial S\nT : Type u_2\nw✝⁴ : CommRing T\nw✝³ : Algebra S T\nw✝² : Module.Finite S T\nw✝¹ : Module.Free S T\nw✝ : No... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 90,
"column": 31
} | {
"line": 90,
"column": 42
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsDomain R\ninst✝ : FaithfulSMul R S\np✝ : S[X]\nhp✝ : IsAlmostIntegral R[X] p✝\ni : ℕ\nq : S[X]\np : R[X]\nhp : p ∈ R[X]⁰\nhp' : ∀ (n : ℕ), p • q ^ n ∈ (algebraMap R[X] S[X]).range\n⊢ p.leadingCoeff ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 17
} | [
{
"pp": "case pos\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsDomain R\ninst✝ : FaithfulSMul R S\ni : ℕ\nH : ∀ {q : S[X]}, IsAlmostIntegral R[X] q → IsAlmostIntegral R q.leadingCoeff\nn : ℕ\nIH : ∀ m < n, ∀ {p : S[X]}, IsAlmostIntegral R[X] p → p.natDe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 114,
"column": 20
} | {
"line": 114,
"column": 31
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsDomain R\ninst✝ : FaithfulSMul R S\ni : ℕ\nH : ∀ {q : S[X]}, IsAlmostIntegral R[X] q → IsAlmostIntegral R q.leadingCoeff\nn : ℕ\nIH : ∀ m < n, ∀ {p : S[X]}, IsAlmostIntegral R[X] p → p.natDegree = m →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 41
} | [
{
"pp": "case neg\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsDomain R\ninst✝ : FaithfulSMul R S\ni : ℕ\nH : ∀ {q : S[X]}, IsAlmostIntegral R[X] q → IsAlmostIntegral R q.leadingCoeff\nn : ℕ\nIH : ∀ m < n, ∀ {p : S[X]}, IsAlmostIntegral R[X] p → p.natDe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 137,
"column": 10
} | {
"line": 137,
"column": 21
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : S[X]\nhp : IsIntegral R[X] p\ni : ℕ\na✝¹ : Nontrivial R\na✝ : Nontrivial S\nhp0 : p ≠ 0\nq : R[X][X] := minpoly R[X] p\nm : ℕ := (q.support.sup fun i ↦ (q.coeff i).natDegree) + p.natDegree + 1\nhm₁ : ∀ (i : ℕ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 40
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : S[X]\nhp : IsIntegral R[X] p\ni : ℕ\na✝¹ : Nontrivial R\na✝ : Nontrivial S\nhp0 : p ≠ 0\nq : R[X][X] := minpoly R[X] p\nm : ℕ := (q.support.sup fun i ↦ (q.coeff i).natDegree) + p.natDegree + 1\nhm₁ : ∀ (i : ℕ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 67
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : S[X]\nhp : IsIntegral R[X] p\ni : ℕ\na✝¹ : Nontrivial R\na✝ : Nontrivial S\nhp0 : p ≠ 0\nq : R[X][X] := minpoly R[X] p\nm : ℕ := (q.support.sup fun i ↦ (q.coeff i).natDegree) + p.natDegree + 1\nhm₁ : ∀ (i : ℕ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 35
} | [
{
"pp": "case inr.inl\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : S[X]\nhp : IsIntegral R[X] p\ni : ℕ\na✝¹ : Nontrivial R\na✝ : Nontrivial S\nhp0 : p ≠ 0\nq : R[X][X] := minpoly R[X] p\nm : ℕ := (q.support.sup fun i ↦ (q.coeff i).natDegree) + p.natDegree + 1\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 177,
"column": 8
} | {
"line": 177,
"column": 48
} | [
{
"pp": "R : Type u_4\nA : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nx : A\np : A[X]\nmonic : p.Monic\ndeg : p.natDegree ≠ 0\nhx : IsIntegral R (eval x p)\nhp : ∀ (i : ℕ), IsIntegral R (p.coeff i)\nq : (↥(integralClosure R A))[X]\nhqp : Polynomial.map (algebraMap (↥(integralClosur... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 229,
"column": 4
} | {
"line": 229,
"column": 91
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ : Type w\nhσ : Finite σ\nα β : Type w\ne : α ≃ β\nIH :\n ∀ {f : MvPolynomial α S},\n (algebraMap (MvPolynomial α R) (MvPolynomial α S)).IsIntegralElem f → ∀ (n : α →₀ ℕ), IsIntegral R (coeff n f)\nf : MvPol... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 238,
"column": 6
} | {
"line": 238,
"column": 74
} | [
{
"pp": "case h.e'_6\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ : Type w\nhσ : Finite σ\nf : MvPolynomial PEmpty.{w + 1} S\nH : (algebraMap (MvPolynomial PEmpty.{w + 1} R) (MvPolynomial PEmpty.{w + 1} S)).IsIntegralElem f\nthis : constantCoeff = (isEmptyAlgEqui... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Basic | {
"line": 408,
"column": 30
} | {
"line": 408,
"column": 45
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP Q : Ideal S\ninst✝² : P.IsPrime\ninst✝¹ : Q.IsPrime\nh₁ : P ≤ Q\ninst✝ : QuasiFiniteAt R Q\nf : Localization.AtPrime Q →ₐ[R] Localization.AtPrime P := IsLocalization.liftAlgHom ⋯\nx s : S\nhs : s ∈ P.primeComp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Basic | {
"line": 445,
"column": 36
} | {
"line": 445,
"column": 47
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\np : Ideal S\ninst✝³ : p.IsPrime\ninst✝² : IsArtinianRing R\ninst✝¹ : EssFiniteType R S\ninst✝ : QuasiFiniteAt R p\n⊢ ∀ {x₁ x₂ : Localization.AtPrime p}, LinearMap.id x₁ = LinearMap.id x₂ → ∃ c, c • x₁ = c • x₂",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Basic | {
"line": 492,
"column": 6
} | {
"line": 492,
"column": 44
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsArtinianRing R\np : PrimeSpectrum S\ninst✝ : FiniteType R S\nH✝ : IsClopen {p}\nthis✝ : IsNoetherianRing S\nthis : IsJacobsonRing S\ne : S\nhe : IsIdempotentElem e\nH : {p} = ↑(PrimeSpectrum.basicOpen... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Basic | {
"line": 538,
"column": 6
} | {
"line": 538,
"column": 17
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\ninst✝⁴ : p.IsPrime\nq : Ideal S\ninst✝³ : q.IsPrime\ninst✝² : q.LiesOver p\ninst✝¹ : EssFiniteType R S\ninst✝ : QuasiFiniteAt R q\ne : ↑(PrimeSpectrum.comap (algebraMap R S) ⁻¹' {{ asIdeal := p, isP... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Basic | {
"line": 539,
"column": 6
} | {
"line": 539,
"column": 83
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\ninst✝⁴ : p.IsPrime\nq : Ideal S\ninst✝³ : q.IsPrime\ninst✝² : q.LiesOver p\ninst✝¹ : EssFiniteType R S\ninst✝ : QuasiFiniteAt R q\ne : ↑(PrimeSpectrum.comap (algebraMap R S) ⁻¹' {{ asIdeal := p, isP... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Polynomial | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 26
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nf : R[X] →ₐ[R] S\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : Algebra.WeaklyQuasiFiniteAt R P\n⊢ Ideal.map C (Ideal.under R P) < Ideal.comap (↑f) P",
"usedConstants": [
"CommSemiring.toSemiring",
"P... | algebraize [f.toRingHom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.QuasiFinite.Polynomial | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 26
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nf : R[X] →ₐ[R] S\nhf : Function.Surjective ⇑f\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : Algebra.WeaklyQuasiFiniteAt R P\nH : RingHom.ker f ≤ Ideal.map C (Ideal.under R P)\n⊢ False",
"usedConstants": [
... | algebraize [f.toRingHom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.QuasiFinite.Polynomial | {
"line": 76,
"column": 4
} | {
"line": 76,
"column": 57
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nf : R[X] →ₐ[R] S\nhf : Function.Surjective ⇑f\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : Algebra.WeaklyQuasiFiniteAt R P\nH : RingHom.ker f ≤ Ideal.map C (Ideal.under R P)\nalgInst✝ : Algebra R[X] S := f.toAlgebr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 97,
"column": 33
} | {
"line": 97,
"column": 69
} | [
{
"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\np : Ideal S\ninst✝² : p.IsPrime\ninst✝¹ : QuasiFinite R (Localization.AtPrime p ⧸ Ideal.map (algebraMap R (Localization.AtPrime p)) (Ideal.under R p))\nq ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 150,
"column": 8
} | {
"line": 150,
"column": 26
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP Q : Ideal S\ninst✝² : P.IsPrime\ninst✝¹ : Q.IsPrime\nh₁ : P ≤ Q\nh₂ : Ideal.under R P = Ideal.under R Q\ninst✝ : WeaklyQuasiFiniteAt R Q\n⊢ RingHom.ker (Ideal.Quotient.mk (Ideal.map (algebraMap R S) (Ideal.und... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 40
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP Q : Ideal S\ninst✝² : P.IsPrime\ninst✝¹ : Q.IsPrime\nh₁ : P ≤ Q\nh₂ : Ideal.under R P = Ideal.under R Q\ninst✝ : WeaklyQuasiFiniteAt R Q\nthis✝¹ : (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R S) (Ide... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 216,
"column": 4
} | {
"line": 217,
"column": 52
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\nq : Ideal T\ninst✝¹ : q.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R q\nthis : QuasiFiniteAt S (Ideal.map (I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 239,
"column": 4
} | {
"line": 240,
"column": 63
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝⁴ : q.IsPrime\ninst✝³ : p.IsPrime\ninst✝² : q.LiesOver p\nQ : Ideal (p.Fiber S)\ninst✝¹ : Q.IsPrime\nhQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q\ninst✝ : QuasiFinit... | rw [← IsLocalization.AtPrime.map_eq_maximalIdeal p, Ideal.map_le_iff_le_comap,
← Ideal.comap_coe (F := AlgHom _ _ _), Ideal.comap_comap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 241,
"column": 4
} | {
"line": 241,
"column": 47
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝⁴ : q.IsPrime\ninst✝³ : p.IsPrime\ninst✝² : q.LiesOver p\nQ : Ideal (p.Fiber S)\ninst✝¹ : Q.IsPrime\nhQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q\ninst✝ : QuasiFinit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 254,
"column": 8
} | {
"line": 254,
"column": 58
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝⁴ : q.IsPrime\ninst✝³ : p.IsPrime\ninst✝² : q.LiesOver p\nQ : Ideal (p.Fiber S)\ninst✝¹ : Q.IsPrime\nhQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q\ninst✝ : QuasiFinit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 263,
"column": 6
} | {
"line": 263,
"column": 24
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝⁴ : q.IsPrime\ninst✝³ : p.IsPrime\ninst✝² : q.LiesOver p\nQ : Ideal (p.Fiber S)\ninst✝¹ : Q.IsPrime\nhQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q\nins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.ResidueField.Polynomial | {
"line": 60,
"column": 20
} | {
"line": 60,
"column": 64
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nI : Ideal R\ninst✝⁴ : I.IsPrime\nJ : Ideal R[X]\ninst✝³ : J.IsPrime\ninst✝² : J.LiesOver I\ninst✝¹ : Algebra (Localization.AtPrime I) (Localization.AtPrime J)\ninst✝ : Localization.AtPrime.IsLiesOverAlgebra I J\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.ResidueField.Polynomial | {
"line": 61,
"column": 6
} | {
"line": 61,
"column": 65
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nI : Ideal R\ninst✝⁴ : I.IsPrime\nJ : Ideal R[X]\ninst✝³ : J.IsPrime\ninst✝² : J.LiesOver I\ninst✝¹ : Algebra (Localization.AtPrime I) (Localization.AtPrime J)\ninst✝ : Localization.AtPrime.IsLiesOverAlgebra I J\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.ResidueField.Polynomial | {
"line": 64,
"column": 6
} | {
"line": 64,
"column": 86
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nI : Ideal R\ninst✝⁴ : I.IsPrime\nJ : Ideal R[X]\ninst✝³ : J.IsPrime\ninst✝² : J.LiesOver I\ninst✝¹ : Algebra (Localization.AtPrime I) (Localization.AtPrime J)\ninst✝ : Localization.AtPrime.IsLiesOverAlgebra I J\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.ResidueField.Polynomial | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 57
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nI : Ideal R\ninst✝⁴ : I.IsPrime\nJ : Ideal R[X]\ninst✝³ : J.IsPrime\ninst✝² : J.LiesOver I\ninst✝¹ : Algebra (Localization.AtPrime I) (Localization.AtPrime J)\ninst✝ : Localization.AtPrime.IsLiesO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 39
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : LocallyQuasiFinite f\ny : ↥Y\n⊢ IsDiscrete (⇑f ⁻¹' {y})",
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"CategoryTheory.ConcreteCategory.hom",
"CommRingCat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 39
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : LocallyQuasiFinite f\ninst✝ : QuasiCompact f\ny : ↥Y\n⊢ (⇑f ⁻¹' {y}).Finite",
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.carrier",
"CategoryTheory.ConcreteCategory.hom",
"CommRingCat",
"TopCat.instCategory",
"ContinuousMap",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.ResidueField.Polynomial | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 15
} | [
{
"pp": "case refine_4.h.H\nR : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nI : Ideal R\ninst✝⁵ : I.IsPrime\nJ : Ideal R[X]\ninst✝⁴ : J.IsPrime\ninst✝³ : J.LiesOver I\ninst✝² : Algebra (Localization.AtPrime I) (Localization.AtPrime J)\ninst✝¹ : Localization.AtPrime.Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.ResidueField.Polynomial | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 52
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝¹ : CommRing R\nP : Ideal R\ninst✝ : P.IsPrime\nI : Ideal R[X]\np : P.ResidueField[X]\nhp : Ideal.map (mapRingHom (algebraMap R P.ResidueField)) I = Ideal.span {p}\nthis✝¹ : Algebra (R ⧸ P)[X] P.ResidueField[X] := (mapRingHom (algebraMap (R ⧸ P) P.ResidueField)).toAlge... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski | {
"line": 190,
"column": 58
} | {
"line": 190,
"column": 69
} | [
{
"pp": "X : Scheme\nU : X.AffineZariskiSite\ns : Set ↑Γ(X, U.toOpens)\nx : ↥X\nhx : x ∈ U.toOpens\nf : ↑Γ(X, ↑U)\nhfs : f ∈ s\nhxf : x ∈ X.basicOpen f\n⊢ x ∈ (U.basicOpen (f * 1)).toOpens",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski | {
"line": 334,
"column": 2
} | {
"line": 334,
"column": 13
} | [
{
"pp": "X : Scheme\nF : X.AffineZariskiSiteᵒᵖ ⥤ CommRingCat\nα : (toOpensFunctor X).op ⋙ X.presheaf ⟶ F\nH : NatTrans.Coequifibered α\nU : X.AffineZariskiSite\n⊢ (relativeGluingData H).toBase ⁻¹ᵁ ↑U = Hom.opensRange ((relativeGluingData H).cover.f U)",
"usedConstants": [
"AlgebraicGeometry.Scheme",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 162,
"column": 4
} | {
"line": 163,
"column": 35
} | [
{
"pp": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\np : FractionRing K[X]\nh : { toFractionRing := p } ≠ 0\nthis : p ≠ 0\n⊢ { toFractionRing := p } * { toFractionRing := p }⁻¹ = 1",
"usedConstants": [
"RatFunc.ofFractionRing.injEq",
"Eq.mpr",
"Polynomial.instOne",
"IsDomain... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 332,
"column": 6
} | {
"line": 332,
"column": 32
} | [
{
"pp": "K : Type u\ninst✝⁶ : CommRing K\nG₀ : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst✝⁵ : CommGroupWithZero G₀\ninst✝⁴ : Field L\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : FunLike F R[X] S[X]\ninst✝ : MonoidHomClass F R[X] S[X]\nφ : F\nhφ : R[X]⁰ ≤ Submonoid.comap φ S[X]... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 345,
"column": 42
} | {
"line": 345,
"column": 53
} | [
{
"pp": "K : Type u\ninst✝⁶ : CommRing K\nG₀ : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst✝⁵ : CommGroupWithZero G₀\ninst✝⁴ : Field L\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : FunLike F R[X] S[X]\ninst✝ : MonoidHomClass F R[X] S[X]\nφ : F\nhφ : R[X]⁰ ≤ Submonoid.comap φ S[X]... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 362,
"column": 2
} | {
"line": 364,
"column": 45
} | [
{
"pp": "case H.H\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : FunLike F R[X] S[X]\ninst✝ : MonoidHomClass F R[X] S[X]\nφ : F\nhφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰\nhf : Function.Injective ⇑φ\ny✝¹ y✝ : R[X] × ↥R[X]⁰\nh :\n (map φ hφ) { toFractionRing := Localiza... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 605,
"column": 2
} | {
"line": 605,
"column": 13
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\nL : Type u_1\ninst✝ : Field L\nφ : K[X] →+* L\nhφ : K[X]⁰ ≤ Submonoid.comap φ L⁰\nx : K[X]\n⊢ (liftRingHom φ hφ) ((algebraMap K[X] K⟮X⟯) x) = φ x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 714,
"column": 35
} | {
"line": 714,
"column": 46
} | [
{
"pp": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nP : K⟮X⟯ → Prop\nx : K⟮X⟯\nf : ∀ (p q : K[X]), q ≠ 0 → P ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q)\np q : K[X]\nhq : q ≠ 0\n⊢ P (RatFunc.mk p q)",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"Algebra.algebraMap",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 914,
"column": 43
} | {
"line": 914,
"column": 81
} | [
{
"pp": "K : Type u\ninst✝ : Field K\n⊢ num 1 = 1",
"usedConstants": [
"RatFunc.num_div",
"Polynomial.monic_one._simp_1",
"Eq.mpr",
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"False"... | convert! num_div (1 : K[X]) 1 <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 914,
"column": 43
} | {
"line": 914,
"column": 81
} | [
{
"pp": "K : Type u\ninst✝ : Field K\n⊢ num 1 = 1",
"usedConstants": [
"RatFunc.num_div",
"Polynomial.monic_one._simp_1",
"Eq.mpr",
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"False"... | convert! num_div (1 : K[X]) 1 <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 914,
"column": 43
} | {
"line": 914,
"column": 81
} | [
{
"pp": "K : Type u\ninst✝ : Field K\n⊢ num 1 = 1",
"usedConstants": [
"RatFunc.num_div",
"Polynomial.monic_one._simp_1",
"Eq.mpr",
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"False"... | convert! num_div (1 : K[X]) 1 <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 924,
"column": 4
} | {
"line": 924,
"column": 71
} | [
{
"pp": "K : Type u\ninst✝ : Field K\np q : K[X]\nhq : q ≠ 0\n⊢ (q / gcd p q).leadingCoeff⁻¹ ≠ 0",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"instHDiv",
"GroupWithZero.toDivisionMonoid",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOne... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 930,
"column": 61
} | {
"line": 930,
"column": 72
} | [
{
"pp": "K : Type u\ninst✝ : Field K\np q : K[X]\nhq : q ≠ 0\n⊢ C (q / gcd p q).leadingCoeff⁻¹ * (p / gcd p q) ∣ p",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 973,
"column": 4
} | {
"line": 973,
"column": 71
} | [
{
"pp": "case neg\nK : Type u\ninst✝ : Field K\np q : K[X]\nhq : ¬q = 0\n⊢ (q / gcd p q).leadingCoeff⁻¹ ≠ 0",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"instHDiv",
"GroupWithZero.toDivisionMonoid",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMono... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 979,
"column": 2
} | {
"line": 979,
"column": 66
} | [
{
"pp": "case f\nK : Type u\ninst✝ : Field K\np q : K[X]\nhq : q ≠ 0\n⊢ (algebraMap K[X] K⟮X⟯) ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q).num /\n (algebraMap K[X] K⟮X⟯) ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q).denom =\n (algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q",
"u... | have q_div_ne_zero : q / gcd p q ≠ 0 := right_div_gcd_ne_zero hq | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 1040,
"column": 4
} | {
"line": 1040,
"column": 26
} | [
{
"pp": "case mpr\nK : Type u\ninst✝ : Field K\np : K[X]\nhp : p ≠ 0\nq : K[X]\nhq : q ≠ 0\n⊢ ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q).num ∣ p",
"usedConstants": [
"RatFunc.num_div_dvd"
]
}
] | exact num_div_dvd p hq | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 90,
"column": 35
} | {
"line": 90,
"column": 61
} | [
{
"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\nx : S\nh : P.HasMap x\nf : S →ₐ[R] T\n⊢ IsUnit ((aeval (f x)) P.g)",
"usedConstants": [
"Eq.mpr",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 96,
"column": 11
} | {
"line": 96,
"column": 28
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : StandardEtalePair R\nx : S\nh : P.HasMap x\np₁ p₂ : R[X]\nn : ℕ\ne : derivative P.f * p₁ + P.f * p₂ = P.g ^ n\n⊢ (aeval x) P.g ^ n = (aeval x) (derivative P.f) * ?m.78",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 18
} | [
{
"pp": "case e_self.e_a.huv\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : StandardEtalePair R\nf g : P.Ring →ₐ[R] S\nH✝ : f P.X = g P.X\nH :\n (f.comp (Ideal.Quotient.mkₐ R (Ideal.span {C P.f, Y * C P.g - 1}))).comp CAlgHom =\n (g.comp (Ideal.Quotient.mkₐ R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 67
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : StandardEtalePair R\nI : Ideal S\nhI : I ^ 2 = ⊥\nx : S\nhx : P.HasMap ((Ideal.Quotient.mk I) x)\nhf : (aeval x) P.f ∈ I\na : S\nha : (aeval x) P.g * a - 1 ∈ I\np₁ p₂ : R[X]\nn : ℕ\ne : (aeval ... | rw [Polynomial.aeval_add_of_sq_eq_zero _ _ _ (by grind)]; grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 67
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : StandardEtalePair R\nI : Ideal S\nhI : I ^ 2 = ⊥\nx : S\nhx : P.HasMap ((Ideal.Quotient.mk I) x)\nhf : (aeval x) P.f ∈ I\na : S\nha : (aeval x) P.g * a - 1 ∈ I\np₁ p₂ : R[X]\nn : ℕ\ne : (aeval ... | rw [Polynomial.aeval_add_of_sq_eq_zero _ _ _ (by grind)]; grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 189,
"column": 52
} | {
"line": 189,
"column": 63
} | [
{
"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\nS : Type u_1\nx✝¹ : CommRing S\nx✝ : Algebra R S\nI : Ideal S\nhI : I ^ 2 = ⊥\nx : S\nhx : P.HasMap ((Ideal.Quotient.mk I) x)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 209,
"column": 4
} | {
"line": 209,
"column": 63
} | [
{
"pp": "case refine_3\nR : 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⊢ Submonoid.powers ((AdjoinRoot.mk P.f) P.g) ≤\n Submonoid.comap (AdjoinRoot.liftAlgHom P.f (Algebra.ofId R P.Ri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 226,
"column": 4
} | {
"line": 226,
"column": 63
} | [
{
"pp": "case refine_3\nR : 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⊢ Submonoid.powers P.g ≤ Submonoid.comap (aeval P.X) (IsUnit.submonoid P.Ring)",
"usedConstants": [
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 228,
"column": 4
} | {
"line": 228,
"column": 31
} | [
{
"pp": "case refine_4\nR : 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 {(algebraMap R[X] (Localization.Away P.g)) P.f} ≤ RingHom.ker (IsLocalization.liftAlgHom ⋯)",
"use... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 335,
"column": 2
} | {
"line": 335,
"column": 88
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : StandardEtalePresentation R S\nx : Localization.Away ((AdjoinRoot.mk P.f) P.g)\nn : ℕ\np : R[X]\ne :\n x *\n (algebraMap (AdjoinRoot P.f) (Localization.Away ((AdjoinRoot.mk P.f) P.g)))\n ↑((Adjoi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 365,
"column": 6
} | {
"line": 365,
"column": 44
} | [
{
"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\nP : StandardEtalePresentation R S\nalgInst✝ : Algebra R (P.map (algebraMap R T)).Ring :=\n ((algebraMap T (P.map (algebraMap R T... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 385,
"column": 27
} | {
"line": 385,
"column": 70
} | [
{
"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✝¹ : StandardEtalePair R\nP✝ : StandardEtalePresentation R S\nP : StandardEtalePair R\n⊢ Function.Bijective ⇑(P.lift P.X ⋯)",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 428,
"column": 21
} | {
"line": 430,
"column": 74
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : IsStandardEtale R S\nSₛ : Type u_4\ninst✝⁴ : CommRing Sₛ\ninst✝³ : Algebra S Sₛ\ninst✝² : Algebra R Sₛ\ninst✝¹ : IsScalarTower R S Sₛ\ns : S\ninst✝ : IsLocalization.Away s Sₛ\nP : StandardEtalePresentat... | by
simp [IsScalarTower.algebraMap_apply R S' (Localization.Away _),
- AlgEquiv.symm_toRingEquiv, IsScalarTower.algebraMap_eq R S Sₛ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 19
} | [
{
"pp": "case h.e'_3.h.e'_8.h₂\nR : Type u_1\nS : Type u_3\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nP : Ideal R\ninst✝ : P.IsPrime\nx : S\nhx : R[x] = ⊤\nhx' : Function.Surjective ⇑(aeval x)\nI : Ideal R[X] := ⋯\ne : P.Fiber S ≃ₐ[P.ResidueField] P.ResidueField[X] ⧸ Ideal.map (mapRingHom ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 15
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nP : Ideal R\ninst✝⁵ : P.IsPrime\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Q.LiesOver P\ninst✝² : IsUnramifiedAt R Q\nx : S\np : R[X]\ninst✝¹ : Algebra (Localization.AtPrime P) (Localization.AtPrime Q)\ninst✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 155,
"column": 27
} | {
"line": 155,
"column": 38
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : Module.Finite R S\nQ : Ideal S\ninst✝¹ : Q.IsPrime\ninst✝ : IsUnramifiedAt R Q\nh✝ : Nontrivial S\nthis✝ : Nontrivial R\nP✝ : Ideal R := Ideal.under R Q\nthis : Algebra (Localization.AtPrime P✝) (Locali... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 15
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : Module.Finite R S\nQ : Ideal S\ninst✝¹ : Q.IsPrime\ninst✝ : IsUnramifiedAt R Q\nh✝ : Nontrivial S\nthis✝¹ : Nontrivial R\nP✝ : Ideal R := Ideal.under R Q\nthis✝ : Algebra (Localization.AtPrime P✝) (Loca... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 224,
"column": 28
} | {
"line": 224,
"column": 48
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Module.Finite R S\ninst✝² : IsUnramifiedAt R Q\ninst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)\ninst✝ : Localization.AtPrime.IsLiesOv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 242,
"column": 4
} | {
"line": 242,
"column": 55
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Module.Finite R S\ninst✝² : IsUnramifiedAt R Q\ninst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)\ninst✝ : Localization.AtPrime.IsLiesOv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 244,
"column": 4
} | {
"line": 244,
"column": 46
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Module.Finite R S\ninst✝² : IsUnramifiedAt R Q\ninst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)\ninst✝ : Localization.AtPrime.IsLiesOv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 260,
"column": 6
} | {
"line": 260,
"column": 31
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Module.Finite R S\ninst✝² : IsUnramifiedAt R Q\ninst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)\ninst✝ : Localization.AtPrime.IsLiesOv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 15
} | [
{
"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\ny : B\nhy : IsIntegral R y\nh✝ : Nontrivial B\nhf : Monic 1\nhf' : ∀ (i : ℕ), IsIntegral R (coeff 1 i)\nhfx ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 66
} | [
{
"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\ny : B\nhy : IsIntegral R y\nh✝ : Nontrivial B\nhf : Monic 1\nhf' : ∀ (i : ℕ), IsIntegral R (coeff 1 i)\nhfx ... | simpa using (RingHom.ker_ne_top φ.toRingHom).symm.trans_eq hfx | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 220,
"column": 6
} | {
"line": 220,
"column": 29
} | [
{
"pp": "case refine_2\nR : 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.toRing... | · exact isIntegral_zero | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 221,
"column": 8
} | {
"line": 221,
"column": 19
} | [
{
"pp": "case refine_2\nR : 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.toRing... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 343,
"column": 6
} | {
"line": 344,
"column": 91
} | [
{
"pp": "case inr.refine_1\nR : Type u_1\nS : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nQ : Ideal S\ninst✝² : Q.IsPrime\ninst✝¹ : FiniteType R S\ninst✝ : IsUnramifiedAt R Q\nthis :\n ∀ {R : Type u_1} {S : Type u_3} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 68
} | [
{
"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 using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 398,
"column": 27
} | {
"line": 400,
"column": 16
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\np : Ideal S\ninst✝² : p.IsPrime\ninst✝¹ : FinitePresentation R S\ninst✝ : IsSmoothAt R p\nf : S\nhfp : f ∉ p\nH✝ : IsStandardSmooth R (Localization.Away f)\nn : ℕ\nφ : MvPolynomial (Fin n) R →+* Localization.Awa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 256,
"column": 6
} | {
"line": 256,
"column": 17
} | [
{
"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 using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 53
} | [
{
"pp": "case a\nR : Type u_1\ninst✝ : Semiring R\nf g : R[X]\nm n : ℕ\ni j : Fin (m + n)\n⊢ f.sylvester g m n i j =\n (Matrix.reindex (finCongr ⋯) (finSumFinEquiv.symm.trans ((Equiv.sumComm (Fin n) (Fin m)).trans finSumFinEquiv)))\n (g.sylvester f n m) i j",
"usedConstants": [
"Fin.addCases_l... | induction j using Fin.addCases <;> simp [sylvester] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 257,
"column": 4
} | {
"line": 257,
"column": 19
} | [
{
"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 using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 15
} | [
{
"pp": "case refine_1\nR : 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\ninst✝¹ : IsScalarTower R S T\np : Ideal T\ninst✝ : p.IsPrime\ns : S\nhsp : s ∉ Ideal.under S p\nhs : IsIntegral R s\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 270,
"column": 6
} | {
"line": 270,
"column": 88
} | [
{
"pp": "case refine_2\nR : 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.toRing... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 25
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nn m : ℕ\nthis : ∀ (i : Fin (m + 1 + n)), f.sylvester 0 (m + 1) n i ⟨0, ⋯⟩ = 0\n⊢ f.resultant 0 (m + 1) n = 0 ^ (m + 1) * f.coeff 0 ^ n",
"usedConstants": [
"Eq.mpr",
"False",
"Nat.instMulZeroClass",
"HMul.hMul",
"P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 13
} | [
{
"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 : R[X]\nht : φ.IsIntegralElem t\nhpm : p.Monic\nhpr : r.natDegree < p.natDegree\nhp : φ p * t = φ r\nSt : Type u_2 := Localization.Away t\nt' : St := IsLocalization.Away.invSelf t\nht... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 72
} | [
{
"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✝ : ... | generalize IsLocalization.integerNormalization (.powers a) q = q' at e | Lean.Elab.Tactic.evalGeneralize | Lean.Parser.Tactic.generalize |
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