module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 272,
"column": 25
} | {
"line": 272,
"column": 33
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝¹ : AddMonoid k\nS : Type u_3\ninst✝ : SMulZeroClass S k\ns : S\na : G\nb : k\n⊢ (s • single a b).toFinsupp = (single a (s • b)).toFinsupp",
"usedConstants": [
"Finsupp.smulZeroClass",
"SkewMonoidAlgebra.toFinsupp_smul",
"instHSMul",
"congrAr... | by simp; | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 449,
"column": 2
} | {
"line": 449,
"column": 64
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : AddCommMonoid k\nN : Type u_3\ninst✝¹ : AddCommMonoid N\ninst✝ : DecidableEq G\nf : SkewMonoidAlgebra k G\na : G\nb : G → k → N\n⊢ (f.sum fun x v ↦ if x = a then b x v else 0) = if a ∈ f.support then b a (f.coeff a) else 0",
"usedConstants": [
"Finset.sum_... | simp only [sum_def', f.toFinsupp.support.sum_ite_eq', support] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 449,
"column": 2
} | {
"line": 449,
"column": 64
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : AddCommMonoid k\nN : Type u_3\ninst✝¹ : AddCommMonoid N\ninst✝ : DecidableEq G\nf : SkewMonoidAlgebra k G\na : G\nb : G → k → N\n⊢ (f.sum fun x v ↦ if x = a then b x v else 0) = if a ∈ f.support then b a (f.coeff a) else 0",
"usedConstants": [
"Finset.sum_... | simp only [sum_def', f.toFinsupp.support.sum_ite_eq', support] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 449,
"column": 2
} | {
"line": 449,
"column": 64
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : AddCommMonoid k\nN : Type u_3\ninst✝¹ : AddCommMonoid N\ninst✝ : DecidableEq G\nf : SkewMonoidAlgebra k G\na : G\nb : G → k → N\n⊢ (f.sum fun x v ↦ if x = a then b x v else 0) = if a ∈ f.support then b a (f.coeff a) else 0",
"usedConstants": [
"Finset.sum_... | simp only [sum_def', f.toFinsupp.support.sum_ite_eq', support] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 694,
"column": 20
} | {
"line": 694,
"column": 36
} | [
{
"pp": "case single\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Monoid G\ninst✝ : MulSemiringAction G k\ng h : SkewMonoidAlgebra k G\nx : G\na : k\n⊢ single x a * g * h = single x a * (g * h)",
"usedConstants": [
"add_mul",
"Distrib.leftDistribClass",
"NonAssocSemiring.toAd... | induction g with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 37
} | [
{
"pp": "J : Type u'\ninst✝⁴ : Category.{v', u'} J\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasColimitsOfShape J TopCat\ninst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ns : Cocone F\nU V : (Opens ↑↑s.pt)ᵒᵖ\ni : U ⟶ V\nj : Jᵒᵖ\nw :... | simp only [Opens.map_comp_map] at w | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits | {
"line": 340,
"column": 6
} | {
"line": 340,
"column": 24
} | [
{
"pp": "J : Type u'\ninst✝⁴ : Category.{v', u'} J\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasColimitsOfShape J TopCat\ninst✝¹ : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\nU : Opens ↑↑(Limits.colimit F)\nj : J\n⊢ (colimitPresheafObjI... | ← Iso.eq_inv_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.RingedSpace.PresheafedSpace | {
"line": 323,
"column": 9
} | {
"line": 325,
"column": 15
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : PresheafedSpace C\n⊢ X.presheaf = (Presheaf.pushforward C (X.ofRestrict ⋯).base).obj (X.restrict ⋯).presheaf",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"TopCat.Presheaf.Pushforward.comp_eq",
"AlgebraicGeometry.... | by
rw [restrict_top_presheaf, ← Presheaf.Pushforward.comp_eq]
tauto | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Scheme | {
"line": 233,
"column": 12
} | {
"line": 233,
"column": 20
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV V' : X.Opens\ne : V ≤ f ⁻¹ᵁ U\ne₂ : V = V'\nP : {R S : CommRingCat} → (R ⟶ S) → Prop\n⊢ P (appLE f U V e) ↔ P (appLE f U V' ⋯)",
"usedConstants": []
}
] | subst e₂ | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.AlgebraicGeometry.Sites.MorphismProperty | {
"line": 84,
"column": 33
} | {
"line": 84,
"column": 53
} | [
{
"pp": "P : MorphismProperty Scheme\nS : Scheme\nι : Type u_1\nX : ι → Scheme\nf : (i : ι) → X i ⟶ S\n⊢ Presieve.ofArrows X f ∈ (precoverage P).coverings S ↔\n (∀ (x : ↥S), ∃ i, x ∈ Set.range ⇑(ConcreteCategory.hom (forget.map (f i)))) ∧ ∀ (i : ι), P (f i)",
"usedConstants": [
"AlgebraicGeometry.S... | ← Scheme.forget_obj, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 665,
"column": 19
} | {
"line": 665,
"column": 45
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\nH : IsOpenImmersion f\ninst✝ : IsOpenImmersion g\ne : Set.range ⇑f = Set.range ⇑g\n⊢ lift g f ⋯ ≫ lift f g ⋯ = 𝟙 X",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"AlgebraicGeometry.Sche... | rw [← cancel_mono f]; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 665,
"column": 19
} | {
"line": 665,
"column": 45
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\nH : IsOpenImmersion f\ninst✝ : IsOpenImmersion g\ne : Set.range ⇑f = Set.range ⇑g\n⊢ lift g f ⋯ ≫ lift f g ⋯ = 𝟙 X",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"AlgebraicGeometry.Sche... | rw [← cancel_mono f]; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 523,
"column": 19
} | {
"line": 523,
"column": 45
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : PresheafedSpace C\nf : X ⟶ Z\nhf : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\ng : Y ⟶ Z\ns : PullbackCone f g\ninst✝ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw repr... | rw [← cancel_mono f]; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 523,
"column": 19
} | {
"line": 523,
"column": 45
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX Y Z : PresheafedSpace C\nf : X ⟶ Z\nhf : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\ng : Y ⟶ Z\ns : PullbackCone f g\ninst✝ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw repr... | rw [← cancel_mono f]; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Cover.Open | {
"line": 250,
"column": 21
} | {
"line": 260,
"column": 44
} | [
{
"pp": "X : Scheme\nU : X.Opens\ns : ↑Γ(X, U)\n𝒰 : X.OpenCover\ninst✝ : Finite 𝒰.I₀\nh : ∀ (i : 𝒰.I₀), IsNilpotent ((ConcreteCategory.hom (Hom.app (𝒰.f i) U)) s)\n⊢ IsNilpotent s",
"usedConstants": [
"Finset.mem_univ",
"Eq.mpr",
"RingHom.instRingHomClass",
"Nat.instLattice",
... | by
choose fn hfn using h
have : Fintype 𝒰.I₀ := Fintype.ofFinite 𝒰.I₀
/- the maximum of all `fn i` (exists, because `𝒰.I₀` is finite) -/
let N : ℕ := Finset.sup Finset.univ fn
have hfnleN (i : 𝒰.I₀) : fn i ≤ N := Finset.le_sup (Finset.mem_univ i)
use N
apply zero_of_zero_cover (𝒰 := 𝒰)
on_goal 1 =... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Cover.Open | {
"line": 304,
"column": 4
} | {
"line": 305,
"column": 63
} | [
{
"pp": "case h_open\nX : Scheme\n⊢ ∀ u ∈ {x | ∃ a, x = Set.range ⇑(X.affineBasisCover.f a)}, IsOpen u",
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Scheme",
"CategoryTheory.PreZeroHypercover.f",
"AlgebraicGeometry.Preshe... | rintro _ ⟨a, rfl⟩
exact IsOpenImmersion.isOpen_range (X.affineBasisCover.f a) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Cover.Open | {
"line": 304,
"column": 4
} | {
"line": 305,
"column": 63
} | [
{
"pp": "case h_open\nX : Scheme\n⊢ ∀ u ∈ {x | ∃ a, x = Set.range ⇑(X.affineBasisCover.f a)}, IsOpen u",
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Scheme",
"CategoryTheory.PreZeroHypercover.f",
"AlgebraicGeometry.Preshe... | rintro _ ⟨a, rfl⟩
exact IsOpenImmersion.isOpen_range (X.affineBasisCover.f a) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Submonoid.Inverses | {
"line": 137,
"column": 91
} | {
"line": 138,
"column": 26
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nS : Submonoid M\nx y : ↥S.leftInv\n⊢ ↑y * (↑x * (↑(S.fromLeftInv x) * ↑(S.fromLeftInv y))) = 1",
"usedConstants": [
"Submonoid.fromLeftInv",
"Eq.mpr",
"MulOne.toOne",
"Semigroup.toMul",
"HMul.hMul",
"Monoid.toMulOneClass",
... | ←
mul_assoc (x : M), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 576,
"column": 24
} | {
"line": 576,
"column": 66
} | [
{
"pp": "case e_a\nX Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\nU : Opens ↥Z\n⊢ (pullbackRestrictIsoRestrict (f ≫ g) U).inv ≫ pullback.fst (f ≫ g) (Scheme.Opens.ι U) ≫ f =\n (pullbackRestrictIsoRestrict f (g ⁻¹ᵁ U)).inv ≫ pullback.fst f (g ⁻¹ᵁ U).ι ≫ f",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.... | pullbackRestrictIsoRestrict_inv_fst_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 727,
"column": 12
} | {
"line": 727,
"column": 20
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV V' : X.Opens\ne : V ≤ f ⁻¹ᵁ U\ne₂ : V = V'\nP : MorphismProperty Scheme\n⊢ P (resLE f U V e) ↔ P (resLE f U V' ⋯)",
"usedConstants": []
}
] | subst e₂ | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 731,
"column": 55
} | {
"line": 732,
"column": 76
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : X.Opens\ne : V ≤ f ⁻¹ᵁ U\nO : (↑U).Opens\n⊢ resLE f U V e ⁻¹ᵁ O = V.ι ⁻¹ᵁ f ⁻¹ᵁ U.ι ''ᵁ O",
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
... | by
rw [← comp_preimage, ← resLE_comp_ι f e, comp_preimage, preimage_image_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 764,
"column": 24
} | {
"line": 764,
"column": 42
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU U' : Y.Opens\nV V' : X.Opens\ne : V ≤ f ⁻¹ᵁ U\nx : ↥V\n⊢ (U.stalkIso ((resLE f U V e) x)).hom ≫ Y.presheaf.stalkSpecializes ⋯ ≫ stalkMap f ↑x =\n stalkMap (resLE f U V e) x ≫ (V.stalkIso x).hom",
"usedConstants": [
"Eq.mpr... | ← Iso.eq_inv_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Sheaves.CommRingCat | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 59
} | [
{
"pp": "case inst.i\nX : TopCat\nC : Type u\ninst✝ : Category.{v, u} C\nF✝ : Presheaf CommRingCat X\nG : F✝.SubmonoidPresheaf\nF : Sheaf CommRingCat X\nU : (Opens ↑X)ᵒᵖ\nm : ?m.34 := ?m.35\n⊢ Function.Injective\n ⇑(ConcreteCategory.hom\n (CommRingCat.ofHom\n (algebraMap (↑(F.presheaf.obj U))... | change Function.Injective (algebraMap _ (Localization m)) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 226,
"column": 4
} | {
"line": 226,
"column": 27
} | [
{
"pp": "case map_nonunit\nX : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X))\nx : ↑X.toTopCat\np : PrimeSpectrum ↑(Γ.obj (op X)) := X.toΓSpecFun x\nS : CommRingCat := (structureSheaf ↑(Γ.obj (op X))).presheaf.stalk p\nt : ↑S\nht : IsUnit ((CommRingCat.Hom.hom (PresheafedSpace.Hom.stalkMap X.toΓSpecSheafedSpace.hom x... | change t * t' = _ at he | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.CategoryTheory.MorphismProperty.Local | {
"line": 177,
"column": 2
} | {
"line": 178,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nP : MorphismProperty C\nK : Precoverage C\ninst✝¹ : P.IsLocalAtSource K\nX Y : C\nf : X ⟶ Y\n𝒰 : K.ZeroHypercover X\ninst✝ : 𝒰.Small\nh : ∀ (i : 𝒰.I₀), P (𝒰.f i ≫ f)\n⊢ P f",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.PreZeroHypercover.f",
... | rw [IsLocalAtSource.iff_of_zeroHypercover (P := P) 𝒰.restrictIndexOfSmall]
simp [h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.Local | {
"line": 177,
"column": 2
} | {
"line": 178,
"column": 10
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nP : MorphismProperty C\nK : Precoverage C\ninst✝¹ : P.IsLocalAtSource K\nX Y : C\nf : X ⟶ Y\n𝒰 : K.ZeroHypercover X\ninst✝ : 𝒰.Small\nh : ∀ (i : 𝒰.I₀), P (𝒰.f i ≫ f)\n⊢ P f",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.PreZeroHypercover.f",
... | rw [IsLocalAtSource.iff_of_zeroHypercover (P := P) 𝒰.restrictIndexOfSmall]
simp [h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.GlueData | {
"line": 243,
"column": 5
} | {
"line": 244,
"column": 48
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v, u₁} C\nC' : Type u₂\ninst✝¹ : Category.{v, u₂} C'\nD : GlueData C\nF : C ⥤ C'\ninst✝ : ∀ (i j k : D.J), PreservesLimit (cospan (D.f i j) (D.f i k)) F\n⊢ ∀ {X Y : WalkingMultispan (MultispanShape.prod D.J)} (f : X ⟶ Y),\n (D.diagram.multispan ⋙ F).map f ≫\n (... | by
rintro (⟨_, _⟩ | _) _ (_ | _ | _) <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Gluing | {
"line": 251,
"column": 38
} | {
"line": 251,
"column": 59
} | [
{
"pp": "case h.e'_3.h.e'_4\nD : GlueData\ni j : D.J\nU : Set ↑(D.U i)\n⊢ ⇑(ConcreteCategory.hom (D.f i j)) ⁻¹' U =\n ⇑(ConcreteCategory.hom (D.t i j)) ⁻¹' ⇑(ConcreteCategory.hom (D.f i j)) ∘ ⇑(ConcreteCategory.hom (D.t j i)) ⁻¹' U",
"usedConstants": [
"CategoryTheory.GlueData.t",
"Eq.mpr",
... | Set.preimage_preimage | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Gluing | {
"line": 336,
"column": 72
} | {
"line": 336,
"column": 90
} | [
{
"pp": "D : GlueData\nh : MkCore\ni j k : h.J\n⊢ (pullbackIsoProdSubtype (h.V i j).inclusion' (h.V i k).inclusion').hom ≫\n ofHom\n {\n toFun := fun x ↦\n ⟨(⟨↑((ConcreteCategory.hom (h.t i j)) (↑x).1), ⋯⟩, (ConcreteCategory.hom (h.t i j)) (↑x).1), ⋯⟩,\n continuo... | ← Iso.eq_inv_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Gluing | {
"line": 345,
"column": 8
} | {
"line": 345,
"column": 26
} | [
{
"pp": "D : GlueData\nh : MkCore\ni j k : h.J\n⊢ (pullbackIsoProdSubtype (h.V i j).inclusion' (h.V i k).inclusion').hom ≫\n ofHom\n {\n toFun := fun x ↦\n ⟨(⟨↑((ConcreteCategory.hom (h.t i j)) (↑x).1), ⋯⟩, (ConcreteCategory.hom (h.t i j)) (↑x).1), ⋯⟩,\n continuo... | ← Iso.eq_inv_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 156,
"column": 6
} | {
"line": 156,
"column": 30
} | [
{
"pp": "X Y : Scheme\ninst✝ : IsAffine Y\nf g : X ⟶ Y\ne : Scheme.Hom.appTop f = Scheme.Hom.appTop g\n⊢ f = g",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Scheme",
"Lattice.toSe... | ← cancel_mono Y.toSpecΓ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 405,
"column": 4
} | {
"line": 405,
"column": 26
} | [
{
"pp": "case right.right\nX : Scheme\n𝒰 : X.OpenCover\nU : Set ↥(gluedCover 𝒰).glued\nhU : ∀ (i : (gluedCover 𝒰).J), IsOpen (⇑((gluedCover 𝒰).ι i) ⁻¹' U)\nx : ↥X\nhx : x ∈ ⇑(fromGlued 𝒰) '' U\n⊢ x ∈ ⇑(fromGlued 𝒰) '' U ∩ Set.range ⇑(𝒰.f (idx 𝒰 x))",
"usedConstants": [
"AlgebraicGeometry.Schem... | exact ⟨hx, 𝒰.covers x⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 405,
"column": 4
} | {
"line": 405,
"column": 26
} | [
{
"pp": "case right.right\nX : Scheme\n𝒰 : X.OpenCover\nU : Set ↥(gluedCover 𝒰).glued\nhU : ∀ (i : (gluedCover 𝒰).J), IsOpen (⇑((gluedCover 𝒰).ι i) ⁻¹' U)\nx : ↥X\nhx : x ∈ ⇑(fromGlued 𝒰) '' U\n⊢ x ∈ ⇑(fromGlued 𝒰) '' U ∩ Set.range ⇑(𝒰.f (idx 𝒰 x))",
"usedConstants": [
"AlgebraicGeometry.Schem... | exact ⟨hx, 𝒰.covers x⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 405,
"column": 4
} | {
"line": 405,
"column": 26
} | [
{
"pp": "case right.right\nX : Scheme\n𝒰 : X.OpenCover\nU : Set ↥(gluedCover 𝒰).glued\nhU : ∀ (i : (gluedCover 𝒰).J), IsOpen (⇑((gluedCover 𝒰).ι i) ⁻¹' U)\nx : ↥X\nhx : x ∈ ⇑(fromGlued 𝒰) '' U\n⊢ x ∈ ⇑(fromGlued 𝒰) '' U ∩ Set.range ⇑(𝒰.f (idx 𝒰 x))",
"usedConstants": [
"AlgebraicGeometry.Schem... | exact ⟨hx, 𝒰.covers x⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 560,
"column": 4
} | {
"line": 560,
"column": 82
} | [
{
"pp": "case a.a\nJ : Type w\ninst✝² : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝¹ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝ : (F ⋙ forget).IsLocallyDirected\ni j k : J\nx : ↥(pullback (V F i j).ι (V F i k).ι)\nk₁ : (k : J) × (k ⟶ i) × (k ⟶ j)\nk₂ : (k_1 : J) × (k_1 ⟶ i) × (k_1 ⟶ k)\nl : J\nhli... | rw [← Scheme.Hom.comp_apply, ← Scheme.Hom.comp_apply, pullback.lift_fst_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 329,
"column": 8
} | {
"line": 329,
"column": 29
} | [
{
"pp": "case left.e_unop.h\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : GlueData C\ninst✝ : HasLimits C\ni : D.J\nU : Opens ↑↑(D.U i)\nj k : D.J\n⊢ ⇑(ConcreteCategory.hom (D.f j k).base) ⁻¹'\n ⇑(ConcreteCategory.hom (D.ι j).base) ⁻¹' ⇑(ConcreteCategory.hom (D.ι i).base) '' ↑U =\n ⇑(ConcreteCategory.hom... | Set.preimage_preimage | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 405,
"column": 4
} | {
"line": 405,
"column": 57
} | [
{
"pp": "case h\nι : Type u\nf : ι → Scheme\nσ : Type v\ng : σ → Scheme\nX Y : Scheme\nx : ↥X ⊕ ↥Y\n⊢ (coprodMk X Y) x ∈\n Set.range\n ⇑({ I₀ := PUnit.{w + 1} ⊕ PUnit.{w + 1}, X := fun x ↦ Sum.elim (fun x ↦ X) (fun x ↦ Y) x,\n f := fun x ↦ Sum.rec (fun x ↦ coprod.inl) (fun x ↦ coprod.inr) x... | simp only [Sum.elim_inl, Sum.elim_inr, Set.mem_range] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 598,
"column": 6
} | {
"line": 598,
"column": 46
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝ : HasAffineProperty P Q\nX✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nthis : Q.IsLocal := isLocal_affineProperty P\nX Y : Scheme\nf : X ⟶ Y\nι : Type u_1\nU : ι → Y.Opens\nhU : iSup U = ⊤\nH : ∀ (i : ι), P (f ∣_ U i)\n𝒰 : Y.OpenCover := Y.openCover... | exact (Scheme.Opens.opensRange_ι _).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 506,
"column": 33
} | {
"line": 506,
"column": 41
} | [
{
"pp": "X Y Z : Scheme\n𝒰✝ : X.OpenCover\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰✝.I₀), HasPullback (𝒰✝.f i ≫ f✝) g✝\ns : PullbackCone f✝ g✝\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\nx : ↥(pullback f g)\ni : (Cover.ulift 𝒰).I₀\n⊢ pullback.map ((Cover.ulift 𝒰).f i ≫ f) g f g ((Cover.ulift 𝒰).f i) (𝟙 Y) (... | gluing_J | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 56
} | [
{
"pp": "Q : AffineTargetMorphismProperty\ninst✝ : Q.IsLocal\nX Y : Scheme\nx✝ : IsAffine Y\nf : X ⟶ Y\ns : Finset ↑Γ(Y, ⊤)\nhs : Ideal.span ↑s = ⊤\nhs' : ∀ (r : ↥s), Q.diagonal (f ∣_ Y.basicOpen ↑r)\n⊢ Q.diagonal f",
"usedConstants": [
"Iff.mpr",
"AlgebraicGeometry.Scheme",
"CategoryTheor... | refine (diagonal_iff (targetAffineLocally Q)).mpr ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.RingHom.Locally | {
"line": 79,
"column": 66
} | {
"line": 85,
"column": 20
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nR S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\n⊢ Locally (fun {R S} [CommRing R] [CommRing S] ↦ P) f ↔\n ∃ s, ∃ (_ : Ideal.span ↑s = ⊤), ∀ t ∈ s, P ((algebraMap S (Localization.Away t)).comp f)"... | by
constructor
· intro ⟨s, hsone, hs⟩
obtain ⟨s', h₁, h₂⟩ := (Ideal.span_eq_top_iff_finite s).mp hsone
exact ⟨s', h₂, fun t ht ↦ hs t (h₁ ht)⟩
· intro ⟨s, hsone, hs⟩
use s, hsone, hs | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 91
} | [
{
"pp": "case respectsIso.h₁\nX✝ Y✝ : Scheme\nf : X✝ ⟶ Y✝\nX Y Z : Scheme\ne : X ≅ Y\nf✝ : Y ⟶ Z\ninst✝ : IsAffine Z\nH : CompactSpace ↥Y\n⊢ CompactSpace ↥X",
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Scheme",
"AlgebraicGeome... | exacts [@Homeomorph.compactSpace _ _ _ _ H (TopCat.homeoOfIso (asIso e.inv.base)), H] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 227,
"column": 4
} | {
"line": 228,
"column": 41
} | [
{
"pp": "case inr\nX : Scheme\nS : CommRingCat\nf : X ⟶ Spec S\ninst✝ : QuasiCompact f\nZ : Set ↥X\nhZ : IsClosed Z\nH : StableUnderSpecialization (⇑f '' Z)\nthis✝ :\n ∀ {X : Scheme} (S : CommRingCat) (f : X ⟶ Spec S) [QuasiCompact f] (Z : Set ↥X),\n IsClosed Z → StableUnderSpecialization (⇑f '' Z) → (∃ R, ... | simp_rw [Scheme.Hom.comp_base, TopCat.comp_app, ← Set.image_image,
Set.image_preimage_eq _ hg] at this | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 7
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsCompact ↑U\nf : ↑Γ(X, U)\nhf : X.basicOpen f = ⊥\nh : (1 |_ X.basicOpen f) ⋯ = 0\nn : ℕ\nhn : f ^ n = 0\n⊢ IsNilpotent f",
"usedConstants": [
"Opposite",
"CommRingCat.carrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"TopologicalSpace.Ope... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Ideal.Height | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nI J : Ideal R\ne : I ≤ J\ninst✝¹ : J.IsPrime\ninst✝ : J.FiniteHeight\ne' : J.height ≤ I.height\np : Ideal R\nh₁ : p ∈ I.minimalPrimes\nh₂ : p ≤ J\n⊢ J ∈ I.minimalPrimes",
"usedConstants": [
"Eq.mpr",
"outParam",
"Ideal.minimalPrimes",
"Comm... | convert! h₁ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.RingTheory.Ideal.Height | {
"line": 479,
"column": 2
} | {
"line": 486,
"column": 61
} | [
{
"pp": "case a\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\n⊢ ringKrullDim R ≤ ↑(⨆ I, ⨆ (_ : I ≠ ⊤), I.height)",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"PrimeSpectrum.mk",
"instCompleteLatticeWithBot",
"WithBot.some",
"WithBot",
"Preorder.toLT",
... | · refine iSup_le fun p => WithBot.coe_le_coe.mpr (le_trans (b := p.last.asIdeal.height) ?_ ?_)
· rw [height_eq_primeHeight]
apply le_trans (b := ⨆ (_ : p.last ≤ p.last), ↑p.length)
· exact le_iSup (fun _ => (↑p.length : ℕ∞)) le_rfl
· exact le_iSup (fun p' => (⨆ _, p'.length : ℕ∞)) p
· apply le... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 297,
"column": 2
} | {
"line": 297,
"column": 7
} | [
{
"pp": "X : Scheme\nS : ↑X.affineOpens\nU₁ U₂ : X.Opens\nn₁ n₂ : ℕ\ny₁ : ↑Γ(X, U₁)\ny₂ : ↑Γ(X, U₂)\nf : ↑Γ(X, U₁ ⊔ U₂)\nx : ↑Γ(X, X.basicOpen f)\nh₁ : ↑S ≤ U₁\nh₂ : ↑S ≤ U₂\ne₁ :\n (y₁ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ =\n ((f |_ U₁) ⋯ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ ^ n₁ * (x |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.AlgebraicGeometry.Properties | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 36
} | [
{
"pp": "case component_integral\nX : Scheme\ninst✝ : IsReduced X\nH : IrreducibleSpace ↥X\nU : X.Opens\nhU : Nonempty ↥↑U\nthis✝ : ∃ x y, x ≠ y\nthis : NoZeroDivisors ↑(X.presheaf.obj (op U))\n⊢ IsDomain ↑Γ(X, U)",
"usedConstants": [
"Opposite",
"CommRingCat.carrier",
"AlgebraicGeometry.P... | exact NoZeroDivisors.to_isDomain _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.OpenImmersion | {
"line": 81,
"column": 30
} | {
"line": 81,
"column": 64
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n𝒰 : X.OpenCover\nhf : Function.Injective ⇑f\nh𝒰 : ∀ (i : 𝒰.I₀), IsOpenImmersion (𝒰.f i ≫ f)\nU : Set ↥X\nhU : IsOpen U\nx✝¹ : ↥Y\nx✝ : x✝¹ ∈ ⇑f '' U\nx : ↥X\nleft✝ : x ∈ U\nright✝ : f x = x✝¹\n⊢ x✝¹ ∈ (⨆ i, (𝒰.f i ≫ f) ''ᵁ 𝒰.f i ⁻¹ᵁ { carrier := U, is_open' := hU }).carri... | have := 𝒰.exists_eq x; simp; grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.OpenImmersion | {
"line": 81,
"column": 30
} | {
"line": 81,
"column": 64
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n𝒰 : X.OpenCover\nhf : Function.Injective ⇑f\nh𝒰 : ∀ (i : 𝒰.I₀), IsOpenImmersion (𝒰.f i ≫ f)\nU : Set ↥X\nhU : IsOpen U\nx✝¹ : ↥Y\nx✝ : x✝¹ ∈ ⇑f '' U\nx : ↥X\nleft✝ : x ∈ U\nright✝ : f x = x✝¹\n⊢ x✝¹ ∈ (⨆ i, (𝒰.f i ≫ f) ''ᵁ 𝒰.f i ⁻¹ᵁ { carrier := U, is_open' := hU }).carri... | have := 𝒰.exists_eq x; simp; grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 414,
"column": 2
} | {
"line": 414,
"column": 7
} | [
{
"pp": "X : Scheme\nU : Opens ↥X\nhU : IsCompact U.carrier\nhU' : IsQuasiSeparated U.carrier\nf g s : ↑Γ(X, U)\nhfg : (f |_ X.basicOpen s) ⋯ = (g |_ X.basicOpen s) ⋯\nn : ℕ\nhc : s ^ n * f = s ^ n * g\n⊢ ∃ n, s ^ n * f = s ^ n * g",
"usedConstants": [
"Opposite",
"HMul.hMul",
"CommRingCat... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.AlgebraicGeometry.Stalk | {
"line": 113,
"column": 2
} | {
"line": 114,
"column": 75
} | [
{
"pp": "X : Scheme\nU : X.Opens\nx : ↥X\nhxU : x ∈ U\nV : X.Opens\nhV : V ∈ X.affineOpens\nhxV : x ∈ ↑V\nhVU : ↑V ⊆ ↑U\n⊢ Hom.app (X.fromSpecStalk x) U =\n X.presheaf.germ U x hxU ≫\n (ΓSpecIso (X.presheaf.stalk x)).inv ≫ (Spec (X.presheaf.stalk x)).presheaf.map (homOfLE ⋯).op",
"usedConstants": [
... | rw [← hV.fromSpecStalk_eq_fromSpecStalk hxV, IsAffineOpen.fromSpecStalk, Scheme.Hom.comp_app,
hV.fromSpec_app_of_le _ hVU, ← X.presheaf.germ_res (homOfLE hVU) x hxV] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 305,
"column": 12
} | {
"line": 305,
"column": 60
} | [
{
"pp": "case h₀.h₀\nX : Scheme\nI : X.IdealSheafData\ni j k : ↑X.affineOpens\n⊢ pullback.fst (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) ≫\n pullback.snd (I.glueDataObjι i) (X.homOfLE ⋯) ≫ (↑i ⊓ ↑j).ι =\n pullback.fst (pullback.fst (I.glueDataObjι i) ... | pullback.condition_assoc (f := I.glueDataObjι i) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 393,
"column": 4
} | {
"line": 393,
"column": 85
} | [
{
"pp": "case refine_2\nX : Scheme\nI : X.IdealSheafData\nx : ↥X\nhx : x ∈ ↑I.support\nU : TopologicalSpace.Opens ↥X\nhU : U ∈ X.affineOpens\nhxU : x ∈ ↑U\n⊢ x ∈ Set.range ⇑I.gluedTo",
"usedConstants": [
"AlgebraicGeometry.Scheme.IdealSheafData.support",
"AlgebraicGeometry.SheafedSpace.instTopol... | obtain ⟨y, rfl⟩ := (I.range_glueDataObjι_ι_eq_support_inter ⟨U, hU⟩).ge ⟨hx, hxU⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 701,
"column": 4
} | {
"line": 704,
"column": 66
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : ↑Y.affineOpens\ninst✝ : QuasiCompact f\n⊢ DenseRange ⇑(Hom.toImage f)",
"usedConstants": [
"Set.univ_subset_iff",
"Set.range_comp",
"Eq.mpr",
"Set.image_univ",
"AlgebraicGeometry.Scheme.IdealSheafData.support",
"AlgebraicGeometry.Sche... | rw [denseRange_iff_closure_range, f.imageι.isEmbedding.closure_eq_preimage_closure_image,
← Set.univ_subset_iff, ← Set.image_subset_iff, Set.image_univ,
IdealSheafData.range_subschemeι, Hom.support_ker, ← Set.range_comp,
← TopCat.coe_comp, ← Scheme.Hom.comp_base, f.toImage_imageι] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 701,
"column": 4
} | {
"line": 704,
"column": 66
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : ↑Y.affineOpens\ninst✝ : QuasiCompact f\n⊢ DenseRange ⇑(Hom.toImage f)",
"usedConstants": [
"Set.univ_subset_iff",
"Set.range_comp",
"Eq.mpr",
"Set.image_univ",
"AlgebraicGeometry.Scheme.IdealSheafData.support",
"AlgebraicGeometry.Sche... | rw [denseRange_iff_closure_range, f.imageι.isEmbedding.closure_eq_preimage_closure_image,
← Set.univ_subset_iff, ← Set.image_subset_iff, Set.image_univ,
IdealSheafData.range_subschemeι, Hom.support_ker, ← Set.range_comp,
← TopCat.coe_comp, ← Scheme.Hom.comp_base, f.toImage_imageι] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 701,
"column": 4
} | {
"line": 704,
"column": 66
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : ↑Y.affineOpens\ninst✝ : QuasiCompact f\n⊢ DenseRange ⇑(Hom.toImage f)",
"usedConstants": [
"Set.univ_subset_iff",
"Set.range_comp",
"Eq.mpr",
"Set.image_univ",
"AlgebraicGeometry.Scheme.IdealSheafData.support",
"AlgebraicGeometry.Sche... | rw [denseRange_iff_closure_range, f.imageι.isEmbedding.closure_eq_preimage_closure_image,
← Set.univ_subset_iff, ← Set.image_subset_iff, Set.image_univ,
IdealSheafData.range_subschemeι, Hom.support_ker, ← Set.range_comp,
← TopCat.coe_comp, ← Scheme.Hom.comp_base, f.toImage_imageι] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.PullbackCarrier | {
"line": 150,
"column": 2
} | {
"line": 152,
"column": 6
} | [
{
"pp": "X Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nT : Triplet f g\np : ↥(Spec T.tensor)\n⊢ (pullback.fst f g) (T.SpecTensorTo p) = T.x",
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.Hom",
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"AlgebraicGeometry.Spec",
"AlgebraicGeom... | simp only [SpecTensorTo]
rw [← Scheme.Hom.comp_apply]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.PullbackCarrier | {
"line": 150,
"column": 2
} | {
"line": 152,
"column": 6
} | [
{
"pp": "X Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nT : Triplet f g\np : ↥(Spec T.tensor)\n⊢ (pullback.fst f g) (T.SpecTensorTo p) = T.x",
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.Hom",
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"AlgebraicGeometry.Spec",
"AlgebraicGeom... | simp only [SpecTensorTo]
rw [← Scheme.Hom.comp_apply]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.PullbackCarrier | {
"line": 158,
"column": 2
} | {
"line": 160,
"column": 6
} | [
{
"pp": "X Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nT : Triplet f g\np : ↥(Spec T.tensor)\n⊢ (pullback.snd f g) (T.SpecTensorTo p) = T.y",
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.Hom",
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"AlgebraicGeometry.Spec",
"AlgebraicGeom... | simp only [SpecTensorTo]
rw [← Scheme.Hom.comp_apply]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.PullbackCarrier | {
"line": 158,
"column": 2
} | {
"line": 160,
"column": 6
} | [
{
"pp": "X Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nT : Triplet f g\np : ↥(Spec T.tensor)\n⊢ (pullback.snd f g) (T.SpecTensorTo p) = T.y",
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.Hom",
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"AlgebraicGeometry.Spec",
"AlgebraicGeom... | simp only [SpecTensorTo]
rw [← Scheme.Hom.comp_apply]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 56
} | [
{
"pp": "case h.e'_2.a\nX✝ Y Z : Scheme\nf✝ : X✝ ⟶ Y\ng✝ : Y ⟶ Z\nU V X : Scheme\nf : U ⟶ X\ng : V ⟶ X\ninst✝¹ : IsAffineHom f\ninst✝ : IsAffineHom g\nW : X.Opens\nhW : IsAffineOpen W\nthis✝ : IsAffine ↑(f ⁻¹ᵁ W)\nthis : IsAffine ↑(g ⁻¹ᵁ W)\ni : ↑(f ⁻¹ᵁ W) ⨿ ↑(g ⁻¹ᵁ W) ⟶ U ⨿ V := coprod.map (f ⁻¹ᵁ W).ι (g ⁻¹ᵁ W... | obtain ⟨(x | x), rfl⟩ := (coprodMk U V).surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.PullbackCarrier | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 92
} | [
{
"pp": "P X Y Z : Scheme\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\nh : IsPullback fst snd f g\nx : ↥X\ny : ↥Y\nhxy : f x = g y\ne : P ≅ pullback f g := h.isoPullback\n⊢ ∃ p, fst p = x ∧ snd p = y",
"usedConstants": [
"AlgebraicGeometry.Scheme.Pullback.exists_preimage_pullback"
]
}
] | obtain ⟨z, hzl, hzr⟩ := AlgebraicGeometry.Scheme.Pullback.exists_preimage_pullback x y hxy | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Morphisms.AffineAnd | {
"line": 247,
"column": 67
} | {
"line": 255,
"column": 12
} | [
{
"pp": "Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nP : MorphismProperty Scheme\nhA : HasAffineProperty P (affineAnd fun {R S} [CommRing R] [CommRing S] ↦ Q)\n⊢ P ≤ @IsAffineHom",
"usedConstants": [
"AlgebraicGeometry.iSup_affineOpens_eq_top",
"Eq.mpr",
... | by
intro X Y f hf
wlog hY : IsAffine Y
· rw [IsZariskiLocalAtTarget.iff_of_iSup_eq_top (P := @IsAffineHom) _ (iSup_affineOpens_eq_top _)]
intro U
exact this P hA _ (IsZariskiLocalAtTarget.restrict hf _) U.2
rw [HasAffineProperty.iff_of_isAffine (P := P) (Q := (affineAnd Q))] at hf
rw [HasAffinePropert... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.AffineAnd | {
"line": 295,
"column": 35
} | {
"line": 295,
"column": 71
} | [
{
"pp": "Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nP : MorphismProperty Scheme\nhP : HasAffineProperty P (affineAnd fun {R S} [CommRing R] [CommRing S] ↦ Q)\nhQi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q\nhQ :\n ∀ {R S T : Type u} [inst : CommRing ... | by simp [← Scheme.Hom.comp_preimage] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.AffineAnd | {
"line": 296,
"column": 35
} | {
"line": 296,
"column": 71
} | [
{
"pp": "Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nP : MorphismProperty Scheme\nhP : HasAffineProperty P (affineAnd fun {R S} [CommRing R] [CommRing S] ↦ Q)\nhQi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q\nhQ :\n ∀ {R S T : Type u} [inst : CommRing ... | by simp [← Scheme.Hom.comp_preimage] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 59
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\ns : Set (PrimeSpectrum R)\nhs : IsConstructible s\n⊢ ∃ S x f, Set.range (comap f) = s",
"usedConstants": [
"Iff.mpr",
"PrimeSpectrum.ConstructibleSetData",
"Exists",
"PrimeSpectrum.exists_constructibleSetData_iff",
"CommRing.toCommSemiri... | obtain ⟨s, rfl⟩ := exists_constructibleSetData_iff.mpr hs | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Spectrum.Prime.Chevalley | {
"line": 97,
"column": 8
} | {
"line": 97,
"column": 12
} | [
{
"pp": "K : Type u_3\nA : Type u_4\nB : Type u_5\ninst✝⁴ : Field K\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra K A\ninst✝ : Algebra K B\nB' : Subalgebra K B\nhB : B'.FG\nf : A ⊗[K] ↥B'\nhU : IsOpen ↑(basicOpen ((Algebra.TensorProduct.map (AlgHom.id A A) B'.val).toRingHom f))\nthis : Algebra.Fin... | hψeq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Functorial | {
"line": 60,
"column": 45
} | {
"line": 65,
"column": 10
} | [
{
"pp": "X Y Z : Scheme\nI : Z.IdealSheafData\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ I.comap (f ≫ g) = (I.comap g).comap f",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"CategoryTheory.Limits.pullback",
"CategoryTheory.Limits.Cone.π",
"CategoryTheory.Functor",
"Algebr... | by
let e : pullback f (I.comap g).subschemeι ≅ pullback (f ≫ g) I.subschemeι :=
asIso (pullback.map _ _ _ _ (𝟙 _) (I.comapIso g).hom (𝟙 _) (by simp) (by simp)) ≪≫
pullbackRightPullbackFstIso _ _ _
rw [comap, comap, ← Scheme.Hom.ker_comp_of_isIso e.hom]
simp [e] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.IdealSheaf.Functorial | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 39
} | [
{
"pp": "Z : Scheme\nI : Z.IdealSheafData\n⊢ Hom.ker (inv (pullback.snd (𝟙 Z) I.subschemeι) ≫ pullback.fst (𝟙 Z) I.subschemeι) = I",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.pullback_inv_snd_fst_of_left_isIso",
"CategoryTheory.Limits.pullback",
"CategoryTheory.Limits.has... | pullback_inv_snd_fst_of_left_isIso, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.HopkinsLevitzki | {
"line": 86,
"column": 27
} | {
"line": 86,
"column": 40
} | [
{
"pp": "case refine_1\nR₀ : Type u_1\nR : Type u_2\nM✝ : Type u\ninst✝⁹ : Ring R₀\ninst✝⁸ : Ring R\ninst✝⁷ : Module R₀ R\ninst✝⁶ : AddCommGroup M✝\ninst✝⁵ : Module R₀ M✝\ninst✝⁴ : Module R M✝\ninst✝³ : IsScalarTower R₀ R M✝\ninst✝² : IsSemiprimaryRing R\ninst✝¹ : IsScalarTower R₀ R R\ninst✝ : Module.Finite R₀ ... | this.out 2 0, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 357,
"column": 6
} | {
"line": 357,
"column": 24
} | [
{
"pp": "case refine_2\nX Y S : Scheme\nsX : X ⟶ S\nsY : Y ⟶ S\ninst✝¹ : LocallyOfFiniteType sY\nx : ↥X\ninst✝ : X.IsGermInjectiveAt x\ny : ↥Y\ne : sX x = sY y\nφ : Y.presheaf.stalk y ⟶ X.presheaf.stalk x\nh : Scheme.Hom.stalkMap sY y ≫ φ = S.presheaf.stalkSpecializes ⋯ ≫ Scheme.Hom.stalkMap sX x\nU : Topologic... | ← Iso.eq_inv_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 27
} | [
{
"pp": "case a.a\nX Y : Scheme\nf : X ⟶ Y\nhf : diagonal (@Surjective) f\nx₁ x₂ : ↥X\ne : f x₁ = f x₂\nt : ↥X\nht₁ : (pullback.fst f f) ((pullback.diagonal f) t) = x₁\nht₂ : (pullback.snd f f) ((pullback.diagonal f) t) = x₂\n⊢ (𝟙 X) t = (pullback.diagonal f ≫ pullback.snd f f) t",
"usedConstants": [
... | pullback.diagonal_snd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ColimitsOver | {
"line": 196,
"column": 6
} | {
"line": 196,
"column": 82
} | [
{
"pp": "case refine_1.refine_1\nP : MorphismProperty Scheme\ninst✝⁸ : P.IsStableUnderBaseChange\ninst✝⁷ : P.IsMultiplicative\nS : Scheme\nJ : Type u_1\ninst✝⁶ : Category.{v_1, u_1} J\nD : J ⥤ P.Over ⊤ S\n𝒰 : S.OpenCover\ninst✝⁵ : Category.{v_2, ?u.92037} 𝒰.I₀\ninst✝⁴ : LocallyDirected 𝒰\nd : ColimitGluingDa... | exact ((d.cocone i).ι.app a).left ≫ colimit.ι d.relativeGluingData.functor i | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Cover.QuasiCompact | {
"line": 197,
"column": 10
} | {
"line": 197,
"column": 51
} | [
{
"pp": "P : MorphismProperty Scheme\nS : Scheme\n𝒰 : Scheme.Cover (Scheme.precoverage P) S\ninst✝² : P.RespectsLeft IsOpenImmersion\ninst✝¹ : CompactSpace ↥S\ninst✝ : QuasiCompactCover 𝒰.toPreZeroHypercover\nn : ℕ\nf : Fin n → 𝒰.I₀\nV : (i : Fin n) → (𝒰.X (f i)).Opens\nhV : ∀ (i : Fin n), IsAffineOpen (V i... | use (hV _).isoSpec.hom.base ⟨x s, hmem s⟩ | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 232,
"column": 59
} | {
"line": 232,
"column": 95
} | [
{
"pp": "I : Type u\ninst✝¹ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\nc : Cone D\nhc : IsLimit c\ninst✝ : IsCofiltered I\ni : I\nU : (D.obj i).Opens\nj : Over i\n⊢ (Cone.whisker (Over.forget i) c).π.app j ⁻¹ᵁ Scheme.Hom.opensRange ((opensDiagramι D i U).app j... | by simp [← Scheme.Hom.comp_preimage] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 438,
"column": 4
} | {
"line": 438,
"column": 85
} | [
{
"pp": "case hP₁\nR✝ : Type u_2\ninst✝³ : CommRing R✝\nn : ℕ\nR : Type u_2\ninst✝² : CommRing R\nR₀ : Type u_1\ninst✝¹ : CommRing R₀\ninst✝ : Algebra R₀ R\nf : R[X]\n⊢ ∃ T,\n comap Polynomial.C '' (zeroLocus (Set.range { val := 0 }.val) \\ zeroLocus {f}) = T.toSet ∧\n ∀ C ∈ T,\n C.n ≤ { val := 0... | refine ⟨(Finset.range (f.natDegree + 2)).image fun j ↦ ⟨f.coeff j, 0, 0⟩, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 467,
"column": 8
} | {
"line": 467,
"column": 24
} | [
{
"pp": "case pos\nR✝ : Type u_2\ninst✝² : CommRing R✝\nn : ℕ\nR : Type u_2\ninst✝¹ : CommRing R\ng : InductionObj R n\ni : Fin n\nhi : (g.val i).Monic\nhi_min : ∀ (j : Fin n), j ≠ i → g.val j = 0\nR₀✝ : Type u_1\nR₀ : CommRing R₀✝\ninst✝ : Algebra R₀✝ R\nf : R[X]\nM : Type u_2 := R[X] ⧸ Ideal.span {g.val i}\nt... | · subst hij; rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 608,
"column": 2
} | {
"line": 608,
"column": 28
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\ninst✝⁴ : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝³ : LocallyOfFiniteType f\ninst✝² : IsCofiltered I\ninst✝¹ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\nA : ExistsHomHomCompEqCompAux ... | refine ⟨k.left, k.hom, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula | {
"line": 195,
"column": 90
} | {
"line": 199,
"column": 7
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nx₁ x₂ y₁ y₂ : F\nhx : x₁ = x₂\nhy : y₁ ≠ W.negY x₂ y₂\n⊢ W.slope x₁ x₂ y₁ y₂ = -evalEval x₁ y₁ W.polynomialX / evalEval x₁ y₁ W.polynomialY",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.R... | by
rw [slope_of_Y_ne hx hy, evalEval_polynomialX, neg_sub]
congr 1
rw [negY, evalEval_polynomialY]
ring1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 577,
"column": 4
} | {
"line": 578,
"column": 62
} | [
{
"pp": "case refine_2.hab\nR : Type u_6\ninst✝ : CommRing R\nM : Submodule ℤ R\nhM : 1 ∈ M\nS : ConstructibleSetData R[X]\nhS : ∀ C ∈ S, ∀ (j : Fin C.n) (k : ℕ), (C.g j).coeff k ∈ M\nf : BasicConstructibleSetData R[X] → ConstructibleSetData R\nhf₁ :\n ∀ (C : BasicConstructibleSetData R[X]),\n comap Polynom... | · refine Submodule.span_le.mpr ?_
simp [Set.subset_def, hM, forall_comm (α := R), hS y hy] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 722,
"column": 28
} | {
"line": 722,
"column": 42
} | [
{
"pp": "I : Type u\ninst✝² : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝¹ : IsCofiltered I\ninst✝ : ∀ (i : I), IsAffine (D.obj i)\ni : I\ns : ↑Γ(D.obj i, ⊤)\nhs : (ConcreteCategory.hom (Scheme.Hom.appTop (c.π.app i))) s = 0\nthis : ∀ (i : Iᵒᵖ), IsAffine (Opposite.unop (D.op.obj i))\nj ... | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 722,
"column": 28
} | {
"line": 722,
"column": 42
} | [
{
"pp": "I : Type u\ninst✝² : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝¹ : IsCofiltered I\ninst✝ : ∀ (i : I), IsAffine (D.obj i)\ni : I\ns : ↑Γ(D.obj i, ⊤)\nhs : (ConcreteCategory.hom (Scheme.Hom.appTop (c.π.app i))) s = 0\nthis : ∀ (i : Iᵒᵖ), IsAffine (Opposite.unop (D.op.obj i))\nj ... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 722,
"column": 28
} | {
"line": 722,
"column": 42
} | [
{
"pp": "I : Type u\ninst✝² : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝¹ : IsCofiltered I\ninst✝ : ∀ (i : I), IsAffine (D.obj i)\ni : I\ns : ↑Γ(D.obj i, ⊤)\nhs : (ConcreteCategory.hom (Scheme.Hom.appTop (c.π.app i))) s = 0\nthis : ∀ (i : Iᵒᵖ), IsAffine (Opposite.unop (D.op.obj i))\nj ... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 435,
"column": 2
} | {
"line": 439,
"column": 28
} | [
{
"pp": "X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\ninst✝ : Flat f\nhUS : IsAffineOpen US\nhUT : IsAffineOpen UT\nhUX : IsCompact ↑... | obtain ⟨I, hI, e⟩ := isCompact_iff_finite_and_eq_biUnion_affineOpens.mp hUX
have := hI.to_subtype
exact mono_pushoutSection_of_iSup_eq (ι := I) H hUST hUSX hUY (·) (by rwa [iSup_subtype, eq_comm])
(fun i ↦ have := isIso_pushoutSection_of_isAffineOpen H hUST _ rfl hUS hUT i.1.2; inferInstance)
(f.flat_appLE ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 435,
"column": 2
} | {
"line": 439,
"column": 28
} | [
{
"pp": "X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\ninst✝ : Flat f\nhUS : IsAffineOpen US\nhUT : IsAffineOpen UT\nhUX : IsCompact ↑... | obtain ⟨I, hI, e⟩ := isCompact_iff_finite_and_eq_biUnion_affineOpens.mp hUX
have := hI.to_subtype
exact mono_pushoutSection_of_iSup_eq (ι := I) H hUST hUSX hUY (·) (by rwa [iSup_subtype, eq_comm])
(fun i ↦ have := isIso_pushoutSection_of_isAffineOpen H hUST _ rfl hUS hUT i.1.2; inferInstance)
(f.flat_appLE ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 446,
"column": 8
} | {
"line": 446,
"column": 58
} | [
{
"pp": "X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\ninst✝ : Flat iX\nhUS : IsAffineOpen US\nhUX : IsAffineOpen UX\nhUT : IsCompact ... | ← mono_comp_iff_of_isIso (pushoutSymmetry _ _).hom | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 487,
"column": 8
} | {
"line": 487,
"column": 58
} | [
{
"pp": "X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\ninst✝ : Flat f\nhUS : IsAffineOpen US\nhUT : IsCompact ↑UT\nhUX : IsCompact ↑UX... | ← mono_comp_iff_of_isIso (pushoutSymmetry _ _).hom | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.EvenOddRec | {
"line": 43,
"column": 87
} | {
"line": 45,
"column": 10
} | [
{
"pp": "P : ℕ → Sort u_1\nh0 : P 0\nh_even : (i : ℕ) → P i → P (2 * i)\nh_odd : (i : ℕ) → P i → P (2 * i + 1)\nH : h_even 0 h0 = h0\nn : ℕ\n⊢ evenOddRec h0 h_even h_odd (2 * n + 1) = h_odd n (evenOddRec h0 h_even h_odd n)",
"usedConstants": [
"Nat.bit",
"Nat.evenOddRec.match_1",
"congrArg... | by
apply binaryRec_eq true n
simp [H] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 183,
"column": 6
} | {
"line": 183,
"column": 11
} | [
{
"pp": "R : Type r\nS : Type s\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nW' : Affine R\nf : R →+* S\nx : R[X]\ny : W'.CoordinateRing\n⊢ (map W' f) (x • y) = Polynomial.map f x • (map W' f) y",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"AdjoinRoot.instSMulAdjoinRoot",
"Weierstrass... | smul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 247,
"column": 2
} | {
"line": 247,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\nn : ℕ\n⊢ W.ΨSq ↑n = W.preΨ' n ^ 2 * if Even n then W.Ψ₂Sq else 1",
"usedConstants": [
"Polynomial.instOne",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"WeierstrassCurve.Ψ₂Sq",
"mul_ite",
"Int.... | simp [ΨSq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 247,
"column": 2
} | {
"line": 247,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\nn : ℕ\n⊢ W.ΨSq ↑n = W.preΨ' n ^ 2 * if Even n then W.Ψ₂Sq else 1",
"usedConstants": [
"Polynomial.instOne",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"WeierstrassCurve.Ψ₂Sq",
"mul_ite",
"Int.... | simp [ΨSq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 247,
"column": 2
} | {
"line": 247,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\nn : ℕ\n⊢ W.ΨSq ↑n = W.preΨ' n ^ 2 * if Even n then W.Ψ₂Sq else 1",
"usedConstants": [
"Polynomial.instOne",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"WeierstrassCurve.Ψ₂Sq",
"mul_ite",
"Int.... | simp [ΨSq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.ΨSq 0 = 0",
"usedConstants": [
"False",
"Polynomial.instOne",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"MulZeroClass.zero_mul",
"AddMonoid.toAddZeroClass",
"Nat.instAtLeastTwoHAddO... | simp [ΨSq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.ΨSq 0 = 0",
"usedConstants": [
"False",
"Polynomial.instOne",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"MulZeroClass.zero_mul",
"AddMonoid.toAddZeroClass",
"Nat.instAtLeastTwoHAddO... | simp [ΨSq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.ΨSq 0 = 0",
"usedConstants": [
"False",
"Polynomial.instOne",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"MulZeroClass.zero_mul",
"AddMonoid.toAddZeroClass",
"Nat.instAtLeastTwoHAddO... | simp [ΨSq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.ΨSq 1 = 1",
"usedConstants": [
"one_pow",
"MulOne.toOne",
"Polynomial.instOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"WeierstrassCurve.Ψ₂Sq",
"Int... | simp [ΨSq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.ΨSq 1 = 1",
"usedConstants": [
"one_pow",
"MulOne.toOne",
"Polynomial.instOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"WeierstrassCurve.Ψ₂Sq",
"Int... | simp [ΨSq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.ΨSq 1 = 1",
"usedConstants": [
"one_pow",
"MulOne.toOne",
"Polynomial.instOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"WeierstrassCurve.Ψ₂Sq",
"Int... | simp [ΨSq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 12
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\n⊢ W.ΨSq 2 = W.Ψ₂Sq",
"usedConstants": [
"one_pow",
"MulOne.toOne",
"Polynomial.instOne",
"HMul.hMul",
"even_two._simp_1",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"Add... | simp [ΨSq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
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