module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 883,
"column": 6
} | {
"line": 883,
"column": 15
} | {
"line": 883,
"column": 16
} | [
{
"pp": "K : Type u\ninst✝ : Field K\np q : K[X]\nhq : q ≠ 0\n⊢ ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q).numDenom =\n (C (q / gcd p q).leadingCoeff⁻¹ * (p / gcd p q), C (q / gcd p q).leadingCoeff⁻¹ * (q / gcd p q))",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"K : Type u\ninst✝ : Field K\np q : K[X]\nhq : q ≠ 0\n⊢ ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q).liftOn'\n (fun p q ↦\n if q = 0 then (0, 1)\n else\n have r := gcd p q;\n (C (q / r).leadingCoeff⁻¹ * (p / r), C (q / r).leadingCoeff⁻¹ * (q / r)))\n ⋯ =\n (C (q... | numDenom, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 24
} | {
"line": 160,
"column": 2
} | [
{
"pp": "R : 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 : derivative P.f * p₁ + ... | [
"R : 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 x) (derivative P.f * p₁ + P... | apply_fun aeval x at e | Mathlib.Tactic._aux_Mathlib_Tactic_ApplyFun___elabRules_Mathlib_Tactic_applyFun_1 | Mathlib.Tactic.applyFun |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 391,
"column": 51
} | {
"line": 391,
"column": 59
} | {
"line": 391,
"column": 59
} | [
{
"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 := { f := X, monic_f := ⋯, g := 1, cond := ⋯ }\nthis : P.X = 0\n⊢ (a... | [] | simp [P] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 391,
"column": 51
} | {
"line": 391,
"column": 59
} | {
"line": 391,
"column": 59
} | [
{
"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 := { f := X, monic_f := ⋯, g := 1, cond := ⋯ }\nthis : P.X = 0\n⊢ (a... | [] | simp [P] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 391,
"column": 51
} | {
"line": 391,
"column": 59
} | {
"line": 391,
"column": 59
} | [
{
"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 := { f := X, monic_f := ⋯, g := 1, cond := ⋯ }\nthis : P.X = 0\n⊢ (a... | [] | simp [P] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 391,
"column": 64
} | {
"line": 391,
"column": 72
} | {
"line": 391,
"column": 72
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP✝¹ : StandardEtalePair R\nP✝ : StandardEtalePresentation R S\nP : StandardEtalePair R := { f := X, monic_f := ⋯, g := 1, cond := ⋯ }\nthis : P.X = 0\n⊢ Is... | [] | simp [P] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 391,
"column": 64
} | {
"line": 391,
"column": 72
} | {
"line": 391,
"column": 72
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP✝¹ : StandardEtalePair R\nP✝ : StandardEtalePresentation R S\nP : StandardEtalePair R := { f := X, monic_f := ⋯, g := 1, cond := ⋯ }\nthis : P.X = 0\n⊢ Is... | [] | simp [P] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 391,
"column": 64
} | {
"line": 391,
"column": 72
} | {
"line": 391,
"column": 72
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP✝¹ : StandardEtalePair R\nP✝ : StandardEtalePresentation R S\nP : StandardEtalePair R := { f := X, monic_f := ⋯, g := 1, cond := ⋯ }\nthis : P.X = 0\n⊢ Is... | [] | simp [P] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 65,
"column": 2
} | {
"line": 72,
"column": 72
} | {
"line": 74,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nf g : S\nH : HasStandardEtaleSurjectionOn R f\nh : f ∣ g\n⊢ HasStandardEtaleSurjectionOn R g",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"IsLocalization.Away.mapₐ_surjective_of_sur... | [] | obtain ⟨P, φ, hsurj⟩ := H
obtain ⟨g, rfl⟩ := h
obtain ⟨a, ha⟩ := hsurj (algebraMap _ _ g)
have : IsLocalization.Away (f * g) (Localization.Away (φ a)) :=
ha ▸ .mul' (Localization.Away f) _ _ _
have : IsStandardEtale R (Localization.Away a) := .of_isLocalizationAway a
exact .mk _ (IsLocalization.Away.mapₐ_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 65,
"column": 2
} | {
"line": 72,
"column": 72
} | {
"line": 74,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nf g : S\nH : HasStandardEtaleSurjectionOn R f\nh : f ∣ g\n⊢ HasStandardEtaleSurjectionOn R g",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"IsLocalization.Away.mapₐ_surjective_of_sur... | [] | obtain ⟨P, φ, hsurj⟩ := H
obtain ⟨g, rfl⟩ := h
obtain ⟨a, ha⟩ := hsurj (algebraMap _ _ g)
have : IsLocalization.Away (f * g) (Localization.Away (φ a)) :=
ha ▸ .mul' (Localization.Away f) _ _ _
have : IsStandardEtale R (Localization.Away a) := .of_isLocalizationAway a
exact .mk _ (IsLocalization.Away.mapₐ_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 82
} | {
"line": 228,
"column": 4
} | [
{
"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... | [
"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.IsLiesOverAlgebra (I... | obtain ⟨x, hx⟩ := Field.exists_primitive_element p.ResidueField Q.ResidueField | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 332,
"column": 8
} | {
"line": 332,
"column": 66
} | {
"line": 333,
"column": 4
} | [
{
"pp": "case e'_2\nR : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : Algebra.IsStandardEtale R S\n𝓟 : StandardEtalePresentation R S\nn : ℕ\n𝓟' : StandardEtalePresentation B (B ⊗[R] S) := 𝓟.baseChange\n... | [] | simp [← map_pow, 𝓟', StandardEtalePresentation.baseChange] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 332,
"column": 8
} | {
"line": 332,
"column": 66
} | {
"line": 333,
"column": 4
} | [
{
"pp": "case e'_3\nR : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : Algebra.IsStandardEtale R S\n𝓟 : StandardEtalePresentation R S\nn : ℕ\n𝓟' : StandardEtalePresentation B (B ⊗[R] S) := 𝓟.baseChange\n... | [] | simp [← map_pow, 𝓟', StandardEtalePresentation.baseChange] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 486,
"column": 16
} | {
"line": 486,
"column": 21
} | {
"line": 487,
"column": 2
} | [
{
"pp": "case h.zero.zero\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\ninst✝⁵ : Algebra R✝ S✝\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nx : S\nhx : R[x] = ⊤\nH : integralClosure R S = ⊥\nH₀ : F... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 486,
"column": 16
} | {
"line": 486,
"column": 21
} | {
"line": 487,
"column": 2
} | [
{
"pp": "case h.zero.succ\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\ninst✝⁵ : Algebra R✝ S✝\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nx : S\nhx : R[x] = ⊤\nH : integralClosure R S = ⊥\nH₀ : F... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 210,
"column": 8
} | {
"line": 210,
"column": 44
} | {
"line": 211,
"column": 8
} | [
{
"pp": "case succ.inl\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nr : R\nm n k : ℕ\nhk : k + m ≤ n\nhf : f.natDegree ≤ m\nhm : m ≠ 0\nM₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (g + f * (monomial k) r) m n\nM₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n\nM : ℕ → Matrix (Fin (m ... | [] | induction j₂ using Fin.addCases with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 69
} | {
"line": 244,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf g p : R[X]\nm n : ℕ\nhp : p.natDegree + m ≤ n\nhf : f.natDegree ≤ m\nH : p.support ⊆ Finset.range (n - m + 1)\n⊢ f.resultant (g + f * p) m n = f.resultant g m n",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toMo... | [
"R : Type u_1\ninst✝ : CommRing R\nf g p : R[X]\nm n : ℕ\nhp : p.natDegree + m ≤ n\nhf : f.natDegree ≤ m\nH : p.support ⊆ Finset.range (n - m + 1)\n⊢ f.resultant (g + f * ∑ n ∈ Finset.range (n - m + 1), (monomial n) (p.coeff n)) m n = f.resultant g m n"
] | rw [← p.sum_monomial_eq, Polynomial.sum_eq_of_subset _ (by simp) H] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 15
} | {
"line": 247,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf g p : R[X]\nm n : ℕ\nhp : p.natDegree + m ≤ n\nhf : f.natDegree ≤ m\nk : ℕ := n - m + 1\nH : k ≤ n - m + 1\n⊢ f.resultant (g + f * ∑ n ∈ Finset.range k, (monomial n) (p.coeff n)) m n = f.resultant g m n",
"ppTerm": "?m.120",
"assigned": true,
"usedConstan... | [
"R : Type u_1\ninst✝ : CommRing R\nf g p : R[X]\nm n : ℕ\nhp : p.natDegree + m ≤ n\nhf : f.natDegree ≤ m\nk : ℕ\nH : k ≤ n - m + 1\n⊢ f.resultant (g + f * ∑ n ∈ Finset.range k, (monomial n) (p.coeff n)) m n = f.resultant g m n"
] | clear_value k | Lean.Elab.Tactic.evalClearValue | Lean.Parser.Tactic.clearValue |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 281,
"column": 8
} | {
"line": 281,
"column": 44
} | {
"line": 282,
"column": 8
} | [
{
"pp": "case succ.inl\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nm n : ℕ\nr : R\nM₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (C r * g) m n\nM₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n\nM : ℕ → Matrix (Fin (m + n)) (Fin (m + n)) R := fun i ↦ Matrix.of fun j₁ j₂ ↦ if ↑j₂ < i t... | [] | induction j₂ using Fin.addCases with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 143,
"column": 92
} | {
"line": 156,
"column": 5
} | {
"line": 158,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\nU V : Y.AffineZariskiSite\ni : U ⟶ V\n⊢ Cover.trans (Precoverage.ZeroHypercover.pullback₁ f (directedCover Y)) i ≫\n (pullbackRestrictIsoRestrict f ↑V).hom ≫\n (f ⁻¹ᵁ ↑V).toSpecΓ ≫\n Spec.map (CommRingCat.o... | [] | by
have : (pullbackRestrictIsoRestrict f U.1).inv ≫
Cover.trans ((directedCover Y).pullback₁ f) i ≫
(pullbackRestrictIsoRestrict f V.1).hom = X.homOfLE
(f.preimage_mono (toOpens_mono i.1.1)) := by
rw [← cancel_mono (Scheme.Opens.ι _)]
simp +instances [Cover.trans, Cover.locallyDirectedPull... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 438,
"column": 79
} | {
"line": 438,
"column": 84
} | {
"line": 438,
"column": 84
} | [
{
"pp": "n : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multiset.map (fu... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 438,
"column": 79
} | {
"line": 438,
"column": 84
} | {
"line": 438,
"column": 84
} | [
{
"pp": "n : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multiset.map (fu... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 438,
"column": 79
} | {
"line": 438,
"column": 84
} | {
"line": 438,
"column": 84
} | [
{
"pp": "n : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multiset.map (fu... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 308,
"column": 8
} | {
"line": 308,
"column": 45
} | {
"line": 308,
"column": 45
} | [
{
"pp": "case hfg\nX Y : Scheme\nf : X ⟶ Y\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\ninst✝ : IsAffineHom f\n⊢ IsAffineHom (toNormalization f ≫ fromNormalization f)",
"ppTerm": "?hfg",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.fromNormalization",
"Eq.mp... | [
"case hfg\nX Y : Scheme\nf : X ⟶ Y\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\ninst✝ : IsAffineHom f\n⊢ IsAffineHom f"
] | Hom.toNormalization_fromNormalization | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 315,
"column": 8
} | {
"line": 315,
"column": 45
} | {
"line": 315,
"column": 45
} | [
{
"pp": "case hfg\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\n⊢ QuasiCompact (toNormalization f ≫ fromNormalization f)",
"ppTerm": "?hfg",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.fromNormalization",
"Eq.mpr",
"AlgebraicGeo... | [
"case hfg\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\n⊢ QuasiCompact f"
] | Hom.toNormalization_fromNormalization | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 321,
"column": 6
} | {
"line": 321,
"column": 43
} | {
"line": 321,
"column": 43
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\n⊢ QuasiSeparated (toNormalization f ≫ fromNormalization f)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.fromNormalization",
"Eq.mpr",
"AlgebraicGeometry.S... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\n⊢ QuasiSeparated f"
] | Hom.toNormalization_fromNormalization | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 446,
"column": 46
} | {
"line": 446,
"column": 90
} | {
"line": 446,
"column": 90
} | [
{
"pp": "n : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multiset.map (fu... | [] | simpa [r, natDegree_C_mul, hr₀] using hrd.le | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 446,
"column": 46
} | {
"line": 446,
"column": 90
} | {
"line": 446,
"column": 90
} | [
{
"pp": "n : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multiset.map (fu... | [] | simpa [r, natDegree_C_mul, hr₀] using hrd.le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 446,
"column": 46
} | {
"line": 446,
"column": 90
} | {
"line": 446,
"column": 90
} | [
{
"pp": "n : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multiset.map (fu... | [] | simpa [r, natDegree_C_mul, hr₀] using hrd.le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 385,
"column": 4
} | {
"line": 385,
"column": 16
} | {
"line": 386,
"column": 4
} | [
{
"pp": "case refine_2\nX Y : Scheme\nf : X ⟶ Y\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\nT : Scheme\nf₁ : X ⟶ T\nf₂ : T ⟶ Y\ninst✝ : IsIntegralHom f₂\nH : f = f₁ ≫ f₂\n⊢ ∀ ⦃X_1 Y_1 : (directedCover Y).I₀⦄ (f_1 : X_1 ⟶ Y_1),\n (normalizationGlueData f).functor.map f_1 ≫\n Spec.map\n ... | [
"case refine_2\nX Y : Scheme\nf : X ⟶ Y\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\nT : Scheme\nf₁ : X ⟶ T\nf₂ : T ⟶ Y\ninst✝ : IsIntegralHom f₂\nH : f = f₁ ≫ f₂\nU V : (directedCover Y).I₀\ni : U ⟶ V\n⊢ (normalizationGlueData f).functor.map i ≫\n Spec.map\n (CommRingCat.ofHom\n ((... | intros U V i | Lean.Elab.Tactic.evalIntros | Lean.Parser.Tactic.intros |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 142,
"column": 50
} | {
"line": 142,
"column": 70
} | {
"line": 142,
"column": 71
} | [
{
"pp": "case refine_2\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\nn m k : ℕ\nhn✝ hn : n = m + k\na✝¹ : Nontrivial R\na✝ : Nontrivial (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)\n⊢ (freeMonic R m).n... | [
"case refine_2\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\nn m k : ℕ\nhn✝ hn : n = m + k\na✝¹ : Nontrivial R\na✝ : Nontrivial (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)\n⊢ m + k = n"
] | natDegree_freeMonic, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 194,
"column": 66
} | {
"line": 194,
"column": 71
} | {
"line": 195,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nf : MvPolynomial (Fin m ⊕ Fin k) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R :=\n eval₂Hom (↑(universalFactorizationMap R n m k hn)) (Sum.elim (fun x ↦ X x ⊗ₜ[R] 1) fun x ↦ 1 ⊗ₜ[R] X x)\ni : Fin m ⊕ Fin k\... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 256,
"column": 6
} | {
"line": 256,
"column": 54
} | {
"line": 257,
"column": 6
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\ni : Fin m\nj : Fin n\nh : ↑j < ↑i\n⊢ ∀ x ∈ Finset.antidiagonal ↑j,\n (if h : x.2 < k then if x.1 < m ∧ x.1 = ↑i then X (Sum.inr ⟨x.2, h⟩) else 0\n else if x.2 = k ∧ x.1 < m ∧ x.1 = ↑i then 1 else 0) =\n 0",
"ppTerm"... | [
"case inl\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\ni : Fin m\nj : Fin n\nh : ↑j < ↑i\n⊢ ∀ (a b : ℕ),\n a + b = ↑j →\n (if h : b < k then if a < m ∧ a = ↑i then X (Sum.inr ⟨b, ⋯⟩) else 0\n else if b = k ∧ a < m ∧ a = ↑i then 1 else 0) =\n 0"
] | simp only [Finset.mem_antidiagonal, Prod.forall] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 611,
"column": 4
} | {
"line": 611,
"column": 9
} | {
"line": 612,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\ng : R[X]\nH : ∃ r, IsUnit r ∧ C r = g\n⊢ ∃ n, resultant 0 g ∣ leadingCoeff 0 ^ n",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"MulOne.toOne",
"Dvd.dvd",
"Poly... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 614,
"column": 4
} | {
"line": 614,
"column": 9
} | {
"line": 615,
"column": 2
} | [
{
"pp": "case inr.inl\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf : R[X]\nhf : f ≠ 0\nH : ∃ r, IsUnit r ∧ C r = f\n⊢ ∃ n, f.resultant 0 ∣ f.leadingCoeff ^ n",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"MulOne.toOne",
"D... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 636,
"column": 4
} | {
"line": 636,
"column": 9
} | {
"line": 637,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\ng : R[X]\nH : ∃ r, IsUnit r ∧ C r = g\n⊢ resultant 0 g ≠ 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"IsDomain.to_noZeroDivisors",
"Polynomial.resultant",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 649,
"column": 53
} | {
"line": 649,
"column": 58
} | {
"line": 650,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nι : Type u_3\nf : ι → R[X]\ng : R[X]\nn : ℕ\nhn : g.natDegree ≤ n\na : ι\ns : Finset ι\nhas : a ∉ s\nIH :\n ∏ i ∈ s, (f i).leadingCoeff ≠ 0 →\n (∏ i ∈ s, f i).resultant g (∏ i ∈ s, f i).natDegree n = ∏ i ∈ s, (f i).resultant g (f i).natDegree n\nhf : ∏ i ∈ insert a... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 716,
"column": 4
} | {
"line": 716,
"column": 57
} | {
"line": 717,
"column": 4
} | [
{
"pp": "case inr.inr.injective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH :\n ∀ (g : S[X]) (r : S),\n map φ f ≠ 0 →\n g ≠ 0 →\n ((map φ f).scaleRoots r).resultant (g.scaleRoots r) (map φ f)... | [
"case inr.inr.injective.h\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH :\n ∀ (g : S[X]) (r : S),\n map φ f ≠ 0 →\n g ≠ 0 →\n ((map φ f).scaleRoots r).resultant (g.scaleRoots r) (map φ f).natDegree... | · simpa [natDegree_map_eq_of_injective hφ] using this | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 729,
"column": 40
} | {
"line": 729,
"column": 45
} | {
"line": 729,
"column": 45
} | [
{
"pp": "R✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Surjective ⇑φ\nf : S[X]\nIH :\n ∀ (q g : R[X]) (r : R),\n q ≠ 0 →\n g ≠ 0 →\n (q.scaleRoots r).resultant (g.scaleRoots r) q.natDegree g.natDegree =\n r ^ (q.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 729,
"column": 40
} | {
"line": 729,
"column": 45
} | {
"line": 729,
"column": 45
} | [
{
"pp": "R✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Surjective ⇑φ\nf : S[X]\nIH :\n ∀ (q g : R[X]) (r : R),\n q ≠ 0 →\n g ≠ 0 →\n (q.scaleRoots r).resultant (g.scaleRoots r) q.natDegree g.natDegree =\n r ^ (q.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 729,
"column": 40
} | {
"line": 729,
"column": 45
} | {
"line": 729,
"column": 45
} | [
{
"pp": "R✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Surjective ⇑φ\nf : S[X]\nIH :\n ∀ (q g : R[X]) (r : R),\n q ≠ 0 →\n g ≠ 0 →\n (q.scaleRoots r).resultant (g.scaleRoots r) q.natDegree g.natDegree =\n r ^ (q.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 729,
"column": 51
} | {
"line": 729,
"column": 56
} | {
"line": 729,
"column": 56
} | [
{
"pp": "R✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Surjective ⇑φ\nf : S[X]\nIH :\n ∀ (q g : R[X]) (r : R),\n q ≠ 0 →\n g ≠ 0 →\n (q.scaleRoots r).resultant (g.scaleRoots r) q.natDegree g.natDegree =\n r ^ (q.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 729,
"column": 51
} | {
"line": 729,
"column": 56
} | {
"line": 729,
"column": 56
} | [
{
"pp": "R✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Surjective ⇑φ\nf : S[X]\nIH :\n ∀ (q g : R[X]) (r : R),\n q ≠ 0 →\n g ≠ 0 →\n (q.scaleRoots r).resultant (g.scaleRoots r) q.natDegree g.natDegree =\n r ^ (q.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 729,
"column": 51
} | {
"line": 729,
"column": 56
} | {
"line": 729,
"column": 56
} | [
{
"pp": "R✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Surjective ⇑φ\nf : S[X]\nIH :\n ∀ (q g : R[X]) (r : R),\n q ≠ 0 →\n g ≠ 0 →\n (q.scaleRoots r).resultant (g.scaleRoots r) q.natDegree g.natDegree =\n r ^ (q.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 529,
"column": 29
} | {
"line": 529,
"column": 49
} | {
"line": 529,
"column": 50
} | [
{
"pp": "case inr.inr.inr\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\np : MonicDegreeEq R n\nh✝² : Nontrivial 𝓡\nh✝¹ : Nontrivial (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)\nh✝ : Nontrivial R\nthis✝ : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k)... | [
"case inr.inr.inr\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\np : MonicDegreeEq R n\nh✝² : Nontrivial 𝓡\nh✝¹ : Nontrivial (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)\nh✝ : Nontrivial R\nthis✝ : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (M... | natDegree_freeMonic, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 601,
"column": 12
} | {
"line": 601,
"column": 31
} | {
"line": 601,
"column": 31
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nn m k : ℕ\nhn : n = m + k\np : MonicDegreeEq R n\nq : { q // ↑q.1 * ↑q.2 = map (algebraMap R S) ↑p ∧ IsCoprime ↑q.1 ↑q.2 }\nf : failed to pretty print expr... | [] | by cases n <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 913,
"column": 48
} | {
"line": 913,
"column": 53
} | {
"line": 914,
"column": 2
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\ng : K[X]\nhg : g ≠ 0\n⊢ (∀ (x : K), C x ≠ g) ↔ ∀ (x : K), x ≠ 0 → C x ≠ g",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"RingHom.instRingHomClass",
"False",
"eq_false",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 913,
"column": 48
} | {
"line": 913,
"column": 53
} | {
"line": 914,
"column": 2
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\ng : K[X]\nhg : g ≠ 0\n⊢ (∀ (x : K), C x ≠ g) ↔ ∀ (x : K), x ≠ 0 → C x ≠ g",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"RingHom.instRingHomClass",
"False",
"eq_false",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 913,
"column": 48
} | {
"line": 913,
"column": 53
} | {
"line": 914,
"column": 2
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\ng : K[X]\nhg : g ≠ 0\n⊢ (∀ (x : K), C x ≠ g) ↔ ∀ (x : K), x ≠ 0 → C x ≠ g",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"RingHom.instRingHomClass",
"False",
"eq_false",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 42
} | {
"line": 187,
"column": 2
} | [
{
"pp": "case refine_2\nX Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nx : ↥X\nhx : QuasiFiniteAt f x\nT : Scheme\nfT : T ⟶ Y\nleft✝¹ : Etale fT\nu : ↥T\nhu : fT u = f x\nV W : (pullback f fT).Opens\nv : ↥V\nhVW : IsCompl V W\nleft✝ : IsFinite (V.ι ≫ pul... | [] | · simp [← cancel_mono U'.ι, fTnU, fTn] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Ideal.CotangentBaseChange | {
"line": 100,
"column": 43
} | {
"line": 100,
"column": 79
} | {
"line": 100,
"column": 79
} | [
{
"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\nI : Ideal S\ninst✝ : Module.Flat R T\na : S →+* T ⊗[R] S := Algebra.TensorProduct.includeRight.toRingHom\nf : (map a I).Cotangent →ₗ[T] T ⊗[R] S ⧸ map a I... | [
"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\nI : Ideal S\ninst✝ : Module.Flat R T\na : S →+* T ⊗[R] S := Algebra.TensorProduct.includeRight.toRingHom\nf : (map a I).Cotangent →ₗ[T] T ⊗[R] S ⧸ map a I ^ 2 := ↑T (... | Ideal.toCotangent_to_quotient_square | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Cotangent.BaseChange | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 50
} | {
"line": 79,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Extension R S\nT : Type u_3\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nt : T\ns : S\nx : Ω[P.Ring⁄R]\n⊢ x ∈ Submodule.span P.Ring (Set.range ⇑(KaehlerDifferential.D R P.Ring))",
"ppTerm": "?m.100",
"... | [
"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Extension R S\nT : Type u_3\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nt : T\ns : S\nx : Ω[P.Ring⁄R]\n⊢ x ∈ ⊤"
] | rw [KaehlerDifferential.span_range_derivation] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 395,
"column": 2
} | {
"line": 395,
"column": 7
} | {
"line": 397,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nthis : Mono f → IsProper f → LocallyQuasiFinite f\n⊢ (IsProper f ∧ LocallyQuasiFinite f) ∧ Mono f ↔ IsProper f ∧ Mono f",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme",
"CategoryTheory.Mono",
"AlgebraicGeometry.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Extension.Cotangent.BaseChange | {
"line": 212,
"column": 2
} | {
"line": 213,
"column": 50
} | {
"line": 215,
"column": 0
} | [
{
"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✝ : Module.Flat R T\nt : T\nx : H1Cotangent R S\n⊢ t •\n (Extension.H1Cotangent.map\n ((Generators.baseChange T (Generators.self R S)).d... | [] | rw [← Extension.H1Cotangent.map_comp_apply, ← Extension.H1Cotangent.map_comp_apply,
H1Cotangent.map, Extension.H1Cotangent.map_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Etale.Descent | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 63
} | {
"line": 97,
"column": 2
} | [
{
"pp": "⊢ CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ Smooth) fun {R S} [CommRing R] [CommRing S] ↦ FaithfullyFlat",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"CommRing",
"Algebra.to_smulCommClass",
"RingHom.CodescendsAlong.mk",
"Algebra.algebraMap",
... | [
"R S T : Type u_1\nx✝⁴ : CommRing R\nx✝³ : CommRing S\nx✝² : CommRing T\nx✝¹ : Algebra R S\nx✝ : Algebra R T\nh : (algebraMap R S).FaithfullyFlat\nh' : (algebraMap S (S ⊗[R] T)).Smooth\n⊢ (algebraMap R T).Smooth"
] | refine .mk _ Smooth.respectsIso fun R S T _ _ _ _ _ h h' ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 320,
"column": 4
} | {
"line": 320,
"column": 20
} | {
"line": 321,
"column": 4
} | [
{
"pp": "R : Type u_2\nS : Type u_3\nR' : Type u_4\nR'' : Type u_5\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra R S\ninst✝⁸ : FiniteType R S\ninst✝⁷ : CommRing R'\ninst✝⁶ : Algebra R R'\ninst✝⁵ : CommRing R''\ninst✝⁴ : Algebra R R''\ninst✝³ : Algebra R'' S\ninst✝² : Algebra.IsIntegral R R''\nin... | [] | induction a with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Tactic.CategoryTheory.Bicategory.PureCoherence | {
"line": 73,
"column": 6
} | {
"line": 73,
"column": 12
} | {
"line": 73,
"column": 12
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c d : B\np : a ⟶ b\nf : b ⟶ c\ng h : c ⟶ d\npf : a ⟶ c\npfg : a ⟶ d\nη : g ≅ h\nη_f : p ≫ f ≅ pf\nη_fg : pf ≫ g ≅ pfg\nη_fh : pf ≫ h ≅ pfg\nih_η : pf ◁ η ≪≫ η_fh = η_fg\n⊢ p ◁ f ◁ η ≪≫ normalizeIsoComp η_f η_fh = normalizeIsoComp η_f η_fg",
"ppTerm": "?m.91",
... | [
"B : Type u\ninst✝ : Bicategory B\na b c d : B\np : a ⟶ b\nf : b ⟶ c\ng h : c ⟶ d\npf : a ⟶ c\npfg : a ⟶ d\nη : g ≅ h\nη_f : p ≫ f ≅ pf\nη_fg : pf ≫ g ≅ pfg\nη_fh : pf ≫ h ≅ pfg\nih_η : pf ◁ η ≪≫ η_fh = η_fg\n⊢ p ◁ f ◁ η ≪≫ normalizeIsoComp η_f η_fh = normalizeIsoComp η_f (pf ◁ η ≪≫ η_fh)"
] | ← ih_η | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Tactic.CategoryTheory.Bicategory.PureCoherence | {
"line": 80,
"column": 6
} | {
"line": 80,
"column": 12
} | {
"line": 80,
"column": 12
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c d : B\np : a ⟶ b\nf g : b ⟶ c\nh : c ⟶ d\npf : a ⟶ c\npfh : a ⟶ d\nη : f ≅ g\nη_f : p ≫ f ≅ pf\nη_g : p ≫ g ≅ pf\nη_fh : pf ≫ h ≅ pfh\nih_η : p ◁ η ≪≫ η_g = η_f\n⊢ p ◁ η ▷ h ≪≫ normalizeIsoComp η_g η_fh = normalizeIsoComp η_f η_fh",
"ppTerm": "?m.91",
"as... | [
"B : Type u\ninst✝ : Bicategory B\na b c d : B\np : a ⟶ b\nf g : b ⟶ c\nh : c ⟶ d\npf : a ⟶ c\npfh : a ⟶ d\nη : f ≅ g\nη_f : p ≫ f ≅ pf\nη_g : p ≫ g ≅ pf\nη_fh : pf ≫ h ≅ pfh\nih_η : p ◁ η ≪≫ η_g = η_f\n⊢ p ◁ η ▷ h ≪≫ normalizeIsoComp η_g η_fh = normalizeIsoComp (p ◁ η ≪≫ η_g) η_fh"
] | ← ih_η | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Sheaves.Over | {
"line": 45,
"column": 29
} | {
"line": 45,
"column": 34
} | {
"line": 45,
"column": 34
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\nU : Opens X\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nW : Opens ↥U\nx✝¹ : X\nx✝ : x✝¹ ∈ { carrier := Subtype.val '' ↑W, is_open' := ⋯ }\n⊢ x✝¹ ∈ U",
"ppTerm": "?m.160",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"SetLike.mem_coe.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Sheaves.Over | {
"line": 45,
"column": 29
} | {
"line": 45,
"column": 34
} | {
"line": 45,
"column": 34
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\nU : Opens X\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nW : Opens ↥U\nx✝¹ : X\nx✝ : x✝¹ ∈ { carrier := Subtype.val '' ↑W, is_open' := ⋯ }\n⊢ x✝¹ ∈ U",
"ppTerm": "?m.160",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"SetLike.mem_coe.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sheaves.Over | {
"line": 45,
"column": 29
} | {
"line": 45,
"column": 34
} | {
"line": 45,
"column": 34
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\nU : Opens X\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nW : Opens ↥U\nx✝¹ : X\nx✝ : x✝¹ ∈ { carrier := Subtype.val '' ↑W, is_open' := ⋯ }\n⊢ x✝¹ ∈ U",
"ppTerm": "?m.160",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"SetLike.mem_coe.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sheaves.Over | {
"line": 53,
"column": 56
} | {
"line": 53,
"column": 61
} | {
"line": 53,
"column": 61
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\nU : Opens X\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nV : Opens ↥U\n⊢ ({ obj := fun W ↦ Over.mk (homOfLE ⋯), map := fun {X_1 Y} f ↦ Over.homMk (homOfLE ⋯) ⋯, map_id := ⋯, map_comp := ⋯ } ⋙\n { obj := fun V ↦ { carrier := Subtype.val ⁻¹' ↑V.left, is_... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Sheaves.Over | {
"line": 53,
"column": 56
} | {
"line": 53,
"column": 61
} | {
"line": 53,
"column": 61
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\nU : Opens X\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nV : Opens ↥U\n⊢ ({ obj := fun W ↦ Over.mk (homOfLE ⋯), map := fun {X_1 Y} f ↦ Over.homMk (homOfLE ⋯) ⋯, map_id := ⋯, map_comp := ⋯ } ⋙\n { obj := fun V ↦ { carrier := Subtype.val ⁻¹' ↑V.left, is_... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sheaves.Over | {
"line": 53,
"column": 56
} | {
"line": 53,
"column": 61
} | {
"line": 53,
"column": 61
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\nU : Opens X\nA : Type u_1\ninst✝ : Category.{v_1, u_1} A\nV : Opens ↥U\n⊢ ({ obj := fun W ↦ Over.mk (homOfLE ⋯), map := fun {X_1 Y} f ↦ Over.homMk (homOfLE ⋯) ⋯, map_id := ⋯, map_comp := ⋯ } ⋙\n { obj := fun V ↦ { carrier := Subtype.val ⁻¹' ↑V.left, is_... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Bicategory.Adjunction.Basic | {
"line": 159,
"column": 15
} | {
"line": 159,
"column": 29
} | {
"line": 159,
"column": 29
} | [
{
"pp": "B : Type u₁\ninst✝ : Bicategory B\na b c : B\nf₁ : a ⟶ b\ng₁ : b ⟶ a\nf₂ : b ⟶ c\ng₂ : c ⟶ b\nadj₁ : f₁ ⊣ g₁\nadj₂ : f₂ ⊣ g₂\n⊢ 𝟙 ((g₂ ≫ g₁) ≫ 𝟙 a) ⊗≫\n g₂ ◁ rightZigzag adj₁.unit adj₁.counit ⊗≫ rightZigzag adj₂.unit adj₂.counit ▷ g₁ ⊗≫ 𝟙 (𝟙 c ≫ g₂ ≫ g₁) =\n (ρ_ (g₂ ≫ g₁)).hom ≫ (λ_ (g₂ ≫ g... | [
"B : Type u₁\ninst✝ : Bicategory B\na b c : B\nf₁ : a ⟶ b\ng₁ : b ⟶ a\nf₂ : b ⟶ c\ng₂ : c ⟶ b\nadj₁ : f₁ ⊣ g₁\nadj₂ : f₂ ⊣ g₂\n⊢ 𝟙 ((g₂ ≫ g₁) ≫ 𝟙 a) ⊗≫ g₂ ◁ ((ρ_ g₁).hom ≫ (λ_ g₁).inv) ⊗≫ ((ρ_ g₂).hom ≫ (λ_ g₂).inv) ▷ g₁ ⊗≫ 𝟙 (𝟙 c ≫ g₂ ≫ g₁) =\n (ρ_ (g₂ ≫ g₁)).hom ≫ (λ_ (g₂ ≫ g₁)).inv"
] | right_triangle | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 585,
"column": 4
} | {
"line": 587,
"column": 32
} | {
"line": 588,
"column": 4
} | [
{
"pp": "case h.succ\nR : Type u\nS : Type (max u v)\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\np : Ideal R\ninst✝ : p.IsPrime\nthis✝⁴ : IsArtinianRing (p.ResidueField ⊗[R] S)\nhpSfin : (p.primesOver S).Finite\nn✝ : ℕ\nIH :\n ∀ m < n✝ + 1,\n ∀ {R : Type u} {... | [
"case h.succ\nR : Type u\nS : Type (max u v)\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\np : Ideal R\ninst✝ : p.IsPrime\nthis✝⁵ : IsArtinianRing (p.ResidueField ⊗[R] S)\nhpSfin : (p.primesOver S).Finite\nn✝ : ℕ\nIH :\n ∀ m < n✝ + 1,\n ∀ {R : Type u} {S : Type (ma... | have := H' ((P''.map (Ideal.Quotient.mk (.span {φ e}))).comap e₁) inferInstance
(inferInstanceAs <| ((P''.map (Ideal.Quotient.mk (.span {φ e}))).comap
e₁.toAlgHom).LiesOver Q) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Bicategory.Adjunction.Mate | {
"line": 422,
"column": 41
} | {
"line": 424,
"column": 12
} | {
"line": 426,
"column": 0
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\nc d : B\nl₁ l₂ : c ⟶ d\nr₁ r₂ : d ⟶ c\nadj₁ : l₁ ⊣ r₁\nadj₂ : l₂ ⊣ r₂\nα : r₁ ⟶ r₂\n⊢ (conjugateEquiv adj₁ adj₂).symm α =\n (λ_ l₂).inv ≫ adj₁.unit ▷ l₂ ≫ (α_ l₁ r₁ l₂).hom ≫ l₁ ◁ α ▷ l₂ ≫ l₁ ◁ adj₂.counit ≫ (ρ_ l₁).hom",
"ppTerm": "?m.127",
"assigned": true... | [] | by
rw [conjugateEquiv_symm_apply, mateEquiv_symm_apply']
bicategory | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Flat.Rank | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 93
} | {
"line": 91,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : Flat R S\ninst✝¹ : Module.Finite R S\nh : ∀ (p : PrimeSpectrum R), rankAtStalk S p ≤ 1\np : Ideal R\ninst✝ : p.IsMaximal\nhr : rankAtStalk S { asIdeal := p, isPrime := ⋯ } = 0\n⊢ Module.rank (Localizati... | [] | simp [← finrank_eq_rank, ← rankAtStalk_eq_finrank_tensorProduct ⟨p, inferInstance⟩, hr] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Morphisms.FlatRank | {
"line": 176,
"column": 4
} | {
"line": 180,
"column": 28
} | {
"line": 181,
"column": 2
} | [
{
"pp": "case inr\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : Flat f\ninst✝ : IsFinite f\nx : ↥X\nthis : ∀ {X Y : Scheme} (f : X ⟶ Y) [Flat f] [IsFinite f] (x : ↥X), (∃ R, Y = Spec R) → 1 ≤ finrank f (f x)\nhY : ¬∃ R, Y = Spec R\n⊢ 1 ≤ finrank f (f x)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
... | [] | obtain ⟨R, g, hg, y, hy⟩ := Y.exists_Spec_apply_eq (f x)
rw [← hy, ← finrank_pullback_snd]
obtain ⟨z, hzl, hzr⟩ := Scheme.Pullback.exists_preimage_pullback (f := f) (g := g) x y hy.symm
rw [hzr.symm]
refine this _ _ ⟨_, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.FlatRank | {
"line": 176,
"column": 4
} | {
"line": 180,
"column": 28
} | {
"line": 181,
"column": 2
} | [
{
"pp": "case inr\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : Flat f\ninst✝ : IsFinite f\nx : ↥X\nthis : ∀ {X Y : Scheme} (f : X ⟶ Y) [Flat f] [IsFinite f] (x : ↥X), (∃ R, Y = Spec R) → 1 ≤ finrank f (f x)\nhY : ¬∃ R, Y = Spec R\n⊢ 1 ≤ finrank f (f x)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
... | [] | obtain ⟨R, g, hg, y, hy⟩ := Y.exists_Spec_apply_eq (f x)
rw [← hy, ← finrank_pullback_snd]
obtain ⟨z, hzl, hzr⟩ := Scheme.Pullback.exists_preimage_pullback (f := f) (g := g) x y hy.symm
rw [hzr.symm]
refine this _ _ ⟨_, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.FlatRank | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 37
} | {
"line": 261,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsIso f\nR : CommRingCat\ng : Spec R ⟶ Y\nw✝ : IsOpenImmersion g\ny : ↥(Spec R)\n⊢ finrank f (g y) = 1 (g y)",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Nontrivial",
"CommRingCat.carrier",
"CommRingCat.instCommRingObjForgetRi... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsIso f\nR : CommRingCat\ng : Spec R ⟶ Y\nw✝ : IsOpenImmersion g\ny : ↥(Spec R)\nthis : Nontrivial ↑R\n⊢ finrank f (g y) = 1 (g y)"
] | have : Nontrivial R := y.nontrivial | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Morphisms.FlatRank | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 60
} | {
"line": 281,
"column": 2
} | [
{
"pp": "case inr\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : Flat f\ninst✝ : IsFinite f\nh : finrank f = 1\nthis : ∀ {X Y : Scheme} (f : X ⟶ Y) [Flat f] [IsFinite f], finrank f = 1 → (∃ R, Y = Spec R) → IsIso f\nhY : ¬∃ R, Y = Spec R\ni : Y.affineCover.toPreZeroHypercover.1\ny : ↥(Y.affineCover.X i)\n⊢ finrank (pullbac... | [] | rw [finrank_pullback_snd, h, Pi.one_apply, Pi.one_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.OrderOfVanishing.Basic | {
"line": 219,
"column": 2
} | {
"line": 220,
"column": 33
} | {
"line": 221,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsPrincipalIdealRing R\nϖ : R\nhϖ : Irreducible ϖ\nthis : (span {ϖ}).IsMaximal\n⊢ IsSimpleModule R (R ⧸ span {ϖ})",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"IsScalarTower.right",
... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsPrincipalIdealRing R\nϖ : R\nhϖ : Irreducible ϖ\nthis : (span {ϖ}).IsMaximal\n⊢ IsSimpleModule (R ⧸ span {ϖ}) (R ⧸ span {ϖ})"
] | rw [isSimpleModule_iff_isSimpleModule_of_algebraMap_surjective (S := R ⧸ Ideal.span {ϖ})
Ideal.Quotient.mk_surjective] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.OrderOfVanishing.Basic | {
"line": 315,
"column": 2
} | {
"line": 315,
"column": 53
} | {
"line": 317,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsNoetherianRing R\ninst✝¹ : KrullDimLE 1 R\ninst✝ : Nontrivial R\ny : ↥(nonZeroDivisors R)\n⊢ IsUnit ((ordMonoidWithZeroHom R) ↑y)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Multiplicative.group",
"GroupWithZero.toMono... | [] | simp [ordMonoidWithZeroHom_eq_zero_iff, ord_ne_top] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.OrderOfVanishing.Basic | {
"line": 315,
"column": 2
} | {
"line": 315,
"column": 53
} | {
"line": 317,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsNoetherianRing R\ninst✝¹ : KrullDimLE 1 R\ninst✝ : Nontrivial R\ny : ↥(nonZeroDivisors R)\n⊢ IsUnit ((ordMonoidWithZeroHom R) ↑y)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Multiplicative.group",
"GroupWithZero.toMono... | [] | simp [ordMonoidWithZeroHom_eq_zero_iff, ord_ne_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.OrderOfVanishing.Basic | {
"line": 315,
"column": 2
} | {
"line": 315,
"column": 53
} | {
"line": 317,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsNoetherianRing R\ninst✝¹ : KrullDimLE 1 R\ninst✝ : Nontrivial R\ny : ↥(nonZeroDivisors R)\n⊢ IsUnit ((ordMonoidWithZeroHom R) ↑y)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Multiplicative.group",
"GroupWithZero.toMono... | [] | simp [ordMonoidWithZeroHom_eq_zero_iff, ord_ne_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.OrderOfVanishing | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 57
} | {
"line": 86,
"column": 2
} | [
{
"pp": "case neg\nX : Scheme\ninst✝¹ : IsIntegral X\ninst✝ : IsLocallyNoetherian X\nx : ↥X\nf g : ↑X.functionField\nhf : f ≠ 0\nhg : g ≠ 0\nhx : coheight x = 1\n⊢ ord (f * g) x = ord f x + ord g x",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.irreducibleSpace_of... | [
"case neg\nX : Scheme\ninst✝¹ : IsIntegral X\ninst✝ : IsLocallyNoetherian X\nx : ↥X\nf g : ↑X.functionField\nhf : f ≠ 0\nhg : g ≠ 0\nhx : coheight x = 1\n⊢ (ordHom x hx) (f * g) = ↑(Multiplicative.ofAdd (ord f x + ord g x))"
] | rw [ord_eq_iff hx <| (mul_ne_zero_iff_right hg).mpr hf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology | {
"line": 430,
"column": 30
} | {
"line": 430,
"column": 68
} | {
"line": 430,
"column": 69
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nx y : ProjectiveSpectrum 𝒜\n⊢ x.asHomogeneousIdeal ≤ y.asHomogeneousIdeal ↔ y ∈ closure {x}",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
... | [
"A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nx y : ProjectiveSpectrum 𝒜\n⊢ x.asHomogeneousIdeal ≤ y.asHomogeneousIdeal ↔ y ∈ zeroLocus 𝒜 ↑(vanishingIdeal {x})"
] | ← zeroLocus_vanishingIdeal_eq_closure, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.GradedAlgebra.Radical | {
"line": 110,
"column": 10
} | {
"line": 110,
"column": 44
} | {
"line": 111,
"column": 8
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : LinearOrder ι\ninst✝³ : IsOrderedCancelAddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\nI_ne_top : I ≠ ⊤\nhomogene... | [] | cases H₁ ⟨rfl, add_left_cancel H₄⟩ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.RingTheory.GradedAlgebra.Radical | {
"line": 56,
"column": 4
} | {
"line": 133,
"column": 26
} | {
"line": 133,
"column": 26
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : LinearOrder ι\ninst✝³ : IsOrderedCancelAddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\nI_ne_top : I ≠ ⊤\nhomogene... | [] | classical
/-
The idea of the proof is the following :
since `x * y ∈ I` and `I` homogeneous, then `proj i (x * y) ∈ I` for any `i : ι`.
Then consider two sets `{i ∈ x.support | xᵢ ∉ I}` and `{j ∈ y.support | yⱼ ∉ J}`;
let `max₁, max₂` be the maximum of the two sets, then `proj (max... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf | {
"line": 324,
"column": 2
} | {
"line": 325,
"column": 49
} | {
"line": 326,
"column": 2
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nx : ↑(ProjectiveSpectrum.top 𝒜)\nU : Opens ↑(ProjectiveSpectrum.top 𝒜)\nhxU : x ∈ U\ns : ToType ((Proj.structureSheaf 𝒜).presheaf.obj (op U))\nV : Opens ↑(Projecti... | [
"A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nx : ↑(ProjectiveSpectrum.top 𝒜)\nU : Opens ↑(ProjectiveSpectrum.top 𝒜)\nhxU : x ∈ U\ns : ToType ((Proj.structureSheaf 𝒜).presheaf.obj (op U))\nV : Opens ↑(ProjectiveSpectrum.t... | change ((Proj.structureSheaf 𝒜).presheaf.map (homOfLE <| fun _ h' ↦ h ⟨_, h'⟩).op) _ =
((Proj.structureSheaf 𝒜).presheaf.map i.op) s | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 88,
"column": 39
} | {
"line": 94,
"column": 43
} | {
"line": 96,
"column": 0
} | [
{
"pp": "σ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nι : Type u_2\nf : ι → A\nhf : (HomogeneousIdeal.irrelevant 𝒜).toIdeal ≤ Ideal.span (Set.range f)\n⊢ ⨆ i, basicOpen 𝒜 (f i) = ⊤",
"ppTerm": "?m.51",
"assigned"... | [] | by
classical
refine top_le_iff.mp fun x hx ↦ TopologicalSpace.Opens.mem_iSup.mpr ?_
by_contra! H
simp only [mem_basicOpen, Decidable.not_not] at H
refine x.not_irrelevant_le (hf.trans ?_)
rwa [Ideal.span_le, Set.range_subset_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 382,
"column": 18
} | {
"line": 382,
"column": 23
} | {
"line": 382,
"column": 23
} | [
{
"pp": "σ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nX : Scheme\nf : A →+* ↑Γ(X, ⊤)\nx x' : ↑Γ(X, ⊤)\nt t' : A\nd d' : ℕ\nH : f t = x\nh0d : 0 < d\nhd : t ∈ 𝒜 d\nH' : f t' = x'\nh0d' : 0 < d'\nhd' : t' ∈ 𝒜 d'\ns : A\nht... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 382,
"column": 18
} | {
"line": 382,
"column": 23
} | {
"line": 382,
"column": 23
} | [
{
"pp": "σ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nX : Scheme\nf : A →+* ↑Γ(X, ⊤)\nx x' : ↑Γ(X, ⊤)\nt t' : A\nd d' : ℕ\nH : f t = x\nh0d : 0 < d\nhd : t ∈ 𝒜 d\nH' : f t' = x'\nh0d' : 0 < d'\nhd' : t' ∈ 𝒜 d'\ns : A\nht... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 382,
"column": 18
} | {
"line": 382,
"column": 23
} | {
"line": 382,
"column": 23
} | [
{
"pp": "σ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nX : Scheme\nf : A →+* ↑Γ(X, ⊤)\nx x' : ↑Γ(X, ⊤)\nt t' : A\nd d' : ℕ\nH : f t = x\nh0d : 0 < d\nhd : t ∈ 𝒜 d\nH' : f t' = x'\nh0d' : 0 < d'\nhd' : t' ∈ 𝒜 d'\ns : A\nht... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 505,
"column": 2
} | {
"line": 513,
"column": 16
} | {
"line": 515,
"column": 0
} | [
{
"pp": "R : CommRingCat\nM : ModuleCat ↑R\n⊢ IsLocalizing (modulesSpecToSheaf.obj (tilde M))",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"CategoryTheory.Functor",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCar... | [] | intro f
-- We can't rewrite with `tilde.toOpen_res` below, because of def-eq abuse between
-- `Spec R` and `PrimeSpectrum R`.
have heq : tilde.toOpen M ⊤ ≫ (modulesSpecToSheaf.obj (tilde M)).obj.map (basicOpen f).leTop.op =
tilde.toOpen M (basicOpen f) :=
tilde.toOpen_res _ _ _ _
rw [← IsLocalizedModu... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 505,
"column": 2
} | {
"line": 513,
"column": 16
} | {
"line": 515,
"column": 0
} | [
{
"pp": "R : CommRingCat\nM : ModuleCat ↑R\n⊢ IsLocalizing (modulesSpecToSheaf.obj (tilde M))",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"CategoryTheory.Functor",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCar... | [] | intro f
-- We can't rewrite with `tilde.toOpen_res` below, because of def-eq abuse between
-- `Spec R` and `PrimeSpectrum R`.
have heq : tilde.toOpen M ⊤ ≫ (modulesSpecToSheaf.obj (tilde M)).obj.map (basicOpen f).leTop.op =
tilde.toOpen M (basicOpen f) :=
tilde.toOpen_res _ _ _ _
rw [← IsLocalizedModu... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.LocalRing.LocalSubring | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 67
} | {
"line": 111,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nK : Type u_3\ninst✝¹ : Field K\nA : Subring K\nP : Ideal ↥A\ninst✝ : P.IsPrime\nx y : Localization.AtPrime P\ne : (IsLocalization.liftAlgHom ⋯) x = (IsLocalization.liftAlgHom ⋯) y\n⊢ x = y",
"ppTerm": "?m.55",
"assigned": tru... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nK : Type u_3\ninst✝¹ : Field K\nA : Subring K\nP : Ideal ↥A\ninst✝ : P.IsPrime\ny : Localization.AtPrime P\nx : ↥A\ns : ↥P.primeCompl\ne : (IsLocalization.liftAlgHom ⋯) (IsLocalization.mk' (Localization.AtPrime P) x s) = (IsLocalization.liftAlgH... | obtain ⟨x, s, rfl⟩ := IsLocalization.exists_mk'_eq P.primeCompl x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.LocalRing.LocalSubring | {
"line": 112,
"column": 65
} | {
"line": 112,
"column": 70
} | {
"line": 113,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nK : Type u_3\ninst✝¹ : Field K\nA : Subring K\nP : Ideal ↥A\ninst✝ : P.IsPrime\nx : ↥A\ns : ↥P.primeCompl\ny : ↥A\nt : ↥P.primeCompl\ne :\n (IsLocalization.liftAlgHom ⋯) (IsLocalization.mk' (Localization.AtPrime P) x s) =\n (IsLo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 612,
"column": 2
} | {
"line": 614,
"column": 27
} | {
"line": 616,
"column": 0
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nx : ↑(ProjectiveSpectrum.top 𝒜)\nhx : x ∈ pbo f\n⊢ awayToSection 𝒜 f ≫ (structureSheaf 𝒜).presheaf.germ (pbo f) x hx =\n CommRingCat.ofHom (HomogeneousLo... | [] | ext z
apply (Proj.stalkIso' 𝒜 x).eq_symm_apply.mpr
apply Proj.stalkIso'_germ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 612,
"column": 2
} | {
"line": 614,
"column": 27
} | {
"line": 616,
"column": 0
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nx : ↑(ProjectiveSpectrum.top 𝒜)\nhx : x ∈ pbo f\n⊢ awayToSection 𝒜 f ≫ (structureSheaf 𝒜).presheaf.germ (pbo f) x hx =\n CommRingCat.ofHom (HomogeneousLo... | [] | ext z
apply (Proj.stalkIso' 𝒜 x).eq_symm_apply.mpr
apply Proj.stalkIso'_germ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 679,
"column": 2
} | {
"line": 679,
"column": 15
} | {
"line": 681,
"column": 0
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nx : ↑(Proj.restrict ⋯).toTopCat\nz : NumDenSameDeg 𝒜 (Submonoid.powers f)\n⊢ ↑{ deg := z.deg, num := ⟨(GradedRingHom.id 𝒜) ↑z.num, ⋯⟩, den := ⟨(GradedRingHom... | [] | exact not_not | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.ValuativeCriterion | {
"line": 179,
"column": 51
} | {
"line": 179,
"column": 63
} | {
"line": 179,
"column": 63
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nR : Type u\ncommRing✝ : CommRing R\ndomain✝ : IsDomain R\nvaluationRing✝ : ValuationRing R\nK : Type u\nfield✝ : Field K\nalgebra✝ : Algebra R K\nisFractionRing✝ : IsFractionRing R K\ni₁ : Spec (CommRingCat.of K) ⟶ X\ni₂ : Spec (CommRingCat.of R) ⟶ Y\nw : i₁ ≫ f = Spec.map (Com... | [
"X Y : Scheme\nf : X ⟶ Y\nR : Type u\ncommRing✝ : CommRing R\ndomain✝ : IsDomain R\nvaluationRing✝ : ValuationRing R\nK : Type u\nfield✝ : Field K\nalgebra✝ : Algebra R K\nisFractionRing✝ : IsFractionRing R K\ni₁ : Spec (CommRingCat.of K) ⟶ X\ni₂ : Spec (CommRingCat.of R) ⟶ Y\nw : i₁ ≫ f = Spec.map (CommRingCat.ofH... | e.hom_inv_id | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 766,
"column": 6
} | {
"line": 766,
"column": 49
} | {
"line": 767,
"column": 4
} | [
{
"pp": "case b_mem\nA : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nx : ↥(pbo f)\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nthis : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAlgebra\ny... | [] | exacts [y.den.2, z.num.2, z.den.2, y.num.2] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 774,
"column": 46
} | {
"line": 774,
"column": 91
} | {
"line": 775,
"column": 6
} | [
{
"pp": "case exists_of_eq.e_6.e_a\nA : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nx : ↥(pbo f)\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nthis : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ... | [
"case exists_of_eq.e_6.e_a\nA : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nx : ↥(pbo f)\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nthis : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAlgebra... | ← tsub_add_cancel_of_le (show 1 ≤ m from hm), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 850,
"column": 91
} | {
"line": 850,
"column": 96
} | {
"line": 851,
"column": 4
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nx : ↑Proj.T\nthis : ∀ (f : A) (m : ℕ), f ∈ 𝒜 m → 0 < m → f ∈ x.asHomogeneousIdeal\nz : A\nhz : z ∈ HomogeneousIdeal.irrelevant 𝒜\nk : ℕ\nhk : k ∈ DFinsupp.support (... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 850,
"column": 91
} | {
"line": 850,
"column": 96
} | {
"line": 851,
"column": 4
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nx : ↑Proj.T\nthis : ∀ (f : A) (m : ℕ), f ∈ 𝒜 m → 0 < m → f ∈ x.asHomogeneousIdeal\nz : A\nhz : z ∈ HomogeneousIdeal.irrelevant 𝒜\nk : ℕ\nhk : k ∈ DFinsupp.support (... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 850,
"column": 91
} | {
"line": 850,
"column": 96
} | {
"line": 851,
"column": 4
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nx : ↑Proj.T\nthis : ∀ (f : A) (m : ℕ), f ∈ 𝒜 m → 0 < m → f ∈ x.asHomogeneousIdeal\nz : A\nhz : z ∈ HomogeneousIdeal.irrelevant 𝒜\nk : ℕ\nhk : k ∈ DFinsupp.support (... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Sites.Small | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 43
} | {
"line": 116,
"column": 4
} | [
{
"pp": "case mp\nP : MorphismProperty Scheme\nS : Scheme\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : P.IsMultiplicative\ninst✝ : P.RespectsIso\nX : Over S\nR : Sieve X\nhR : (Sieve.overEquiv X) R ∈ (grothendieckTopology P) X.left\n𝒰 : Cover (precoverage P) X.left\nhle : Presieve.ofArrows 𝒰.X 𝒰.f ≤ ((Sieve... | [
"case mp\nP : MorphismProperty Scheme\nS : Scheme\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : P.IsMultiplicative\ninst✝ : P.RespectsIso\nX : Over S\nR : Sieve X\nhR : ∃ 𝒰, Presieve.ofArrows 𝒰.X 𝒰.f ≤ ((Sieve.overEquiv X) R).arrows\n𝒰 : Cover (precoverage P) X.left\nhle : Presieve.ofArrows 𝒰.X 𝒰.f ≤ ((Sieve.... | rw [mem_grothendieckTopology_iff] at hR | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Sites.Small | {
"line": 122,
"column": 4
} | {
"line": 125,
"column": 50
} | {
"line": 127,
"column": 0
} | [
{
"pp": "case mpr\nP : MorphismProperty Scheme\nS : Scheme\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : P.IsMultiplicative\ninst✝ : P.RespectsIso\nX : Over S\nR : Sieve X\n⊢ R ∈ (overPretopology P S).toGrothendieck X → (Sieve.overEquiv X) R ∈ (grothendieckTopology P) X.left",
"ppTerm": "?mpr",
"assigne... | [] | rintro ⟨T, ⟨𝒰, h, rfl⟩, hT⟩
rw [mem_grothendieckTopology_iff]
use 𝒰
rwa [Cover.toPresieveOver_le_arrows_iff] at hT | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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