module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.FieldTheory.SeparableDegree | {
"line": 417,
"column": 2
} | {
"line": 417,
"column": 52
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nx : F\nn : ℕ\n⊢ ((X - C x) ^ n).natSepDegree = if n = 0 then 0 else 1",
"usedConstants": [
"Polynomial.C",
"Polynomial.natSepDegree_pow",
"congrArg",
"HSub.hSub",
"RingHom",
"Field.toDivisionRing",
"DivisionRing.toRing",
"... | simp only [natSepDegree_pow, natSepDegree_X_sub_C] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.SeparableDegree | {
"line": 417,
"column": 2
} | {
"line": 417,
"column": 52
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nx : F\nn : ℕ\n⊢ ((X - C x) ^ n).natSepDegree = if n = 0 then 0 else 1",
"usedConstants": [
"Polynomial.C",
"Polynomial.natSepDegree_pow",
"congrArg",
"HSub.hSub",
"RingHom",
"Field.toDivisionRing",
"DivisionRing.toRing",
"... | simp only [natSepDegree_pow, natSepDegree_X_sub_C] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.SeparableDegree | {
"line": 417,
"column": 2
} | {
"line": 417,
"column": 52
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nx : F\nn : ℕ\n⊢ ((X - C x) ^ n).natSepDegree = if n = 0 then 0 else 1",
"usedConstants": [
"Polynomial.C",
"Polynomial.natSepDegree_pow",
"congrArg",
"HSub.hSub",
"RingHom",
"Field.toDivisionRing",
"DivisionRing.toRing",
"... | simp only [natSepDegree_pow, natSepDegree_X_sub_C] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.SeparableDegree | {
"line": 478,
"column": 58
} | {
"line": 485,
"column": 80
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nf : F[X]\nq : ℕ\nhF : ExpChar F q\nn : ℕ\n⊢ ((expand F (q ^ n)) f).natSepDegree = f.natSepDegree",
"usedConstants": [
"Multiset.toFinset",
"Iff.mpr",
"one_pow",
"Eq.mpr",
"MulOne.toOne",
"Polynomial.roots",
"Nat.Prime",
"E... | by
obtain - | hprime := hF
· simp only [one_pow, expand_one]
haveI := Fact.mk hprime
classical
simpa only [natSepDegree_eq_of_isAlgClosed (AlgebraicClosure F), aroots_def, map_expand,
Fintype.card_coe] using Fintype.card_eq.2
⟨(f.map (algebraMap F (AlgebraicClosure F))).rootsExpandPowEquivRoots q n⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Trace.Basic | {
"line": 68,
"column": 88
} | {
"line": 69,
"column": 81
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nh : PowerBasis R S\n⊢ (LinearMap.BilinForm.toMatrix h.basis) (traceForm R S) = of fun i j ↦ (trace R S) (h.gen ^ (↑i + ↑j))",
"usedConstants": [
"Eq.mpr",
"Algebra.to_smulCommClass",
"NonUni... | by
ext; rw [traceForm_toMatrix, of_apply, pow_add, h.basis_eq_pow, h.basis_eq_pow] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Trace.Basic | {
"line": 244,
"column": 30
} | {
"line": 244,
"column": 47
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝¹¹ : Field K\ninst✝¹⁰ : Field L\ninst✝⁹ : Algebra K L\nF : Type u_6\ninst✝⁸ : Field F\ninst✝⁷ : Algebra L F\ninst✝⁶ : Algebra K F\ninst✝⁵ : IsScalarTower K L F\nE : Type u_7\ninst✝⁴ : Field E\ninst✝³ : Algebra K E\ninst✝² : IsAlgClosed E\ninst✝¹ : FiniteDimensional K F\... | Finset.sum_sigma, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Trace.Basic | {
"line": 313,
"column": 26
} | {
"line": 313,
"column": 35
} | [
{
"pp": "case neg.succ\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nH : ¬Algebra.IsSeparable K L\np : ℕ\nhp : ExpChar K p\nthis : p ≠ 0\nx : L\nh₀ : FiniteDimensional K L\nhx : ¬IsSeparable K x\ng : K[X]\nhg₁ : g.Separable\nn : ℕ\nhg₂ : (expand K (p ^ (n + 1))) g = minpo... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.SeparableClosure | {
"line": 270,
"column": 60
} | {
"line": 270,
"column": 78
} | [
{
"pp": "case refine_1\nF : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK✝ : Type w\ninst✝¹ : Field K✝\ninst✝ : Algebra F K✝\nK L : IntermediateField F E\nle : K ≤ L\nx : E\nhx : x ∈ (fun K ↦ IntermediateField.restrictScalars F (separableClosure (↥K) E)) K\n⊢ x ∈ (fun K ↦ Inter... | dsimp only at hx ⊢ | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.FieldTheory.SeparableClosure | {
"line": 270,
"column": 60
} | {
"line": 270,
"column": 78
} | [
{
"pp": "case refine_2\nF : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK✝ : Type w\ninst✝¹ : Field K✝\ninst✝ : Algebra F K✝\nK : IntermediateField F E\nx : E\nhx :\n x ∈\n (fun K ↦ IntermediateField.restrictScalars F (separableClosure (↥K) E))\n ((fun K ↦ Intermediate... | dsimp only at hx ⊢ | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 330,
"column": 2
} | {
"line": 334,
"column": 26
} | [
{
"pp": "case h\nΓ : Type u_1\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nK : Type u_2\ninst✝ : Field K\nv : Valuation K Γ\nhv : v.IsRankOneDiscrete\nr : ↥v.valuationSubring\nhr : r ≠ 0\nπ : v.Uniformizer\nhr₀ : v ↑r ≠ 0\nvr : Γˣ := Units.mk0 (v ↑r) hr₀\nhvr_def : vr = Units.mk0 (v ↑r) hr₀\nm : ℤ\nhm : Units.mk... | have ha₀ : (↑a : K) ≠ 0 := by
simp only [zpow_neg, ne_eq, mul_eq_zero, inv_eq_zero, ZeroMemClass.coe_eq_zero, not_or, ha]
refine ⟨?_, hr⟩
rw [hn, zpow_natCast, pow_eq_zero_iff', not_and_or]
exact Or.inl π.ne_zero | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 347,
"column": 4
} | {
"line": 347,
"column": 26
} | [
{
"pp": "case hJ\nΓ : Type u_1\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nK : Type u_2\ninst✝ : Field K\nv : Valuation K Γ\nhv : v.IsRankOneDiscrete\nπ : v.Uniformizer\nh : IsUnit π.val\n⊢ False",
"usedConstants": [
"Field.toDivisionRing",
"DivisionRing.toRing",
"Valuation.Uniformizer.val... | apply π.2.not_isUnit h | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Algebra.Valued.WithVal | {
"line": 480,
"column": 4
} | {
"line": 480,
"column": 49
} | [
{
"pp": "R : Type u_4\nΓ₀ : Type u_5\nΓ₀' : Type u_6\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ₀'\nv : Valuation R Γ₀\nw : Valuation R Γ₀'\nhval : Valued R Γ₀'\nhv : Valued.v = w\nh : v.IsEquiv w\nγ : (ValueGroup₀ Valued.v)ˣ\nr s : R\nhr₀ : 0 < Valued.... | simp [restrict_pos_iff, h.pos_iff, ← hv, hs₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 141,
"column": 12
} | {
"line": 141,
"column": 36
} | [
{
"pp": "G : Type u_1\nG' : Type u_2\ninst✝⁵ : CommGroup G\ninst✝⁴ : LinearOrder G\ninst✝³ : IsOrderedMonoid G\ninst✝² : CommGroup G'\ninst✝¹ : LinearOrder G'\ninst✝ : IsOrderedMonoid G'\nx : G\ny : G'\nhxy : x = 1 ↔ y = 1\nhx : ¬x = 1\nx' : G := max x x⁻¹\nhx' : x' = max x x⁻¹\nxpos : 1 < x'\ny' : G' := max y ... | simpa [hyc] using a.prop | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 141,
"column": 12
} | {
"line": 141,
"column": 36
} | [
{
"pp": "G : Type u_1\nG' : Type u_2\ninst✝⁵ : CommGroup G\ninst✝⁴ : LinearOrder G\ninst✝³ : IsOrderedMonoid G\ninst✝² : CommGroup G'\ninst✝¹ : LinearOrder G'\ninst✝ : IsOrderedMonoid G'\nx : G\ny : G'\nhxy : x = 1 ↔ y = 1\nhx : ¬x = 1\nx' : G := max x x⁻¹\nhx' : x' = max x x⁻¹\nxpos : 1 < x'\ny' : G' := max y ... | simpa [hyc] using a.prop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 141,
"column": 12
} | {
"line": 141,
"column": 36
} | [
{
"pp": "G : Type u_1\nG' : Type u_2\ninst✝⁵ : CommGroup G\ninst✝⁴ : LinearOrder G\ninst✝³ : IsOrderedMonoid G\ninst✝² : CommGroup G'\ninst✝¹ : LinearOrder G'\ninst✝ : IsOrderedMonoid G'\nx : G\ny : G'\nhxy : x = 1 ↔ y = 1\nhx : ¬x = 1\nx' : G := max x x⁻¹\nhx' : x' = max x x⁻¹\nxpos : 1 < x'\ny' : G' := max y ... | simpa [hyc] using a.prop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 254,
"column": 4
} | {
"line": 262,
"column": 47
} | [
{
"pp": "case neg\nG : Type u_2\ninst✝³ : AddCommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedAddMonoid G\ninst✝ : Archimedean G\nH : ∀ (x : G), ¬IsLeast {y | 0 < y} x\n⊢ Nonempty (G ≃+o ℤ) ∨ DenselyOrdered G",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Math... | refine Or.inr ⟨?_⟩
intro x y hxy
specialize H (y - x)
obtain ⟨z, hz⟩ : ∃ z : G, 0 < z ∧ z < y - x := by
contrapose! H
refine ⟨by simp [hxy], fun _ ↦ H _⟩
refine ⟨x + z, ?_, ?_⟩
· simp [hz.left]
· simpa [lt_sub_iff_add_lt'] using hz.right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 254,
"column": 4
} | {
"line": 262,
"column": 47
} | [
{
"pp": "case neg\nG : Type u_2\ninst✝³ : AddCommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedAddMonoid G\ninst✝ : Archimedean G\nH : ∀ (x : G), ¬IsLeast {y | 0 < y} x\n⊢ Nonempty (G ≃+o ℤ) ∨ DenselyOrdered G",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Math... | refine Or.inr ⟨?_⟩
intro x y hxy
specialize H (y - x)
obtain ⟨z, hz⟩ : ∃ z : G, 0 < z ∧ z < y - x := by
contrapose! H
refine ⟨by simp [hxy], fun _ ↦ H _⟩
refine ⟨x + z, ?_, ?_⟩
· simp [hz.left]
· simpa [lt_sub_iff_add_lt'] using hz.right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyOpen | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 41
} | [
{
"pp": "case inr\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : LocallyOfFinitePresentation f\nhf : GeneralizingMap ⇑f\nthis :\n ∀ {X Y : Scheme} (f : X ⟶ Y) [LocallyOfFinitePresentation f],\n GeneralizingMap ⇑f →\n (∃ R, Y = Spec R) → topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ IsOpenMap)... | dsimp only [Scheme.Cover.pullbackHom] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 118,
"column": 14
} | {
"line": 118,
"column": 23
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nx y : R\nhx : ¬x = 0\nhy : y = 0\n⊢ (if x * 0 = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x * 0})).factors))) =\n (if x = 0 then 0 else exp (-↑((Associates.mk v.a... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 78
} | [
{
"pp": "case h.h\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nhv : Irreducible (Associates.mk v.asIdeal)\nhlt : v.asIdeal ^ 2 < v.asIdeal\nπ : R\nmem : π ∈ v.asIdeal\nnotMem : π ∉ v.asIdeal ^ 2\nhπ : Associates.mk (Ideal.span {π}) ≠ 0\n⊢ (Associates.mk v.asIdeal).cou... | rw [← Ideal.dvd_span_singleton, ← Associates.mk_le_mk_iff_dvd] at mem notMem | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Preserves | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nα : Type u_1\ninst✝ : Small.{w, u_1} α\nX : α → C\nc : Cofan X\nhc : IsColimit c\nthis : HasCoproduct X\nh : (Pi.lift fun i ↦ F.map (c.inj i).op) = F.map (Pi.lift fun i ↦ (c.inj i).op) ≫ piComparison F fun i ↦ op (X i)\n⊢ (piComparison F fun x ↦... | rw [h, ← Category.assoc, ← Functor.map_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Preserves | {
"line": 107,
"column": 2
} | {
"line": 107,
"column": 88
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nα : Type u_1\ninst✝² : Small.{w, u_1} α\nX : α → C\nc : Cofan X\nhc : IsColimit c\ninst✝¹ : (ofArrows X c.inj).HasPairwisePullbacks\ninst✝ : PreservesLimit (Discrete.functor fun x ↦ op (X x)) F\n⊢ IsSheafFor F (ofArrows X c.inj)",
"usedConst... | rw [Equalizer.Presieve.Arrows.sheaf_condition, Limits.Types.type_equalizer_iff_unique] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 634,
"column": 2
} | {
"line": 635,
"column": 13
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\na : adicCompletion K v\nha : a ∈ adicCompletionIntegers K v\n⊢ ∃ b ∈ R⁰, a * ↑b ∈ adicCompletionIntegers K v",
"usedCo... | · use 1
simp [ha] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Canonical | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 27
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nP✝ : Cᵒᵖ ⥤ Type v\nX✝ : C\nJ : GrothendieckTopology C\nP : Cᵒᵖ ⥤ Type v\nX Y : C\nf : Y ⟶ X\n⊢ Presieve.IsSheafFor P (Sieve.pullback f ⊤).arrows",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"co... | rw [Sieve.pullback_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Smooth.Basic | {
"line": 468,
"column": 62
} | {
"line": 468,
"column": 91
} | [
{
"pp": "R✝ : Type u\nA✝ : Type v\ninst✝¹⁴ : CommRing R✝\ninst✝¹³ : CommRing A✝\ninst✝¹² : Algebra R✝ A✝\nB✝ : Type u_1\nP : Type u_2\nC✝ : Type u_3\ninst✝¹¹ : CommRing B✝\ninst✝¹⁰ : Algebra R✝ B✝\ninst✝⁹ : CommRing C✝\ninst✝⁸ : Algebra R✝ C✝\ninst✝⁷ : CommRing P\ninst✝⁶ : Algebra R✝ P\nR : Type u_4\ninst✝⁵ : C... | by simpa [Algebra.ofId_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Smooth.Basic | {
"line": 460,
"column": 2
} | {
"line": 469,
"column": 87
} | [
{
"pp": "R✝ : Type u\nA✝ : Type v\ninst✝¹⁴ : CommRing R✝\ninst✝¹³ : CommRing A✝\ninst✝¹² : Algebra R✝ A✝\nB✝ : Type u_1\nP : Type u_2\nC : Type u_3\ninst✝¹¹ : CommRing B✝\ninst✝¹⁰ : Algebra R✝ B✝\ninst✝⁹ : CommRing C\ninst✝⁸ : Algebra R✝ C\ninst✝⁷ : CommRing P\ninst✝⁶ : Algebra R✝ P\nR : Type u_4\ninst✝⁵ : Comm... | refine .of_comp_surjective fun C _ _ I hI f ↦ ?_
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) ?_, ?_⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp Tens... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Smooth.Basic | {
"line": 460,
"column": 2
} | {
"line": 469,
"column": 87
} | [
{
"pp": "R✝ : Type u\nA✝ : Type v\ninst✝¹⁴ : CommRing R✝\ninst✝¹³ : CommRing A✝\ninst✝¹² : Algebra R✝ A✝\nB✝ : Type u_1\nP : Type u_2\nC : Type u_3\ninst✝¹¹ : CommRing B✝\ninst✝¹⁰ : Algebra R✝ B✝\ninst✝⁹ : CommRing C\ninst✝⁸ : Algebra R✝ C\ninst✝⁷ : CommRing P\ninst✝⁶ : Algebra R✝ P\nR : Type u_4\ninst✝⁵ : Comm... | refine .of_comp_surjective fun C _ _ I hI f ↦ ?_
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) ?_, ?_⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp Tens... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 424,
"column": 2
} | {
"line": 424,
"column": 80
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nι' : Type w₃\nσ : Type w₂\nσ' : Type w₄\nQ : Generators S T ι\nP : Generators R S σ\nQ' : Gen... | simp only [LinearMap.domRestrict_apply, Extension.Cotangent.map_mk, δ_eq_δAux] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.RingHom.Smooth | {
"line": 64,
"column": 2
} | {
"line": 67,
"column": 16
} | [
{
"pp": "⊢ IsStableUnderBaseChange @FormallySmooth",
"usedConstants": [
"Eq.mpr",
"CommRing",
"Algebra.to_smulCommClass",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"RingHom.IsStableUnderBaseChange.mk",
"congrArg",
"CommSe... | refine .mk respectsIso ?_
introv H
rw [formallySmooth_algebraMap] at H ⊢
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RingHom.Smooth | {
"line": 64,
"column": 2
} | {
"line": 67,
"column": 16
} | [
{
"pp": "⊢ IsStableUnderBaseChange @FormallySmooth",
"usedConstants": [
"Eq.mpr",
"CommRing",
"Algebra.to_smulCommClass",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"RingHom.IsStableUnderBaseChange.mk",
"congrArg",
"CommSe... | refine .mk respectsIso ?_
introv H
rw [formallySmooth_algebraMap] at H ⊢
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Smooth.Pi | {
"line": 76,
"column": 6
} | {
"line": 76,
"column": 97
} | [
{
"pp": "R : Type u_1\nI : Type u_2\nA : I → Type u_3\ninst✝³ : CommRing R\ninst✝² : (i : I) → CommRing (A i)\ninst✝¹ : (i : I) → Algebra R (A i)\ninst✝ : Finite I\nval✝ : Fintype I\nH : ∀ (i : I), FormallySmooth R (A i)\nB : Type (max u_1 u_2 u_3)\nx✝¹ : CommRing B\nx✝ : Algebra R B\nJ : Ideal B\nhJ : J ^ 2 = ... | rw [← mul_assoc, ← map_mul, mul_sub, mul_one, (he.idem i).eq, sub_self, map_zero, zero_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Unramified.Locus | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 78
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : FiniteType R A\np : Ideal A\ninst✝¹ : p.IsPrime\ninst✝ : IsUnramifiedAt R p\n⊢ ∃ f ∉ p, Unramified R (Localization.Away f)",
"usedConstants": [
"Algebra.exists_formallyUnramified_of_isUnramifi... | obtain ⟨f, hfp, H⟩ := exists_formallyUnramified_of_isUnramifiedAt (R := R) p | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Unramified.Pi | {
"line": 34,
"column": 4
} | {
"line": 35,
"column": 93
} | [
{
"pp": "case intro.mp\nR : Type u_1\nI : Type u_2\ninst✝³ : Finite I\nf : I → Type u_3\ninst✝² : CommRing R\ninst✝¹ : (i : I) → CommRing (f i)\ninst✝ : (i : I) → Algebra R (f i)\nval✝ : Fintype I\n⊢ FormallyUnramified R ((i : I) → f i) → ∀ (i : I), FormallyUnramified R (f i)",
"usedConstants": [
"Non... | intro _ i
exact FormallyUnramified.of_surjective (Pi.evalAlgHom R f i) (Function.surjective_eval i) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Unramified.Pi | {
"line": 34,
"column": 4
} | {
"line": 35,
"column": 93
} | [
{
"pp": "case intro.mp\nR : Type u_1\nI : Type u_2\ninst✝³ : Finite I\nf : I → Type u_3\ninst✝² : CommRing R\ninst✝¹ : (i : I) → CommRing (f i)\ninst✝ : (i : I) → Algebra R (f i)\nval✝ : Fintype I\n⊢ FormallyUnramified R ((i : I) → f i) → ∀ (i : I), FormallyUnramified R (f i)",
"usedConstants": [
"Non... | intro _ i
exact FormallyUnramified.of_surjective (Pi.evalAlgHom R f i) (Function.surjective_eval i) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.FreeLocus | {
"line": 279,
"column": 24
} | {
"line": 279,
"column": 37
} | [
{
"pp": "case intro\nR : Type uR\ninst✝⁵ : CommRing R\nι : Type u_1\ninst✝⁴ : Finite ι\nM : ι → Type u_2\ninst✝³ : (i : ι) → AddCommGroup (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : ∀ (i : ι), Flat R (M i)\ninst✝ : ∀ (i : ι), Module.Finite R (M i)\np : PrimeSpectrum R\nval✝ : Fintype ι\nf : ((i : ι) → M ... | e.finrank_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Spectrum.Prime.FreeLocus | {
"line": 348,
"column": 44
} | {
"line": 348,
"column": 57
} | [
{
"pp": "case h\nR : Type uR\nM : Type uM\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : Flat R M\ninst✝⁴ : Module.Finite R M\nN : Type u_1\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Module.Finite R N\ninst✝ : Flat R N\np : PrimeSpectrum R\ne : Localization.AtPrime p.a... | e.finrank_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Etale.Kaehler | {
"line": 384,
"column": 2
} | {
"line": 384,
"column": 45
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\nM : Submonoid S\ninst✝ : IsLocalization M T\n⊢ IsLocalizedModule M (map R R S T)",
"usedConstants"... | rw [isLocalizedModule_iff_isBaseChange M T] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Unramified.Finite | {
"line": 75,
"column": 67
} | {
"line": 77,
"column": 62
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : EssFiniteType R S\nthis✝ : ∀ (t : S ⊗[R] S), (TensorProduct.lmul' R) t = 1 ↔ 1 - t ∈ KaehlerDifferential.ideal R S\nt e : S ⊗[R] S\nht₁ : ∀ (s : S), (1 ⊗ₜ[R] s - s ⊗ₜ[R] 1) * (1 - e) = 0\nht₂ : e ∈ Ideal... | by
simpa [IsIdempotentElem, mul_sub, sub_eq_zero, eq_comm, -Ideal.submodule_span_eq,
Submodule.mem_annihilator_span_singleton] using this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Etale.Field | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 20
} | [
{
"pp": "case refine_1\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsSeparable K L\nB : Type (max u_1 u_2)\nx✝¹ : CommRing B\nx✝ : Algebra K B\nI : Ideal B\nh : I ^ 2 = ⊥\nf : L →ₐ[K] B ⧸ I\ng : (k : L) → ↥K⟮k⟯ →ₐ[K] B\nhg₁ : ∀ (k : L), (fun g ↦ (Ideal.... | change g 1 1 = 1 | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 121,
"column": 22
} | {
"line": 121,
"column": 33
} | [
{
"pp": "case smul\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : Algebra S B\ninst✝¹ : IsScalarTower R A B\ninst✝ : IsS... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 150,
"column": 62
} | {
"line": 150,
"column": 71
} | [
{
"pp": "case tmul.zero\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : Algebra S B\ninst✝¹ : IsScalarTower R A B\ninst✝ ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 258,
"column": 2
} | {
"line": 258,
"column": 38
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : Algebra S B\ninst✝¹ : IsScalarTower R A B\ninst✝ : IsScalarTower ... | refine { __ := e₂, map_smul' := ?_ } | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.AdicCompletion.Functoriality | {
"line": 418,
"column": 18
} | {
"line": 418,
"column": 23
} | [
{
"pp": "case right\nR : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nh : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)\nx : M\nn : ℕ\ny : N ⧸ I ^ n • ⊤\ny' : N ⧸ I ^ (n + 1) • ⊤\nhyy' ... | hx'y0 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 61
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : Module.Flat R S\ninst✝⁴ : FinitePresentation R S\np : Ideal R\nq : Ideal S\ninst✝³ : p.IsPrime\ninst✝² : q.IsPrime\ninst✝¹ : q.LiesOver p\ninst✝ : FormallySmooth p.ResidueField (p.Fiber S)\nRp : Type u_... | let Sp := Localization (algebraMapSubmonoid S p.primeCompl) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Smooth.StandardSmoothOfFree | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 94
} | [
{
"pp": "case inr\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FinitePresentation R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : IsSmoothAt R p\nthis✝ :\n ∀ (R : Type u_1) {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]\n ... | refine ⟨g * (IsLocalization.Away.sec g g').1, ?_, .of_algEquiv (e.restrictScalars R).symm⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.RingHom.LocallyStandardSmooth | {
"line": 47,
"column": 6
} | {
"line": 48,
"column": 35
} | [
{
"pp": "R S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nhf : Locally (fun {R S} [CommRing R] [CommRing S] ↦ IsStandardSmooth) f\n⊢ f.Smooth",
"usedConstants": [
"Eq.mpr",
"CommRing",
"RingHom.locally_iff_of_localizationSpanTarget",
"congrArg",
"id",
"... | ← locally_iff_of_localizationSpanTarget Smooth.propertyIsLocal.respectsIso
Smooth.ofLocalizationSpanTarget | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RingHom.StandardSmooth | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 74
} | [
{
"pp": "n : ℕ\nR : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Algebra.IsStandardSmoothOfRelativeDimension n R S\nthis✝ : (α : Type) → [_root_.Finite α] → Fintype α := Fintype.ofFinite\nι σ : Type\nw✝¹ : _root_.Finite σ\nw✝ : _root_.Finite ι\nP : Algebra.Submersi... | have : IsScalarTower R (MvPolynomial (Fin n) R) S := .to₁₂₄ _ _ P.Ring _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 144,
"column": 44
} | {
"line": 144,
"column": 64
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nι : Type u_4\nP : Generators R S ι\nσ : Type u_5\nb : Module.Basis σ S P.toExtension.Cotangent\nD : Aux P b\nx : ↥D.presLeft.toExtension.ker\n⊢ Extension.Cotangent.mk ⟨↑(AlgHom.id R P.Ring) ↑x, ⋯⟩ = Extension.Cot... | AlgHom.id_toRingHom, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.Morphisms.Smooth | {
"line": 77,
"column": 8
} | {
"line": 77,
"column": 19
} | [
{
"pp": "case h\nn m : ℕ\nX✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\nf : X ⟶ Y\n⊢ Smooth f ↔ affineLocally (fun {R S} [CommRing R] [CommRing S] ↦ RingHom.Smooth) f",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"CommRing",
"CommRi... | smooth_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 49
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\na : ι → K\nF : MvPolynomial ι k\nHF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree\nhF0 : F ≠ 0\nhFa : (aeval a) F = 0\nq₁ q₂ : MvPolynomial ι k\ne : F = q₁ * q₂\n... | rw [totalDegree_eq_zero_iff_eq_C.mp this] at ne | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.Proper | {
"line": 45,
"column": 92
} | {
"line": 48,
"column": 5
} | [
{
"pp": "⊢ @IsProper = @IsSeparated ⊓ @UniversallyClosed ⊓ @LocallyOfFiniteType",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"AlgebraicGeometry.Scheme",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.MorphismProperty.instCompl... | by
ext X Y f
rw [isProper_iff, ← and_assoc]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.FormallyUnramified | {
"line": 145,
"column": 2
} | {
"line": 146,
"column": 56
} | [
{
"pp": "X Y : Scheme\nf✝ : X ⟶ Y\nS R : CommRingCat\nf : R ⟶ S\ninst✝¹ : (CommRingCat.Hom.hom f).FormallyUnramified\ninst✝ : (CommRingCat.Hom.hom f).FiniteType\nalgInst✝ : Algebra ↑R ↑S := (CommRingCat.Hom.hom f).toAlgebra\nalgebraizeInst✝¹ : Algebra.FormallyUnramified ↑R ↑S\nalgebraizeInst✝ : Algebra.FiniteTy... | rw [show f = CommRingCat.ofHom (algebraMap R S) from rfl, diagonal_SpecMap R S,
cancel_right_of_respectsIso (P := @IsOpenImmersion)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.Descent | {
"line": 74,
"column": 2
} | {
"line": 91,
"column": 57
} | [
{
"pp": "case H.inr\nP P' : MorphismProperty Scheme\ninst✝² : IsZariskiLocalAtTarget P\ninst✝¹ : P'.IsStableUnderBaseChange\nH : ∀ {R : CommRingCat} {X Y : Scheme} (f : X ⟶ Spec R) (g : Y ⟶ Spec R), P' f → P (pullback.fst f g) → P g\nX Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : HasPullback f g\nh : P' f\nhf : ... | · rw [IsZariskiLocalAtTarget.iff_of_openCover (P := P) Z.affineCover]
intro i
let ι := Z.affineCover.f i
let e : pullback (pullback.snd f ι) (pullback.snd g ι) ≅
pullback (pullback.fst f g) (pullback.fst f ι) :=
pullbackLeftPullbackSndIso f ι (pullback.snd g ι) ≪≫
pullback.congrHom rfl... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Conductor | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 38
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p ∈ comap (algebraMap R S) (conductor R x)\nl : R →₀ S\nH : l ∈ Finsupp.supported S S ↑I\nH' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z\n⊢ (algebraMap R S) p * z ∈ ⇑(alg... | rw [← H', mul_comm, Finsupp.sum_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Conductor | {
"line": 136,
"column": 18
} | {
"line": 136,
"column": 24
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤\nh_alg : Function.Injective ⇑(algebraMap (↥R[x]) S)\nz : S\nhz : z ∈ R[x]\nhy : ⟨z, hz⟩ ∈ comap (algebraMap (↥R[x]) S) (Ideal.map (algebraMa... | ← temp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 243,
"column": 8
} | {
"line": 247,
"column": 15
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nσ : Type w\nhσ : Finite σ\nα : Type w\ninst✝ : Fintype α\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 ... | convert H
· ext i m
· aesop
· cases i <;> aesop
· aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 243,
"column": 8
} | {
"line": 247,
"column": 15
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nσ : Type w\nhσ : Finite σ\nα : Type w\ninst✝ : Fintype α\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 ... | convert H
· ext i m
· aesop
· cases i <;> aesop
· aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 63
} | [
{
"pp": "case pos\nK : Type u\ninst✝⁴ : CommRing K\nR : Type u_1\ninst✝³ : IsDomain K\ninst✝² : Monoid R\ninst✝¹ : DistribMulAction R K[X]\ninst✝ : IsScalarTower R K[X] K[X]\nc : R\np q : K[X]\nthis : SMulZeroClass R (FractionRing K[X]) := inferInstance\nhq : q = 0\n⊢ RatFunc.mk (c • p) q = c • RatFunc.mk p q",... | · rw [hq, mk_zero, mk_zero, ← ofFractionRing_smul, smul_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 206,
"column": 63
} | {
"line": 206,
"column": 74
} | [
{
"pp": "K : Type u\ninst✝⁴ : CommRing K\nR : Type u_1\ninst✝³ : IsDomain K\ninst✝² : Monoid R\ninst✝¹ : DistribMulAction R K[X]\ninst✝ : IsScalarTower R K[X] K[X]\nc : R\np : K[X]\nq✝ : K⟮X⟯\nq r : K[X]\nx✝ : r ≠ 0\n⊢ RatFunc.mk ((c • p) • q) r = c • p • RatFunc.mk q r",
"usedConstants": [
"Eq.mpr",
... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 198,
"column": 2
} | {
"line": 199,
"column": 46
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal S\ninst✝⁴ : p.IsPrime\ninst✝³ : QuasiFiniteAt R (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R S) (Ideal.under R p))) p)\nA : Type u_4\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nq : Ideal (A ⊗... | refine .of_surjectiveOnStalks (q.map φ.toRingHom) e.symm.toAlgHom
e.symm.toRingEquiv.surjectiveOnStalks _ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 503,
"column": 19
} | {
"line": 507,
"column": 61
} | [
{
"pp": "K : Type u\ninst✝³ : CommRing K\ninst✝² : IsDomain K\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R K[X]\nc : R\nx : K⟮X⟯\n⊢ c • x =\n { toFun := fun x ↦ RatFunc.mk ((algebraMap R K[X]) x) 1, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯,\n map_add' := ⋯ }\n c *\n x",
... | by
induction x using RatFunc.induction_on' with | _ p q hq
rw [RingHom.coe_mk, MonoidHom.coe_mk, OneHom.coe_mk, mk_one', ← mk_smul,
mk_def_of_ne (c • p) hq, mk_def_of_ne p hq, ← ofFractionRing_mul,
IsLocalization.mul_mk'_eq_mk'_of_mul, Algebra.smul_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski | {
"line": 227,
"column": 4
} | {
"line": 237,
"column": 7
} | [
{
"pp": "X : Scheme\nU V : (directedCover X).I₀\nx : ↥(pullback ((directedCover X).f U) ((directedCover X).f V))\n⊢ ∃ k hki hkj y, (pullback.lift (X.homOfLE ⋯) (X.homOfLE ⋯) ⋯) y = x",
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.Hom",
"Eq.mpr",
"CategoryTheory.Limits.pullback",
... | let a := (pullback.fst _ _ ≫ U.1.ι) x
have haU : a ∈ U.1 := (pullback.fst U.1.ι V.1.ι x).2
have haV : a ∈ V.1 := by unfold a; rw [pullback.condition]; exact (pullback.snd U.1.ι V.1.ι x).2
obtain ⟨f, g, e, hxf⟩ := exists_basicOpen_le_affine_inter U.2 V.2 _ ⟨haU, haV⟩
refine ⟨U.basicOpen f, homOfLE (U.bas... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski | {
"line": 227,
"column": 4
} | {
"line": 237,
"column": 7
} | [
{
"pp": "X : Scheme\nU V : (directedCover X).I₀\nx : ↥(pullback ((directedCover X).f U) ((directedCover X).f V))\n⊢ ∃ k hki hkj y, (pullback.lift (X.homOfLE ⋯) (X.homOfLE ⋯) ⋯) y = x",
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.Hom",
"Eq.mpr",
"CategoryTheory.Limits.pullback",
... | let a := (pullback.fst _ _ ≫ U.1.ι) x
have haU : a ∈ U.1 := (pullback.fst U.1.ι V.1.ι x).2
have haV : a ∈ V.1 := by unfold a; rw [pullback.condition]; exact (pullback.snd U.1.ι V.1.ι x).2
obtain ⟨f, g, e, hxf⟩ := exists_basicOpen_le_affine_inter U.2 V.2 _ ⟨haU, haV⟩
refine ⟨U.basicOpen f, homOfLE (U.bas... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite | {
"line": 398,
"column": 50
} | {
"line": 398,
"column": 73
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : LocallyOfFiniteType f\nthis :\n ∀ {X Y : Scheme} {f : X ⟶ Y} [LocallyOfFiniteType f] {x : ↥X},\n (∃ R, Y = Spec R) → (QuasiFiniteAt f x ↔ IsOpen {⟨x, ⋯⟩})\nhY : ¬∃ R, Y = Spec R\ni : Y.affineCover.I₀\nx : ↥(pullback f (Y.affineCover.f i))\nhy : (Y.affineCover.f i) (... | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 60
} | [
{
"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⊢ ∀ (a : AdjoinRoot P.f) (b : a ∈ Submonoid.powers ((AdjoinRoot.mk P.f) P.g)),\n IsUnit ((AdjoinRoot.liftAlgHom ... | change Submonoid.powers _ ≤ (IsUnit.submonoid _).comap _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 60
} | [
{
"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⊢ ∀ (a : R[X]) (b : a ∈ Submonoid.powers P.g), IsUnit ((aeval P.X) ↑⟨a, b⟩)",
"usedConstants": [
"CommSem... | change Submonoid.powers _ ≤ (IsUnit.submonoid _).comap _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 253,
"column": 8
} | {
"line": 253,
"column": 17
} | [
{
"pp": "case zero\nR : Type u_1\ninst✝ : CommRing R\nf p : R[X]\nm n : ℕ\nhp : p.natDegree + m ≤ n\nhf : f.natDegree ≤ m\ng : R[X]\nH : 0 ≤ n - m + 1\n⊢ f.resultant (g + f * 0) m n = f.resultant g m n",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hM... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 255,
"column": 31
} | {
"line": 255,
"column": 39
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : CommRing R\nf p : R[X]\nm n : ℕ\nhp : p.natDegree + m ≤ n\nhf : f.natDegree ≤ m\nk : ℕ\nIH :\n ∀ (g : R[X]),\n k ≤ n - m + 1 → f.resultant (g + f * ∑ n ∈ Finset.range k, (monomial n) (p.coeff n)) m n = f.resultant g m n\ng : R[X]\nH : k + 1 ≤ n - m + 1\n⊢ f.resultan... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 74,
"column": 34
} | {
"line": 74,
"column": 78
} | [
{
"pp": "case hf.a\nX Y : Scheme\nf : X ⟶ Y\nU V : (TopologicalSpace.Opens ↥Y)ᵒᵖ\ni : U ⟶ V\nx : ↑(Y.presheaf.obj U)\n⊢ (CommRingCat.Hom.hom\n (Y.presheaf.map i ≫\n CommRingCat.ofHom\n (algebraMap ↑Γ(Y, Opposite.unop V) ↥(integralClosure ↑Γ(Y, Opposite.unop V) ↑Γ(X, f ⁻¹ᵁ Opposite.uno... | exact Subtype.ext congr($(f.naturality i) x) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 399,
"column": 2
} | {
"line": 399,
"column": 54
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R[X]\nm : ℕ\nr : R\nhf : f.natDegree ≤ m\n⊢ f.resultant (X - C r) m 1 = (-1) ^ m * eval r f",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.eval",
"NegZeroClass.toNeg",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | rw [resultant_comm, resultant_X_sub_C_left _ _ _ hf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 600,
"column": 8
} | {
"line": 600,
"column": 67
} | [
{
"pp": "case refine_1.a\nn : ℕ\nIH :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal S) [inst_3 : p.IsPrime]\n [WeaklyQuasiFiniteAt R p] (f : MvPolynomial (Fin n) R →ₐ[R] S), f.Finite → ZariskisMainProperty R p\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Comm... | simp only [Subalgebra.restrictScalars_top, Algebra.map_top] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 442,
"column": 4
} | {
"line": 444,
"column": 52
} | [
{
"pp": "case refine_3\nX Y : Scheme\nf : X ⟶ Y\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\nT : Scheme\nf₁ f₂ : normalization f ⟶ T\ng : T ⟶ Y\ninst✝ : IsAffineHom g\nH₁ : toNormalization f ≫ f₁ = toNormalization f ≫ f₂\nhf₁ : f₁ ≫ g = fromNormalization f\nhf₂ : f₂ ≫ g = fromNormalization f\nU : (norma... | have h₁ : f ⁻¹ᵁ U.1 ≤ f₀ ⁻¹ᵁ g ⁻¹ᵁ U.1 := by
simp only [← Scheme.Hom.comp_preimage, f₀, Category.assoc,
hf₁, toNormalization_fromNormalization]; rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 770,
"column": 4
} | {
"line": 782,
"column": 71
} | [
{
"pp": "case Splits\nR✝ : Type u_1\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : Field R\nf : R[X]\nhf' : f.Splits\ng : R[X]\nr : R\n⊢ ((taylor r) f).resultant ((taylor r) g) = f.resultant g",
"usedConstants": [
"neg_add_rev",
"Polynomial.taylor_eval",
"one_pow",
"Eq.mpr",
"Pol... | induction hf' using Submonoid.closure_induction with
| mem x h =>
obtain (⟨s, rfl⟩ | ⟨s, rfl⟩) := h
· rw [taylor_C]; simp
· nontriviality R
rw [map_add, taylor_X, taylor_C, add_assoc, ← map_add]
simp [-map_add, taylor_eval]
| one => simp
| mul x y hx hy hx' hy' =>
by_... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 770,
"column": 4
} | {
"line": 782,
"column": 71
} | [
{
"pp": "case Splits\nR✝ : Type u_1\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : Field R\nf : R[X]\nhf' : f.Splits\ng : R[X]\nr : R\n⊢ ((taylor r) f).resultant ((taylor r) g) = f.resultant g",
"usedConstants": [
"neg_add_rev",
"Polynomial.taylor_eval",
"one_pow",
"Eq.mpr",
"Pol... | induction hf' using Submonoid.closure_induction with
| mem x h =>
obtain (⟨s, rfl⟩ | ⟨s, rfl⟩) := h
· rw [taylor_C]; simp
· nontriviality R
rw [map_add, taylor_X, taylor_C, add_assoc, ← map_add]
simp [-map_add, taylor_eval]
| one => simp
| mul x y hx hy hx' hy' =>
by_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 770,
"column": 4
} | {
"line": 782,
"column": 71
} | [
{
"pp": "case Splits\nR✝ : Type u_1\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : Field R\nf : R[X]\nhf' : f.Splits\ng : R[X]\nr : R\n⊢ ((taylor r) f).resultant ((taylor r) g) = f.resultant g",
"usedConstants": [
"neg_add_rev",
"Polynomial.taylor_eval",
"one_pow",
"Eq.mpr",
"Pol... | induction hf' using Submonoid.closure_induction with
| mem x h =>
obtain (⟨s, rfl⟩ | ⟨s, rfl⟩) := h
· rw [taylor_C]; simp
· nontriviality R
rw [map_add, taylor_X, taylor_C, add_assoc, ← map_add]
simp [-map_add, taylor_eval]
| one => simp
| mul x y hx hy hx' hy' =>
by_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 580,
"column": 6
} | {
"line": 580,
"column": 47
} | [
{
"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\nΔ : 𝓡 := (presentation m k hn p).jacobian\nhΔ : IsUnit ((algebraMap 𝓡 (Localization.Away Δ)) Δ)\nP : Al... | simp [Algebra.Presentation.dimension, hn] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Group.Abelian | {
"line": 88,
"column": 56
} | {
"line": 88,
"column": 79
} | [
{
"pp": "K : Type u\ninst✝⁴ : Field K\ninst✝³ : IsAlgClosed K\nG : Over (Spec (CommRingCat.of K))\ninst✝² : IsProper G.hom\ninst✝¹ : IsIntegral (G ⊗ G).left\ninst✝ : GrpObj G\nS : Scheme := Spec (CommRingCat.of K)\npoint : ↥S := IsLocalRing.closedPoint K\nhpoint : IsClosed {point}\nthis✝¹⁰ : Nonempty ↥G.left\nt... | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Tactic.CategoryTheory.Bicategory.PureCoherence | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 49
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b : B\nf g f' : a ⟶ b\nη θ : f ≅ g\nη_f : 𝟙 a ≫ f ≅ f'\nη_g : 𝟙 a ≫ g ≅ f'\nh_η : 𝟙 a ◁ η ≪≫ η_g = η_f\nh_θ : 𝟙 a ◁ θ ≪≫ η_g = η_f\n⊢ (λ_ f).inv ≫ η_f.hom ≫ η_g.inv ≫ (λ_ g).hom = θ.hom",
"usedConstants": [
"CategoryTheory.Category.assoc",
"Catego... | simp [← reassoc_of% (congrArg Iso.hom h_θ)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Tactic.CategoryTheory.Bicategory.PureCoherence | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 49
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b : B\nf g f' : a ⟶ b\nη θ : f ≅ g\nη_f : 𝟙 a ≫ f ≅ f'\nη_g : 𝟙 a ≫ g ≅ f'\nh_η : 𝟙 a ◁ η ≪≫ η_g = η_f\nh_θ : 𝟙 a ◁ θ ≪≫ η_g = η_f\n⊢ (λ_ f).inv ≫ η_f.hom ≫ η_g.inv ≫ (λ_ g).hom = θ.hom",
"usedConstants": [
"CategoryTheory.Category.assoc",
"Catego... | simp [← reassoc_of% (congrArg Iso.hom h_θ)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.CategoryTheory.Bicategory.PureCoherence | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 49
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b : B\nf g f' : a ⟶ b\nη θ : f ≅ g\nη_f : 𝟙 a ≫ f ≅ f'\nη_g : 𝟙 a ≫ g ≅ f'\nh_η : 𝟙 a ◁ η ≪≫ η_g = η_f\nh_θ : 𝟙 a ◁ θ ≪≫ η_g = η_f\n⊢ (λ_ f).inv ≫ η_f.hom ≫ η_g.inv ≫ (λ_ g).hom = θ.hom",
"usedConstants": [
"CategoryTheory.Category.assoc",
"Catego... | simp [← reassoc_of% (congrArg Iso.hom h_θ)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology | {
"line": 316,
"column": 8
} | {
"line": 316,
"column": 50
} | [
{
"pp": "case h₁\nA : 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\nt : Set (ProjectiveSpectrum 𝒜)\nx : ProjectiveSpectrum 𝒜\nhx : x ∈ zeroLocus 𝒜 ↑(vanishingIdeal t)\nfs : Set A\nht' : IsClosed (zeroLocus 𝒜 fs)\nht : t ... | subset_zeroLocus_iff_subset_vanishingIdeal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.FunLike.Graded | {
"line": 75,
"column": 25
} | {
"line": 75,
"column": 60
} | [
{
"pp": "case a\nE : Type u_1\nA : Type u_2\nB : Type u_3\nσ : Type u_4\nτ : Type u_5\nι : Type u_6\ninst✝³ : SetLike σ A\ninst✝² : SetLike τ B\n𝒜 : ι → σ\nℬ : ι → τ\ninst✝¹ : EquivLike E A B\ninst✝ : GradedEquivLike E 𝒜 ℬ\ne : E\ni : ι\nx✝ : ↥(ℬ i)\n⊢ ↑(subtypeMap e i ((fun y ↦ ⟨EquivLike.inv e ↑y, ⋯⟩) x✝)) ... | exact EquivLike.apply_inv_apply e _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 288,
"column": 2
} | {
"line": 291,
"column": 5
} | [
{
"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\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nm' : ℕ\ng : A\ng_deg : g ∈ 𝒜 m'\nhm' : 0 < m'\nx : A\nhx : x = f * g\n⊢ (pullbackAwayιIso 𝒜 f_deg hm g_deg hm' hx).hom ≫ S... | rw [← cancel_mono (awayι 𝒜 g g_deg hm'), ← Limits.pullback.condition,
← pullbackAwayιIso_hom_awayι 𝒜 f_deg hm g_deg hm' hx,
Category.assoc, SpecMap_awayMap_awayι]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 288,
"column": 2
} | {
"line": 291,
"column": 5
} | [
{
"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\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nm' : ℕ\ng : A\ng_deg : g ∈ 𝒜 m'\nhm' : 0 < m'\nx : A\nhx : x = f * g\n⊢ (pullbackAwayιIso 𝒜 f_deg hm g_deg hm' hx).hom ≫ S... | rw [← cancel_mono (awayι 𝒜 g g_deg hm'), ← Limits.pullback.condition,
← pullbackAwayιIso_hom_awayι 𝒜 f_deg hm g_deg hm' hx,
Category.assoc, SpecMap_awayMap_awayι]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 637,
"column": 61
} | {
"line": 637,
"column": 70
} | [
{
"pp": "case pos\nι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubgroupClass σ A\ninst✝² : AddCommMonoid ι\ninst✝¹ : DecidableEq ι\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ι\nhf : f ∈ 𝒜 m\nz : Away 𝒜 f\nk : ℕ\nhk : f ^ k = den z\nk' : ℕ\nhk' : k ≤ k'\... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 205,
"column": 8
} | {
"line": 205,
"column": 64
} | [
{
"pp": "R : CommRingCat\nM✝ : ModuleCat ↑R\nM : (Spec (CommRingCat.of ↑R)).Modules\nf : (↑R)ᵒᵖ\n⊢ ∀ (a : ↑R) (b : a ∈ Submonoid.powers (unop f)),\n IsUnit\n ((algebraMap (↑R)\n (Module.End ↑R\n ↑((modulesSpecToSheaf.obj M).obj.obj (op ((inducedFunctor PrimeSpectrum.basicOpen).obj (u... | change Submonoid.powers _ ≤ (IsUnit.submonoid _).comap _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 214,
"column": 10
} | {
"line": 214,
"column": 66
} | [
{
"pp": "case hf.map_unit\nR : CommRingCat\nM✝ : ModuleCat ↑R\nM : (Spec (CommRingCat.of ↑R)).Modules\nf g : (↑R)ᵒᵖ\ni : f ⟶ g\nN : ModuleCat ↑(CommRingCat.of ↑R) := (modulesSpecToSheaf.obj M).presheaf.obj (op ⊤)\n⊢ ∀ (a : ↑R) (b : a ∈ Submonoid.powers (unop f)),\n IsUnit\n ((algebraMap (↑R)\n ... | change Submonoid.powers _ ≤ (IsUnit.submonoid _).comap _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 261,
"column": 6
} | {
"line": 261,
"column": 62
} | [
{
"pp": "case h.a.w.h.hf.map_unit\nR : CommRingCat\nM✝ : ModuleCat ↑R\nM N : (Spec (CommRingCat.of ↑R)).Modules\nf : M ⟶ N\nr : (InducedCategory (TopologicalSpace.Opens ↥(Spec (CommRingCat.of ↑R))) PrimeSpectrum.basicOpen)ᵒᵖ\n⊢ ∀ (a : ↑R) (b : a ∈ Submonoid.powers (unop r)),\n IsUnit\n ((algebraMap (↑R)... | change Submonoid.powers _ ≤ (IsUnit.submonoid _).comap _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 291,
"column": 6
} | {
"line": 291,
"column": 62
} | [
{
"pp": "case h.a.w.h.hf.map_unit\nR : CommRingCat\nM✝ M : ModuleCat ↑R\nr : (InducedCategory (Opens ↥(Spec (CommRingCat.of ↑R))) basicOpen)ᵒᵖ\n⊢ ∀ (a : ↑R) (b : a ∈ Submonoid.powers (unop r)),\n IsUnit\n ((algebraMap (↑R)\n (Module.End ↑R\n ↑(((inducedFunctor basicOpen).op ⋙\n ... | change Submonoid.powers _ ≤ (IsUnit.submonoid _).comap _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 1034,
"column": 6
} | {
"line": 1034,
"column": 90
} | [
{
"pp": "case a.e_y.e_val.inr\nA : Type u_2\nσ : Type u_3\ninst✝⁴ : CommRing A\ninst✝³ : SetLike σ A\ninst✝² : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝¹ : GradedRing 𝒜\nf : A\nd : ℕ\nhf : f ∈ 𝒜 d\nι' : Type u_4\ninst✝ : Fintype ι'\nv : ι' → A\nhx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤\ndv : ι' → ℕ\nhxd :... | rw [← mul_le_mul_iff_of_pos_right hd, ← smul_eq_mul (a := a), ← hai, Finset.sum_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 1038,
"column": 2
} | {
"line": 1038,
"column": 25
} | [
{
"pp": "A : Type u_2\nσ : Type u_3\ninst✝⁴ : CommRing A\ninst✝³ : SetLike σ A\ninst✝² : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝¹ : GradedRing 𝒜\nf : A\nd : ℕ\nhf : f ∈ 𝒜 d\nι' : Type u_4\ninst✝ : Fintype ι'\nv : ι' → A\nhx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤\ndv : ι' → ℕ\nhxd : ∀ (i : ι'), v i ∈ 𝒜 ... | rw [H, SetLike.mem_coe] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 503,
"column": 50
} | {
"line": 507,
"column": 70
} | [
{
"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\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nx : ↑↑(Proj.restrict ⋯).toPresheafedSpace\n⊢ FromSpec.toFun f_deg hm ((ConcreteCategory.hom (toSpec 𝒜 f)) x) = x",
"u... | by
refine Subtype.ext <| ProjectiveSpectrum.ext <| HomogeneousIdeal.ext' ?_
intro i z hzi
refine (FromSpec.mem_carrier_iff_of_mem f_deg hm _ _ hzi).trans ?_
exact (ToSpec.mk_mem_carrier _ _).trans (x.1.2.pow_mem_iff_mem m hm) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 603,
"column": 28
} | {
"line": 603,
"column": 96
} | [
{
"pp": "case h.h.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✝² x✝¹ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nx✝ : failed to pretty print expression (u... | simp only [map_add, HomogeneousLocalization.val_add, Proj.add_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 227,
"column": 2
} | {
"line": 241,
"column": 14
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\nK : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Field K\ninst✝⁵ : IsDomain R\ninst✝⁴ : ValuationRing R\ninst✝³ : IsLocalRing S\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\nf : R →+* S\ng : S →+* K\nh : g.comp f = algebraMap R K\ninst✝ :... | · let V : ValuationSubring K :=
⟨(algebraMap R K).range, ValuationRing.isInteger_or_isInteger R⟩
suffices LocalSubring.range g ≤ V.toLocalSubring by
rintro ⟨_, x, rfl⟩
obtain ⟨y, hy⟩ := this.1 ⟨x, rfl⟩
exact ⟨y, Subtype.ext (by simpa [← h] using hy)⟩
apply V.isMax_toLocalSubring
have... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 140,
"column": 2
} | {
"line": 141,
"column": 49
} | [
{
"pp": "case refine_1\nR : CommRingCat\nH✝ : ∀ (I : Ideal ↑R), I.IsPrime → ∃ f ∉ I, ∀ (x y : ↑R), y * x = 0 → y ∉ I → ∃ n, f ^ n * x = 0\np : ↥(Spec R)\nf : ↑R\nhf : f ∉ p.asIdeal\nH : ∀ (x y : ↑R), y * x = 0 → y ∉ p.asIdeal → ∃ n, f ^ n * x = 0\n⊢ IsAffineOpen (PrimeSpectrum.basicOpen f)",
"usedConstants"... | · rw [← basicOpen_eq_of_affine]
exact (isAffineOpen_top (Spec R)).basicOpen _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.RationalMap | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 63
} | [
{
"pp": "case e_a.e_a.e_f\nX Y : Scheme\nf : X.PartialMap Y\nU : X.Opens\nhU : Dense ↑U\nhU' : U ≤ f.domain\nx : ↥X\nhx : x ∈ U\ne : ⟨x, ⋯⟩ = (X.homOfLE hU') ⟨x, hx⟩\n⊢ Hom.stalkMap f.domain.ι ⟨x, ⋯⟩ ≫ (↑f.domain).presheaf.stalkSpecializes ⋯ ≫ Hom.stalkMap (X.homOfLE hU') ⟨x, hx⟩ =\n X.presheaf.stalkSpeciali... | rw [← Hom.stalkSpecializes_stalkMap_assoc, Hom.stalkMap_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Sites.QuasiCompact | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 77
} | [
{
"pp": "S : Scheme\nX : Scheme\nE : PreZeroHypercover X\nF : (i : E.I₀) → PreZeroHypercover (E.X i)\nhE : qcCoverFamily.property E\nhF : ∀ (i : E.I₀), qcCoverFamily.property (F i)\n⊢ qcCoverFamily.property (E.bind F)",
"usedConstants": [
"CategoryTheory.PreZeroHypercover.bind",
"Eq.mpr",
... | simp only [qcCoverFamily_property, Scheme.quasiCompactCover_iff] at hE hF ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Sites.QuasiCompact | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 77
} | [
{
"pp": "S : Scheme\nX : Scheme\nE F : PreZeroHypercover X\nhE : qcCoverFamily.property E\nhF : qcCoverFamily.property F\n⊢ qcCoverFamily.property (E.sum F)",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.QuasiCompactCover",
"AlgebraicGeometry.Scheme",
"Eq.mp",
"id",
"C... | simp only [qcCoverFamily_property, Scheme.quasiCompactCover_iff] at hE hF ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Sites.QuasiCompact | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 59
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : Surjective f\ninst✝ : QuasiCompact f\nE : Cover (precoverage ⊤) Y := cover f trivial\n⊢ qcCoverFamily.property (cover f trivial).toPreZeroHypercover",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"AlgebraicGeometry.QuasiCompactC... | simp only [qcCoverFamily_property, quasiCompactCover_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 222,
"column": 6
} | {
"line": 222,
"column": 45
} | [
{
"pp": "case e_a\nX Y : Scheme\nx : ↥X\ninst✝ : X.IsGermInjectiveAt x\nf g : X ⟶ Y\ne : f x = g x\nH : Scheme.Hom.stalkMap f x = Y.presheaf.stalkSpecializes ⋯ ≫ Scheme.Hom.stalkMap g x\nV : Y.Opens\nhV : V ∈ Y.affineOpens\nhxV : f x ∈ ↑V\nhxV' : g x ∈ V\nU : X.Opens\nhxU : x ∈ U\nleft✝ : IsAffineOpen U\nhUV : ... | ← cancel_mono (X.presheaf.germ U x hxU) | Lean.Elab.Tactic.evalRewriteSeq | null |
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