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.AlgebraicGeometry.Group.Smooth | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 47
} | [
{
"pp": "K : Type u\ninst✝⁴ : Field K\nG : Scheme\nf : G ⟶ Spec (CommRingCat.of K)\ninst✝³ : LocallyOfFiniteType f\ninst✝² : GrpObj (Over.mk f)\ninst✝¹ : IsReduced G\ninst✝ : IsAlgClosed K\nthis✝ : JacobsonSpace ↥G\nthis : Nonempty ↥G\nH : (↑(Scheme.Hom.smoothLocus f))ᶜ.Nonempty\nx : ↥G\nhx : x ∈ (↑(Scheme.Hom.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.LimitsOver | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 15
} | [
{
"pp": "case left\nX✝ : Scheme\nP : MorphismProperty Scheme\nS : Scheme\nU X Y : P.Over ⊤ S\nf : U ⟶ X\ng : U ⟶ Y\ninst✝¹ : IsOpenImmersion f.left\ninst✝ : IsOpenImmersion g.left\n⊢ Function.Injective\n ⇑(ConcreteCategory.hom\n ((span f g ⋙ MorphismProperty.Over.forget P ⊤ S ⋙ Over.forget S ⋙ Scheme.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.LimitsOver | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 15
} | [
{
"pp": "case right\nX✝ : Scheme\nP : MorphismProperty Scheme\nS : Scheme\nU X Y : P.Over ⊤ S\nf : U ⟶ X\ng : U ⟶ Y\ninst✝¹ : IsOpenImmersion f.left\ninst✝ : IsOpenImmersion g.left\n⊢ Function.Injective\n ⇑(ConcreteCategory.hom\n ((span f g ⋙ MorphismProperty.Over.forget P ⊤ S ⋙ Over.forget S ⋙ Scheme... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.LimitsOver | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 20
} | [
{
"pp": "X : Scheme\nP : MorphismProperty Scheme\ninst✝⁵ : IsZariskiLocalAtSource P\nS : Scheme\nJ : Type u_1\ninst✝⁴ : Category.{v_1, u_1} J\nF : J ⥤ P.Over ⊤ S\ninst✝³ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f).left\ninst✝² : (F ⋙ MorphismProperty.Over.forget P ⊤ S ⋙ Over.forget S ⋙ Scheme.forget).I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Adjunction.Adj | {
"line": 143,
"column": 8
} | {
"line": 143,
"column": 19
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c d : Adj B\nα : a ⟶ b\n⊢ (conjugateEquiv α.adj (𝟙 a ≫ α).adj) (λ_ α.l).hom = (ρ_ α.r).symm.hom",
"usedConstants": [
"Equiv.instEquivLike",
"CategoryTheory.CategoryStruct.toQuiver",
"CategoryTheory.Bicategory.Adj.Hom.l",
"Quiver.Hom",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Adjunction.Adj | {
"line": 143,
"column": 5
} | {
"line": 143,
"column": 72
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c d : Adj B\nα : a ⟶ b\n⊢ (conjugateEquiv α.adj (𝟙 a ≫ α).adj) (λ_ α.l).hom = (ρ_ α.r).symm.hom",
"usedConstants": [
"CategoryTheory.Bicategory.Adjunction.comp",
"Equiv.instEquivLike",
"CategoryTheory.CategoryStruct.toQuiver",
"Category... | by simpa using conjugateEquiv_id_comp_right_apply α.adj α.adj (𝟙 _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Bicategory.Adjunction.Adj | {
"line": 149,
"column": 8
} | {
"line": 149,
"column": 19
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c d : Adj B\nα : a ⟶ b\n⊢ (conjugateEquiv α.adj (α ≫ 𝟙 b).adj) (ρ_ α.l).hom = (λ_ α.r).symm.hom",
"usedConstants": [
"Equiv.instEquivLike",
"CategoryTheory.CategoryStruct.toQuiver",
"CategoryTheory.Bicategory.Adj.Hom.l",
"Quiver.Hom",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Adjunction.Cat | {
"line": 52,
"column": 4
} | {
"line": 52,
"column": 15
} | [
{
"pp": "C✝ D✝ E : Type u\ninst✝² : Category.{v, u} C✝\ninst✝¹ : Category.{v, u} D✝\ninst✝ : Category.{v, u} E\nF✝ : C✝ ⥤ D✝\nG✝ : D✝ ⥤ C✝\nadj✝ : F✝ ⊣ G✝\nF' : D✝ ⥤ E\nG' : E ⥤ D✝\nadj' : F' ⊣ G'\nC D : Cat\nF : C ⟶ D\nG : D ⟶ C\nadj : F ⊣ G\nX : ↑C\n⊢ F.toFunctor.map (adj.unit.toNatTrans.app X) ≫ adj.counit.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Adjunction.Cat | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 15
} | [
{
"pp": "C✝ D✝ E : Type u\ninst✝² : Category.{v, u} C✝\ninst✝¹ : Category.{v, u} D✝\ninst✝ : Category.{v, u} E\nF✝ : C✝ ⥤ D✝\nG✝ : D✝ ⥤ C✝\nadj✝ : F✝ ⊣ G✝\nF' : D✝ ⥤ E\nG' : E ⥤ D✝\nadj' : F' ⊣ G'\nC D : Cat\nF : C ⟶ D\nG : D ⟶ C\nadj : F ⊣ G\nX : ↑D\n⊢ adj.unit.toNatTrans.app (G.toFunctor.obj X) ≫ G.toFunctor.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.ConcreteCategory.WithAlgebraicStructures | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Ring R\nJ : Type w\ninst✝³ : Category.{r, w} J\nF : J ⥤ ModuleCat R\ninst✝² : PreservesColimit F (forget (ModuleCat R))\ninst✝¹ : IsFiltered J\ninst✝ : HasColimit F\nr : R\nj : J\nx : ToType (F.obj j)\nhx : (ModuleCat.Hom.hom (colimit.ι F j)) (r • x) = 0\nj' : J\ni : j ⟶ j'\nh : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.ModuleFinitePresentation | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 28
} | [
{
"pp": "case insert\nR : Type u\nS : Type u_1\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Module.Finite R S\na : S\ns : Finset S\nhas : a ∉ s\nS' : Type u\nw✝⁴ : CommRing S'\nw✝³ : Algebra R S'\nw✝² : Module.Finite R S'\nw✝¹ : Free R S'\nw✝ : Algebra.FinitePresentation R S'\nf : S'... | algebraize [f.toRingHom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.AlgebraicGeometry.Morphisms.FlatRank | {
"line": 131,
"column": 2
} | {
"line": 132,
"column": 60
} | [
{
"pp": "case h.hy\nR S : CommRingCat\nf : R ⟶ S\nhf₁✝ : IsFinite (Spec.map f)\nhf₂✝ : Flat (Spec.map f)\nhf₁ : (CommRingCat.Hom.hom (appTop (Spec.map f))).Finite\nhf₂ : (CommRingCat.Hom.hom (appTop (Spec.map f))).Flat\nx✝ : ↥(Spec R)\nthis : f = (ΓSpecIso R).inv ≫ appTop (Spec.map f) ≫ (ΓSpecIso S).hom\n⊢ x✝ =... | · simp [isoSpec_Spec_hom, SpecMap_ΓSpecIso_hom, ← AlgebraicGeometry.Spec.map_apply,
← Scheme.Hom.comp_apply, toSpecΓ_SpecMap_ΓSpecIso_inv] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 35
} | [
{
"pp": "case h\nι : Type u_2\nR : ι → Type u_1\ninst✝ : (i : ι) → CommRing (R i)\ns : Set ((i : ι) → R i)\nhs : s.Finite\nf : (i : ι) → { x // x ∈ s } →₀ R i\nhf : ∀ (i : ι), ((f i).sum fun i_1 a ↦ a * (⇑(Pi.evalRingHom (fun x ↦ R x) i) ∘ Subtype.val) i_1) = 1\nthis : Fintype ↑s\ni : ι\n⊢ (Finsupp.equivFunOnFi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 52,
"column": 84
} | {
"line": 52,
"column": 95
} | [
{
"pp": "ι : Type u\nR : ι → CommRingCat\nU : (Spec (CommRingCat.of ((i : ι) → ↑(R i)))).Opens\nV : Set ↥(Spec (CommRingCat.of ((i : ι) → ↑(R i))))\nhV : ↑(Scheme.Hom.opensRange (sigmaSpec R)) ⊆ V\nhV' : IsCompact V\nhVU : V ⊆ ↑U\ns : Set ↑(CommRingCat.of ((i : ι) → ↑(R i)))\nhs : U.carrierᶜ = zeroLocus s\nt : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 72
} | [
{
"pp": "ι : Type u\nR : ι → CommRingCat\nU : (Spec (CommRingCat.of ((i : ι) → ↑(R i)))).Opens\nV : Set ↥(Spec (CommRingCat.of ((i : ι) → ↑(R i))))\nhV : ↑(Scheme.Hom.opensRange (sigmaSpec R)) ⊆ V\nhV' : IsCompact V\nhVU : V ⊆ ↑U\ns : Set ↑(CommRingCat.of ((i : ι) → ↑(R i)))\nhs : U.carrierᶜ = zeroLocus s\nt : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 71
} | [
{
"pp": "ι : Type u\nR : ι → CommRingCat\nU : (Spec (CommRingCat.of ((i : ι) → ↑(R i)))).Opens\nV : Set ↥(Spec (CommRingCat.of ((i : ι) → ↑(R i))))\nhV : ↑(Scheme.Hom.opensRange (sigmaSpec R)) ⊆ V\nhV' : IsCompact V\nhVU : V ⊆ ↑U\ns : Set ↑(CommRingCat.of ((i : ι) → ↑(R i)))\nhs : U.carrierᶜ = zeroLocus s\nt : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 71
} | [
{
"pp": "case h\nι : Type u\nR : ι → CommRingCat\nI : Ideal ((i : ι) → ↑(R i))\nf : (∐ fun i ↦ Spec (R i)) ⟶ Spec (CommRingCat.of (((i : ι) → ↑(R i)) ⧸ I))\nhf : f ≫ Spec.map (CommRingCat.ofHom (Ideal.Quotient.mk I)) = sigmaSpec R\nx : (i : ι) → ↑(R i)\nhx : x ∈ I\ni : ι\n⊢ x i = 0 i",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 83,
"column": 6
} | {
"line": 83,
"column": 30
} | [
{
"pp": "case hV\nι : Type u\nR : ι → CommRingCat\nV : Scheme\nf : (∐ fun i ↦ Spec (R i)) ⟶ V\ng : V ⟶ Spec (CommRingCat.of ((i : ι) → ↑(R i)))\ninst✝¹ : IsImmersion g\ninst✝ : CompactSpace ↥V\nhU' : f ≫ g = sigmaSpec R\n⊢ ↑(Scheme.Hom.opensRange (sigmaSpec R)) ⊆ Set.range ⇑g",
"usedConstants": [
"Eq.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 83,
"column": 6
} | {
"line": 83,
"column": 62
} | [
{
"pp": "case hV\nι : Type u\nR : ι → CommRingCat\nV : Scheme\nf : (∐ fun i ↦ Spec (R i)) ⟶ V\ng : V ⟶ Spec (CommRingCat.of ((i : ι) → ↑(R i)))\ninst✝¹ : IsImmersion g\ninst✝ : CompactSpace ↥V\nhU' : f ≫ g = sigmaSpec R\n⊢ ↑(Scheme.Hom.opensRange (sigmaSpec R)) ⊆ Set.range ⇑g",
"usedConstants": [
"Set... | simpa only [← hU'] using Set.range_comp_subset_range f g | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 83,
"column": 6
} | {
"line": 83,
"column": 62
} | [
{
"pp": "case hV\nι : Type u\nR : ι → CommRingCat\nV : Scheme\nf : (∐ fun i ↦ Spec (R i)) ⟶ V\ng : V ⟶ Spec (CommRingCat.of ((i : ι) → ↑(R i)))\ninst✝¹ : IsImmersion g\ninst✝ : CompactSpace ↥V\nhU' : f ≫ g = sigmaSpec R\n⊢ ↑(Scheme.Hom.opensRange (sigmaSpec R)) ⊆ Set.range ⇑g",
"usedConstants": [
"Set... | simpa only [← hU'] using Set.range_comp_subset_range f g | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 83,
"column": 6
} | {
"line": 83,
"column": 62
} | [
{
"pp": "case hV\nι : Type u\nR : ι → CommRingCat\nV : Scheme\nf : (∐ fun i ↦ Spec (R i)) ⟶ V\ng : V ⟶ Spec (CommRingCat.of ((i : ι) → ↑(R i)))\ninst✝¹ : IsImmersion g\ninst✝ : CompactSpace ↥V\nhU' : f ≫ g = sigmaSpec R\n⊢ ↑(Scheme.Hom.opensRange (sigmaSpec R)) ⊆ Set.range ⇑g",
"usedConstants": [
"Set... | simpa only [← hU'] using Set.range_comp_subset_range f g | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 107,
"column": 46
} | {
"line": 107,
"column": 57
} | [
{
"pp": "case h\nι : Type u\nR : ι → CommRingCat\nX : Scheme\ninst✝ : QuasiSeparatedSpace ↥X\nf g : Spec (CommRingCat.of ((i : ι) → ↑(R i))) ⟶ X\ne : pointsPi R X f = pointsPi R X g\ni : ι\n⊢ Sigma.ι (fun i ↦ Spec (R i)) i ≫ sigmaSpec R ≫ f = Sigma.ι (fun i ↦ Spec (R i)) i ≫ sigmaSpec R ≫ g",
"usedConstants... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 105,
"column": 2
} | {
"line": 109,
"column": 58
} | [
{
"pp": "ι : Type u\nR : ι → CommRingCat\nX : Scheme\ninst✝ : QuasiSeparatedSpace ↥X\n⊢ Function.Injective (pointsPi R X)",
"usedConstants": [
"AlgebraicGeometry.instCompactSpaceCarrierCarrierCommRingCatEqualizerSchemeOfQuasiSeparatedSpace",
"Eq.mpr",
"AlgebraicGeometry.Spec",
"Algeb... | rintro f g e
have := isIso_of_comp_eq_sigmaSpec R (V := equalizer f g)
(equalizer.lift (sigmaSpec R) (by ext1 i; simpa using congr_fun e i))
(equalizer.ι f g) (by simp)
rw [← cancel_epi (equalizer.ι f g), equalizer.condition] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 105,
"column": 2
} | {
"line": 109,
"column": 58
} | [
{
"pp": "ι : Type u\nR : ι → CommRingCat\nX : Scheme\ninst✝ : QuasiSeparatedSpace ↥X\n⊢ Function.Injective (pointsPi R X)",
"usedConstants": [
"AlgebraicGeometry.instCompactSpaceCarrierCarrierCommRingCatEqualizerSchemeOfQuasiSeparatedSpace",
"Eq.mpr",
"AlgebraicGeometry.Spec",
"Algeb... | rintro f g e
have := isIso_of_comp_eq_sigmaSpec R (V := equalizer f g)
(equalizer.lift (sigmaSpec R) (by ext1 i; simpa using congr_fun e i))
(equalizer.ι f g) (by simp)
rw [← cancel_epi (equalizer.ι f g), equalizer.condition] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Submodule | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 15
} | [
{
"pp": "ιA : Type u_1\nιM : Type u_2\nσA : Type u_3\nσM : Type u_4\nA : Type u_5\nM : Type u_6\ninst✝¹³ : Semiring A\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : Module A M\n𝒜 : ιA → σA\nℳ : ιM → σM\ninst✝¹⁰ : DecidableEq ιA\ninst✝⁹ : AddMonoid ιA\ninst✝⁸ : SetLike σA A\ninst✝⁷ : AddSubmonoidClass σA A\ninst✝⁶ : Gra... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Adjunction.Mate | {
"line": 591,
"column": 7
} | {
"line": 591,
"column": 48
} | [
{
"pp": "B : Type u\ninst✝¹ : Bicategory B\nc d : B\nl₁ l₂ : c ⟶ d\nr₁ r₂ : d ⟶ c\nadj₁ : l₁ ⊣ r₁\nadj₂ : l₂ ⊣ r₂\nα : l₂ ⟶ l₁\ninst✝ : IsIso ((conjugateEquiv adj₁ adj₂) α)\nthis : IsIso ((conjugateEquiv adj₁ adj₂).symm ((conjugateEquiv adj₁ adj₂) α))\n⊢ IsIso α",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Adjunction.Mate | {
"line": 601,
"column": 7
} | {
"line": 601,
"column": 48
} | [
{
"pp": "B : Type u\ninst✝¹ : Bicategory B\nc d : B\nl₁ l₂ : c ⟶ d\nr₁ r₂ : d ⟶ c\nadj₁ : l₁ ⊣ r₁\nadj₂ : l₂ ⊣ r₂\nα : r₁ ⟶ r₂\ninst✝ : IsIso ((conjugateEquiv adj₁ adj₂).symm α)\nthis : IsIso ((conjugateEquiv adj₁ adj₂) ((conjugateEquiv adj₁ adj₂).symm α))\n⊢ IsIso α",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal | {
"line": 604,
"column": 65
} | {
"line": 604,
"column": 76
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁷ : Semiring A\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : PartialOrder ι\ninst✝³ : CanonicallyOrderedAdd ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nx : A\nhx : x ∈ 𝒜₊.toAddSubmonoid\nj : ι\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 49
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nideal_gc : GaloisConnection Ideal.span SetLike.coe\n⊢ GaloisConnection (fun s ↦ zeroLocus 𝒜 s) fun t ↦ ↑(vanishingIdeal t)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 95
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nI : HomogeneousIdeal 𝒜\nt : (Set (ProjectiveSpectrum 𝒜))ᵒᵈ\n⊢ (fun I ↦ zeroLocus 𝒜 ↑I) I ≤ t ↔ I ≤ (fun t ↦ vanishingIdeal t) t",
"usedConstants": [
"Ch... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 13
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\n⊢ vanishingIdeal ∅ = ⊤",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"CommSemiring.toSemiring",
"Nat.instAddMonoid",
"Project... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology | {
"line": 260,
"column": 22
} | {
"line": 260,
"column": 33
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf g : A\nx : ProjectiveSpectrum 𝒜\n⊢ x ∈ zeroLocus 𝒜 {f * g} ↔ x ∈ zeroLocus 𝒜 {f} ∪ zeroLocus 𝒜 {g}",
"usedConstants": [
"Eq.mpr",
"SetLike.mem_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology | {
"line": 265,
"column": 22
} | {
"line": 265,
"column": 33
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nn : ℕ\nhn : 0 < n\nx : ProjectiveSpectrum 𝒜\n⊢ x ∈ zeroLocus 𝒜 {f ^ n} ↔ x ∈ zeroLocus 𝒜 {f}",
"usedConstants": [
"Eq.mpr",
"SetLike.mem_co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology | {
"line": 373,
"column": 35
} | {
"line": 373,
"column": 46
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nn : ℕ\nhn : 0 < n\n⊢ ↑(basicOpen 𝒜 (f ^ n)) = ↑(basicOpen 𝒜 f)",
"usedConstants": [
"Eq.mpr",
"ProjectiveSpectrum.zariskiTopology",
"c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.FiniteType | {
"line": 37,
"column": 38
} | {
"line": 37,
"column": 49
} | [
{
"pp": "S : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : CommRing S\ninst✝³ : SetLike σ S\ninst✝² : AddSubgroupClass σ S\n𝒜 : ι → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) S\nF : Finset S\nhF : Algebra.adjoin ↥(𝒜 0) ↑F = ⊤\nι₀ : Type (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.FiniteType | {
"line": 41,
"column": 53
} | {
"line": 41,
"column": 64
} | [
{
"pp": "S : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : CommRing S\ninst✝³ : SetLike σ S\ninst✝² : AddSubgroupClass σ S\n𝒜 : ι → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) S\nF : Finset S\nhF : Algebra.adjoin ↥(𝒜 0) ↑F = ⊤\nι₀ : Type (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.FiniteType | {
"line": 47,
"column": 37
} | {
"line": 47,
"column": 48
} | [
{
"pp": "S : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : CommRing S\ninst✝³ : SetLike σ S\ninst✝² : AddSubgroupClass σ S\n𝒜 : ι → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) S\ns : Finset S\nh₁ : Algebra.adjoin ↥(𝒜 0) ↑s = ⊤\nn : S → ι\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.FiniteType | {
"line": 53,
"column": 36
} | {
"line": 53,
"column": 53
} | [
{
"pp": "S : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : CommRing S\ninst✝³ : SetLike σ S\ninst✝² : AddSubgroupClass σ S\n𝒜 : ι → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) S\ns : Finset S\nh₁ : Algebra.adjoin ↥(𝒜 0) ↑s = ⊤\nn : S → ι\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 430,
"column": 42
} | {
"line": 430,
"column": 58
} | [
{
"pp": "σ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nX : Scheme\nf✝ : A →+* ↑Γ(X, ⊤)\nx✝ x' : ↑Γ(X, ⊤)\nt t' : A\nd d' : ℕ\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nx : A\nhx : x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 452,
"column": 4
} | {
"line": 452,
"column": 70
} | [
{
"pp": "σ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nX : Scheme\nf✝ : A →+* ↑Γ(X, ⊤)\nx✝ x' : ↑Γ(X, ⊤)\nt t' : A\nd d' : ℕ\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nx y : (openCov... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 454,
"column": 4
} | {
"line": 454,
"column": 70
} | [
{
"pp": "σ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nX : Scheme\nf✝ : A →+* ↑Γ(X, ⊤)\nx✝ x' : ↑Γ(X, ⊤)\nt t' : A\nd d' : ℕ\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nx y : (openCov... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 471,
"column": 47
} | {
"line": 471,
"column": 80
} | [
{
"pp": "case a\nσ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nX : Scheme\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nr : A\nn : ℕ\nhn : 0 < n\nhr : r ∈ 𝒜 n\ni : (openCoverOfMapIrrel... | ← TopologicalSpace.Opens.map_coe, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 473,
"column": 26
} | {
"line": 473,
"column": 59
} | [
{
"pp": "case a\nσ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nX : Scheme\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nr : A\nn : ℕ\nhn : 0 < n\nhr : r ∈ 𝒜 n\ni : (openCoverOfMapIrrel... | ← TopologicalSpace.Opens.map_coe, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 251,
"column": 23
} | {
"line": 251,
"column": 61
} | [
{
"pp": "case succ\nι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubmonoidClass σ A\n𝒜 : ι → σ\nx : Submonoid A\ninst✝² : AddCommMonoid ι\ninst✝¹ : DecidableEq ι\ninst✝ : GradedRing 𝒜\nc : NumDenSameDeg 𝒜 x\nn : ℕ\nih : ↑(GradedMonoid.GMonoid.gnpow n c.den)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 488,
"column": 6
} | {
"line": 488,
"column": 49
} | [
{
"pp": "σ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nX : Scheme\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nr : A\nn : ℕ\nhn : 0 < n\nhr : r ∈ 𝒜 n\nx : ↥X\nhx : x ∈ X.basicOpen (f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 216,
"column": 12
} | {
"line": 216,
"column": 78
} | [
{
"pp": "R : 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⊢ ∃ n, unop f ∣ unop g ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 593,
"column": 4
} | {
"line": 594,
"column": 11
} | [
{
"pp": "ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : SetLike σ A\ninst✝⁴ : AddSubgroupClass σ A\n𝒜 : ι → σ\nx : Submonoid A\ninst✝³ : AddCommMonoid ι\ninst✝² : DecidableEq ι\ninst✝¹ : GradedRing 𝒜\n𝔭 : Ideal A\ninst✝ : 𝔭.IsPrime\na : AtPrime 𝒜 𝔭\n⊢ IsUnit a ∨ IsUnit (1 - a)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 390,
"column": 5
} | {
"line": 390,
"column": 76
} | [
{
"pp": "R : CommRingCat\nM : ModuleCat ↑R\ns : Set ↑M\nhs : Submodule.span (↑R) s = ⊤\nt : Set (↑s →₀ ↑R)\nht : Submodule.span (↑R) t = (Finsupp.linearCombination (↑R) Subtype.val).ker\nH₁ :\n Function.Exact ⇑(ConcreteCategory.hom (ModuleCat.ofHom (Finsupp.linearCombination (↑R) Subtype.val)))\n ⇑(Concrete... | by simp [← LinearMap.range_eq_top, Finsupp.range_linearCombination, hs] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LocalRing.LocalSubring | {
"line": 114,
"column": 4
} | {
"line": 115,
"column": 73
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nK : Type u_3\ninst✝¹ : Field K\nA : Subring K\nP : Ideal ↥A\ninst✝ : P.IsPrime\nx : ↥A\ns : ↥P.primeCompl\ny : ↥A\nt : ↥P.primeCompl\ne :\n (IsLocalization.liftAlgHom ⋯) (IsLocalization.mk' (Localization.AtPrime P) x s) =\n (IsLo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 414,
"column": 8
} | {
"line": 414,
"column": 52
} | [
{
"pp": "case neg\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\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nq : ↑↑(Spec A⁰_ f).toPresheafedSpace\nx : A\nhx : x ∈ carrier f_deg q\nn : ℕ\na : A\nha : a ∈ 𝒜 n\ni : ℕ\nprodu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 431,
"column": 37
} | {
"line": 431,
"column": 93
} | [
{
"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\nq : ↑↑(Spec A⁰_ f).toPresheafedSpace\ni : ℕ\na : A\nha : a ∈ asIdeal f_deg hm q\nj : ℕ\nh : i = j\n⊢ HomogeneousLocalizati... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 433,
"column": 4
} | {
"line": 434,
"column": 38
} | [
{
"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\nq : ↑↑(Spec A⁰_ f).toPresheafedSpace\ni : ℕ\na : A\nha : a ∈ asIdeal f_deg hm q\nj : ℕ\nh : ¬i = j\n⊢ HomogeneousLocalizat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 476,
"column": 12
} | {
"line": 476,
"column": 44
} | [
{
"pp": "case h.e'_2.h.e'_4\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\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nq : ↑↑(Spec A⁰_ f).toPresheafedSpace\nx y : A\nx✝¹ : IsHomogeneousElem 𝒜 x\nx✝ : IsHomogeneousElem 𝒜... | decompose_of_mem_ne 𝒜 _ hn.symm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 476,
"column": 12
} | {
"line": 476,
"column": 44
} | [
{
"pp": "case h.e'_2.h.e'_4\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\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nq : ↑↑(Spec A⁰_ f).toPresheafedSpace\nx y : A\nx✝¹ : IsHomogeneousElem 𝒜 x\nx✝ : IsHomogeneousElem 𝒜... | decompose_of_mem_ne 𝒜 _ hn.symm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 1044,
"column": 40
} | {
"line": 1044,
"column": 78
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 1080,
"column": 40
} | {
"line": 1080,
"column": 55
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 1085,
"column": 4
} | {
"line": 1085,
"column": 47
} | [
{
"pp": "case refine_2.H.refine_1\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\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 696,
"column": 35
} | {
"line": 698,
"column": 48
} | [
{
"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\nt : NumDenSameDeg 𝒜 (Submonoid.powers f)\n⊢ ⇑(ConcreteCategory.hom (toSpec 𝒜 f).base) ⁻¹' ↑(sbo HomogeneousLocalization.mk t) =\n ↑((Opens.comap { toFun :... | by
convert! (ProjIsoSpecTopComponent.ToSpec.preimage_basicOpen f t)
exact funext fun _ => toSpec_base_apply_eq _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 71,
"column": 20
} | {
"line": 71,
"column": 66
} | [
{
"pp": "K : Type u_3\ninst✝ : Field K\nR : LocalSubring K\nhR : IsMax R\nx : K\nhx : IsIntegral (↥R.toSubring) x\nS : Subalgebra (↥R.toSubring) K := (↥R.toSubring)[x]\nthis : Algebra.IsIntegral ↥R.toSubring ↥S\nQ : Ideal ↥S.toSubring\nhQ : Q.IsMaximal\ne : Ideal.comap (algebraMap ↥R.toSubring ↥S) Q = maximalId... | ← IsLocalization.AtPrime.map_eq_maximalIdeal Q | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 1104,
"column": 40
} | {
"line": 1104,
"column": 61
} | [
{
"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\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) A\nf : A\nd : ℕ\nhf : f ∈ 𝒜 d\ns : Finset A\nhs : Algebra.adjoin ↥(𝒜 0) ↑s = ⊤\nhs' : ∀ i ∈ s, ∃ n, n ≠ 0 ∧ i ∈ 𝒜 n\ndx : ↥s ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 87,
"column": 22
} | {
"line": 87,
"column": 79
} | [
{
"pp": "K : Type u_3\ninst✝ : Field K\nR : ValuationSubring K\nS : LocalSubring K\nhS : R.toLocalSubring ≤ S\nx : K\nhx : x ∈ S.toSubring\nh : x⁻¹ ∈ R.carrier\nh' : x ∉ R.toLocalSubring.toSubring\nhx0 : x ≠ 0\nthis : IsUnit ((Subring.inclusion ⋯) ⟨x⁻¹, h⟩)\nx' : ↥R.toLocalSubring.toSubring\nhx' : ⟨x⁻¹, h⟩ * x'... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 104,
"column": 58
} | {
"line": 104,
"column": 69
} | [
{
"pp": "K : Type u_3\ninst✝ : Field K\nR : LocalSubring K\nhR : IsMax R\nx : K\nhx : x ∉ R.toSubring\nhx0 : x ≠ 0\nthis✝ : Invertible x := invertibleOfNonzero hx0\nS : Subalgebra (↥R.toSubring) K := (↥R.toSubring)[x]\nthis : R.toSubring < S.toSubring\np : Polynomial ↥R.toSubring\nhp : p.leadingCoeff - 1 ∈ maxi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 105,
"column": 59
} | {
"line": 105,
"column": 70
} | [
{
"pp": "K : Type u_3\ninst✝ : Field K\nR : LocalSubring K\nhR : IsMax R\nx : K\nhx : x ∉ R.toSubring\nhx0 : x ≠ 0\nthis✝ : Invertible x := invertibleOfNonzero hx0\nS : Subalgebra (↥R.toSubring) K := (↥R.toSubring)[x]\nthis : R.toSubring < S.toSubring\np : Polynomial ↥R.toSubring\nhp : p.leadingCoeff - 1 ∈ maxi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 85
} | [
{
"pp": "case refine_1.refine_1\nK : Type u_3\ninst✝ : Field K\nA✝ : LocalSubring K\ns : Set (LocalSubring K)\nhs : s ⊆ Set.Ici A✝\nH : IsChain (fun x1 x2 ↦ x1 ≤ x2) s\ny : LocalSubring K\nhys : y ∈ s\ninst : Nonempty ↑s\nhdir : Directed LE.le (toSubring ∘ fun x ↦ ↑x)\na : K\nha : a ∈ ⨆ i, (↑i).toSubring\nb : K... | · exact fun h ↦ h.map (Subring.inclusion (le_iSup (fun i : s ↦ i.1.toSubring) C)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 85
} | [
{
"pp": "case refine_1.refine_2\nK : Type u_3\ninst✝ : Field K\nA✝ : LocalSubring K\ns : Set (LocalSubring K)\nhs : s ⊆ Set.Ici A✝\nH : IsChain (fun x1 x2 ↦ x1 ≤ x2) s\ny : LocalSubring K\nhys : y ∈ s\ninst : Nonempty ↑s\nhdir : Directed LE.le (toSubring ∘ fun x ↦ ↑x)\na : K\nha : a ∈ ⨆ i, (↑i).toSubring\nb : K... | · exact fun h ↦ h.map (Subring.inclusion (le_iSup (fun i : s ↦ i.1.toSubring) C)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 165,
"column": 53
} | {
"line": 165,
"column": 85
} | [
{
"pp": "K : Type u_3\ninst✝¹ : Field K\nx : K\nR : Subring K\nhxR : x ∉ R\ninst✝ : IsIntegrallyClosedIn (↥R) K\nhx0 : x ≠ 0\nthis : Invertible x := invertibleOfNonzero hx0\nB : Subalgebra (↥R) K := (↥R)[x⁻¹]\nxinv : ↥B.toSubring := ⟨x⁻¹, ⋯⟩\neq : Ideal.span {xinv} = ⊤\np : (↥R)[X]\nhp : p.leadingCoeff - 1 ∈ ⊥\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 182,
"column": 60
} | {
"line": 182,
"column": 71
} | [
{
"pp": "K : Type u_3\ninst✝¹ : Field K\nx : K\nR : LocalSubring K\nhxR : x ∉ R.toSubring\ninst✝ : IsIntegrallyClosedIn (↥R.toSubring) K\nhx0 : x ≠ 0\nthis : Invertible x := invertibleOfNonzero hx0\nB : Subalgebra (↥R.toSubring) K := (↥R.toSubring)[x⁻¹]\nxinv : ↥B.toSubring := ⟨x⁻¹, ⋯⟩\neq : Ideal.map (algebraM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 184,
"column": 57
} | {
"line": 184,
"column": 68
} | [
{
"pp": "K : Type u_3\ninst✝¹ : Field K\nx : K\nR : LocalSubring K\nhxR : x ∉ R.toSubring\ninst✝ : IsIntegrallyClosedIn (↥R.toSubring) K\nhx0 : x ≠ 0\nthis : Invertible x := invertibleOfNonzero hx0\nB : Subalgebra (↥R.toSubring) K := (↥R.toSubring)[x⁻¹]\nxinv : ↥B.toSubring := ⟨x⁻¹, ⋯⟩\neq : Ideal.map (algebraM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 233,
"column": 32
} | {
"line": 233,
"column": 49
} | [
{
"pp": "R : 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✝ : IsLocalHom f\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Sites.Small | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 96
} | [
{
"pp": "case h.mpr\nP : MorphismProperty Scheme\nS : Scheme\ninst✝¹ : P.IsStableUnderBaseChange\nX : Over S\n𝒰 : Cover (precoverage P) X.left\ninst✝ : Cover.Over S 𝒰\nV : Scheme\nf : V ⟶ X.left\nY : Scheme\nk : 𝒰.I₀\nh : V ⟶ 𝒰.X k\nhcomp : h ≫ 𝒰.f k = f\nthis : 𝒰.f k ≫ X.hom = 𝒰.X k ↘ S\n⊢ ∃ Y h g, 𝒰.t... | refine ⟨(𝒰.X k).asOver S, Over.homMk h (by simp [← hcomp, this]), (𝒰.f k).asOver S, ⟨k⟩, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.Sites.Small | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 38
} | [
{
"pp": "P Q : MorphismProperty Scheme\nS : Scheme\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : P.IsMultiplicative\ninst✝ : P.RespectsIso\nY X : Over S\nf : X ⟶ Y\n𝒰 : Cover (precoverage P) Y.left\nh : Cover.Over S 𝒰\n⊢ Presieve.pullbackArrows f 𝒰.toPresieveOver = (Cover.pullbackCoverOver' S 𝒰 (Over.Hom.le... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ValuativeCriterion | {
"line": 200,
"column": 45
} | {
"line": 200,
"column": 51
} | [
{
"pp": "case of_isPullback\nX✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nY' X X' Y : Scheme\nY'_to_Y : Y' ⟶ Y\nf : X ⟶ Y\nX'_to_X : X' ⟶ X\nf' : X' ⟶ Y'\nhP : IsPullback X'_to_X f' f Y'_to_Y\nhf : Existence f\n⊢ Existence f'",
"usedConstants": [
"AlgebraicGeometry.ValuativeCommSq"
]
}
] | commSq | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 131,
"column": 24
} | {
"line": 131,
"column": 35
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝² : Category.{v₀, u₀} C₀\ninst✝¹ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nX : C\ndata : F.PreOneHypercoverDenseData X\ni₁ i₂ : data.I₀\nW₀ : C₀\np₁ : W₀ ⟶ data.X i₁\np₂ : W₀ ⟶ data... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 131,
"column": 50
} | {
"line": 131,
"column": 61
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝² : Category.{v₀, u₀} C₀\ninst✝¹ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝ : Category.{v', u'} A\nX : C\ndata : F.PreOneHypercoverDenseData X\ni₁ i₂ : data.I₀\nW₀ : C₀\np₁ : W₀ ⟶ data.X i₁\np₂ : W₀ ⟶ data... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 217,
"column": 2
} | {
"line": 219,
"column": 85
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝² : Category.{v₀, u₀} C₀\ninst✝¹ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\ninst✝ : IsDenseSubsite J₀ J F\nX : C\ndata : F.OneHypercoverDenseData J₀ J X\ni₁ i₂ : data.I₀\nW : C\np₁ : W ⟶ F.obj (data.X i₁)\np₂ : W ⟶ F.obj (da... | let S := Sieve.bind (Sieve.coverByImage F W).arrows
(fun Y f hf ↦ ((F.imageSieve (hf.some.map ≫ p₁) ⊓
F.imageSieve (hf.some.map ≫ p₂)).functorPushforward F).pullback hf.some.lift) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicGeometry.ValuativeCriterion | {
"line": 298,
"column": 10
} | {
"line": 298,
"column": 21
} | [
{
"pp": "case h₀\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : IsSeparated f\nS : ValuativeCommSq f\nl₁ : Spec (CommRingCat.of S.R) ⟶ X\nhl₁ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₁ = S.i₁\nhl₁' : l₁ ≫ f = S.i₂\nl₂ : Spec (CommRingCat.of S.R) ⟶ X\nhl₂ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₂... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ValuativeCriterion | {
"line": 299,
"column": 10
} | {
"line": 299,
"column": 21
} | [
{
"pp": "case h₁\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : IsSeparated f\nS : ValuativeCommSq f\nl₁ : Spec (CommRingCat.of S.R) ⟶ X\nhl₁ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₁ = S.i₁\nhl₁' : l₁ ≫ f = S.i₂\nl₂ : Spec (CommRingCat.of S.R) ⟶ X\nhl₂ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₂... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ValuativeCriterion | {
"line": 300,
"column": 4
} | {
"line": 301,
"column": 29
} | [
{
"pp": "case allEq.l.left\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : IsSeparated f\nS : ValuativeCommSq f\nl₁ : Spec (CommRingCat.of S.R) ⟶ X\nhl₁ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₁ = S.i₁\nhl₁' : l₁ ≫ f = S.i₂\nl₂ : Spec (CommRingCat.of S.R) ⟶ X\nhl₂ : Spec.map (CommRingCat.ofHom (algebraMap S.R ... | have hg : l ≫ g = Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) :=
pullback.lift_snd _ _ _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.Elements | {
"line": 89,
"column": 47
} | {
"line": 89,
"column": 58
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nA : C ⥤ Type w\nI : Type u₁\ninst✝³ : Category.{v₁, u₁} I\ninst✝² : Small.{w, u₁} I\nF : I ⥤ A.Elements\ninst✝¹ : HasLimitsOfShape I C\ninst✝ : PreservesLimitsOfShape I A\ni : I\n⊢ (hom (A.map (limit.π (F ⋙ π A) i))) (((Functor.const I).obj ⟨limit (F ⋙ π A), lift... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Elements | {
"line": 90,
"column": 42
} | {
"line": 90,
"column": 53
} | [
{
"pp": "case w\nC : Type u\ninst✝⁴ : Category.{v, u} C\nA : C ⥤ Type w\nI : Type u₁\ninst✝³ : Category.{v₁, u₁} I\ninst✝² : Small.{w, u₁} I\nF : I ⥤ A.Elements\ninst✝¹ : HasLimitsOfShape I C\ninst✝ : PreservesLimitsOfShape I A\ni i' : I\nf : i ⟶ i'\n⊢ ↑(((Functor.const I).obj ⟨limit (F ⋙ π A), liftedConeElemen... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Elements | {
"line": 101,
"column": 16
} | {
"line": 101,
"column": 27
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nA : C ⥤ Type w\nI : Type u₁\ninst✝³ : Category.{v₁, u₁} I\ninst✝² : Small.{w, u₁} I\nF : I ⥤ A.Elements\ninst✝¹ : HasLimitsOfShape I C\ninst✝ : PreservesLimitsOfShape I A\ns : Cone F\nm : s.pt ⟶ (liftedCone F).pt\nh : ∀ (j : I), m ≫ (liftedCone F).π.app j = s.π.a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 350,
"column": 67
} | {
"line": 350,
"column": 78
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝³ : Category.{v₀, u₀} C₀\ninst✝² : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\nG : Cᵒᵖ ⥤ A\nhG₀ : Presh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 378,
"column": 10
} | {
"line": 378,
"column": 34
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝³ : Category.{v₀, u₀} C₀\ninst✝² : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\nG : Cᵒᵖ ⥤ A\nhG₀ : Presh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 382,
"column": 8
} | {
"line": 382,
"column": 19
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝³ : Category.{v₀, u₀} C₀\ninst✝² : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\nG : Cᵒᵖ ⥤ A\nhG₀ : Presh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 482,
"column": 2
} | {
"line": 482,
"column": 26
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\ninst✝ : HasLimitsOfSize... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Presheaf | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 15
} | [
{
"pp": "case w.h\nC : Type u\ninst✝² : Category.{v, u} C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : HasCoproducts A\nX : C\nM : A\nF : Cᵒᵖ ⥤ A\nf : freeYoneda X M ⟶ F\nY : Cᵒᵖ\nφ : (yoneda.obj X).obj Y\n⊢ Sigma.ι (fun i ↦ M) φ ≫\n ((fun g ↦ { app := fun Y ↦ Sigma.desc fun φ ↦ g ≫ F.map (Quiver.Ho... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.Basic | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 13
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nΦ : J.Point\ninst✝ : LocallySmall.{w, v, u} C\nX Y : C\nf : X ⟶ Y\nx : Φ.fiber.obj X\n⊢ (ConcreteCategory.hom ((shrinkYoneda.{w, v, u}.map f).app (op X))) (shrinkYonedaObjObjEquiv.symm (𝟙 X)) =\n shrinkYonedaObjObjEquiv.symm f",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.Basic | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 75
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\nΦ : J.Point\ninst✝ : LocallySmall.{w, v, u} C\nX Y : C\nf : X ⟶ Y\nx : Φ.fiber.obj X\n⊢ (ConcreteCategory.hom ((shrinkYoneda.{w, v, u}.map f).app (op X))) (shrinkYonedaObjObjEquiv.symm (𝟙 X)) =\n shrinkYonedaObjObjEquiv.symm f",
... | simpa using shrinkYoneda_map_app_shrinkYonedaObjObjEquiv_symm.{w} (𝟙 _) f | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 566,
"column": 2
} | {
"line": 566,
"column": 13
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\ninst✝ : HasLimitsOfSize... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 566,
"column": 52
} | {
"line": 566,
"column": 63
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\ninst✝ : HasLimitsOfSize... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Presheaf | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 49
} | [
{
"pp": "case w.h.a\nC : Type u\ninst✝² : Category.{v, u} C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : HasCoproducts A\nι : Type w\nS : ι → A\nhS : (ObjectProperty.ofObj S).IsSeparating\nF G : Cᵒᵖ ⥤ A\nf g : F ⟶ G\nh :\n ∀ (G_1 : Cᵒᵖ ⥤ A),\n ObjectProperty.ofObj\n (fun x ↦\n match x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Presheaf | {
"line": 89,
"column": 10
} | {
"line": 89,
"column": 21
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nA : Type u'\ninst✝⁴ : Category.{v', u'} A\ninst✝³ : HasCoproducts A\ninst✝² : HasSeparator A\ninst✝¹ : HasZeroMorphisms A\ninst✝ : HasCoproducts A\n⊢ (ObjectProperty.ofObj fun x ↦ separator A).IsSeparating",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Sheaf | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 64
} | [
{
"pp": "case a\nC : Type u\ninst✝³ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : HasCoproducts A\ninst✝ : HasWeakSheafify J A\nι : Type w\nS : ι → A\nhS : (ObjectProperty.ofObj S).IsSeparating\nF G : Sheaf J A\nf g : F ⟶ G\nhfg :\n ∀ (G_1 : Sheaf J A),\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Sheaf | {
"line": 66,
"column": 10
} | {
"line": 66,
"column": 21
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nJ : GrothendieckTopology C\nA : Type u'\ninst✝⁵ : Category.{v', u'} A\ninst✝⁴ : HasCoproducts A\ninst✝³ : HasWeakSheafify J A\ninst✝² : HasSeparator A\ninst✝¹ : Preadditive A\ninst✝ : HasCoproducts A\n⊢ (ObjectProperty.ofObj fun x ↦ separator A).IsSeparating",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Point.Basic | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 15
} | [
{
"pp": "C : Type u\ninst✝⁷ : Category.{v, u} C\nJ : GrothendieckTopology C\nΦ : J.Point\nA : Type u'\ninst✝⁶ : Category.{v', u'} A\ninst✝⁵ : HasColimitsOfSize.{w, w, v', u'} A\nFC : A → A → Type u_1\nCC : A → Type w'\ninst✝⁴ : (X Y : A) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝³ : ConcreteCategory A FC\nP Q : Cᵒ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 616,
"column": 60
} | {
"line": 616,
"column": 71
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\ninst✝ : HasLimitsOfSize... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 52,
"column": 4
} | {
"line": 52,
"column": 20
} | [
{
"pp": "case hx\nσ : Type u_1\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nd e : ℕ\nf : A\nhf : f ∈ 𝒜 d\ng : A\nhg : g ∈ 𝒜 e\nx : A\nhx : x = f * g\nhd : 0 < d\nn : ℕ\na : A\nha : a ∈ 𝒜 n\nj : ℕ\nhb' : (fun x_1 ↦ x ^ x_1) j ∈ 𝒜 ... | · exact hx ▸ hfg | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Hypercover.ZeroFamily | {
"line": 94,
"column": 8
} | {
"line": 94,
"column": 63
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nP : PreZeroHypercoverFamily C\nh : ∀ ⦃X Y : C⦄ (f : X ⟶ Y) [IsIso f], P.property (PreZeroHypercover.singleton f)\nS T : C\nf : S ⟶ T\nhf : IsIso f\n⊢ Presieve.singleton f ∈ P.precoverage.coverings T",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Pr... | ← PreZeroHypercover.presieve₀_singleton.{_, _, max u v} | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Sites.QuasiCompact | {
"line": 183,
"column": 67
} | {
"line": 188,
"column": 16
} | [
{
"pp": "P : MorphismProperty Scheme\nS : Scheme\nR : Presieve S\n⊢ R ∈ (propQCPrecoverage P).coverings S ↔ ∃ 𝒰, QuasiCompactCover 𝒰.toPreZeroHypercover ∧ R = 𝒰.presieve₀",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.QuasiCompactCover",
"AlgebraicGeometry.Scheme",
"CategoryThe... | by
rw [Precoverage.mem_iff_exists_zeroHypercover]
refine ⟨fun ⟨𝒰, h⟩ ↦ ⟨𝒰.weaken propQCPrecoverage_le_precoverage, ?_, h⟩,
fun ⟨𝒰, _, h⟩ ↦ ⟨⟨𝒰.1, ⟨by simpa, 𝒰.mem₀⟩⟩, h⟩⟩
rw [← Scheme.presieve₀_mem_qcPrecoverage_iff]
exact 𝒰.mem₀.1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 140,
"column": 77
} | {
"line": 140,
"column": 88
} | [
{
"pp": "σ : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : SetLike σ A\ninst✝² : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) A\nx : Finset A\nhx : Algebra.adjoin ↥(𝒜 0) ↑x = ⊤\nd : (i : A) → i ∈ x → ℕ\nhd : ∀ (i : A) (a : i ∈ x), d i a ≠ 0\nhxd : ∀ (i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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