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.CategoryTheory.Limits.Constructions.ZeroObjects | {
"line": 40,
"column": 24
} | {
"line": 40,
"column": 35
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX : C\ns : BinaryFan 0 X\nm : s.pt ⟶ X\nx✝ : m ≫ 0 = s.fst\nh₂ : m ≫ 𝟙 X = s.snd\n⊢ m = (fun s ↦ s.snd) s",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Constructions.ZeroObjects | {
"line": 66,
"column": 24
} | {
"line": 66,
"column": 35
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX : C\ns : BinaryFan X 0\nm : s.pt ⟶ X\nh₁ : m ≫ 𝟙 X = s.fst\nx✝ : m ≫ 0 = s.snd\n⊢ m = (fun s ↦ s.fst) s",
"usedConstants": [
"CategoryTheory.Limits.BinaryFan.fst",
"CategoryTheory.Cate... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Constructions.ZeroObjects | {
"line": 92,
"column": 24
} | {
"line": 92,
"column": 35
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX : C\ns : BinaryCofan 0 X\nm : X ⟶ s.pt\nx✝ : 0 ≫ m = s.inl\nh₂ : 𝟙 X ≫ m = s.inr\n⊢ m = (fun s ↦ s.inr) s",
"usedConstants": [
"CategoryTheory.Limits.BinaryCofan.inr",
"CategoryTheory.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Constructions.ZeroObjects | {
"line": 118,
"column": 24
} | {
"line": 118,
"column": 35
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX : C\ns : BinaryCofan X 0\nm : X ⟶ s.pt\nh₁ : 𝟙 X ≫ m = s.inl\nx✝ : 0 ≫ m = s.inr\n⊢ m = (fun s ↦ s.inl) s",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 13
} | [
{
"pp": "case h\nX : Scheme\n𝒰 : X.OpenCover\nx y : (gluedCover 𝒰).J\n⊢ pullback.fst (𝒰.f x) (𝒰.f y) ≫ 𝒰.f ((MultispanShape.prod (gluedCover 𝒰).J).fst (x, y)) =\n ((pullbackSymmetry (𝒰.f x) (𝒰.f y)).hom ≫ pullback.fst (𝒰.f y) (𝒰.f x)) ≫\n 𝒰.f ((MultispanShape.prod (gluedCover 𝒰).J).snd (x, y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 163,
"column": 7
} | {
"line": 163,
"column": 39
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens ↑↑(D.V (i, j))\nthis :\n ∀ (U : (Opens ↑↑(D.U i))ᵒᵖ),\n (D.f i j).c.app U ≫ (pullback.fst (D.f i j) (D.f i k)).c.app (op ((Opens.map (D.f i j).base).obj (unop U))) =\n ((D.f i k).c.app U ≫ (pullback.snd (D.f i j) (D.... | (π₁ i, j, k).c.naturality_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 446,
"column": 2
} | {
"line": 446,
"column": 45
} | [
{
"pp": "case h\nX : Scheme\n𝒰✝ : X.OpenCover\n𝒰 : X.OpenCover\nY : Scheme\nf : (x : 𝒰.I₀) → 𝒰.X x ⟶ Y\nhf : ∀ (x y : 𝒰.I₀), pullback.fst (𝒰.f x) (𝒰.f y) ≫ f x = pullback.snd (𝒰.f x) (𝒰.f y) ≫ f y\ni j : (ulift 𝒰).gluedCover.J\n⊢ pullback.fst ((ulift 𝒰).f i) ((ulift 𝒰).f j) ≫ f (idx 𝒰 i) =\n ((u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 478,
"column": 29
} | {
"line": 478,
"column": 40
} | [
{
"pp": "X Y : Scheme\nf g : X ⟶ Y\nU : ↥X → X.Opens\nhxU : ∀ (x : ↥X), x ∈ U x\nhU : ∀ (x : ↥X), (U x).ι ≫ f = (U x).ι ≫ g\nx : ↥X\n⊢ x ∈ Set.range ⇑({ I₀ := ↥X, X := fun i ↦ ↑(U i), f := fun i ↦ (U i).ι }.f x)",
"usedConstants": [
"CategoryTheory.PreZeroHypercover.mk",
"Eq.mpr",
"SetLike... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 544,
"column": 10
} | {
"line": 544,
"column": 32
} | [
{
"pp": "J : 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)\ny₁ : ↥(F.obj k₁.fst)\nhy₁ : (F.map k₁.snd.1) y₁ = ↑((pullbac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 561,
"column": 4
} | {
"line": 561,
"column": 15
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 584,
"column": 6
} | {
"line": 584,
"column": 17
} | [
{
"pp": "case e_a.e_a.e_self.e_self.e_self\nJ : Type w\ninst✝³ : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝² : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝¹ : (F ⋙ forget).IsLocallyDirected\ninst✝ : Quiver.IsThin J\ni j : J\nk₁ k₂ : (k : J) × (k ⟶ i) × (k ⟶ j)\nU : (F.obj i).Opens\nh₁ : Hom.opensRang... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 86,
"column": 34
} | {
"line": 86,
"column": 91
} | [
{
"pp": "case h₀.h₀\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g\ni : 𝒰.I₀\n⊢ ((t 𝒰 f g i i ≫ pullback.fst (pullback.fst (𝒰.f i ≫ f) g ≫ 𝒰.f i) (𝒰.f i)) ≫ pullback.fst (𝒰.f i ≫ f) g) ≫ 𝒰.f i =\n (pullback.fst (pullback.fst (𝒰.f i ≫ f) g ≫ ... | simp only [pullback.condition, Category.assoc, t_fst_fst] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 592,
"column": 2
} | {
"line": 595,
"column": 39
} | [
{
"pp": "case H\nJ : Type w\ninst✝³ : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝² : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝¹ : (F ⋙ forget).IsLocallyDirected\ninst✝ : Quiver.IsThin J\ni j : J\nk₁ k₂ : (k : J) × (k ⟶ i) × (k ⟶ j)\nU : (F.obj i).Opens\nh₁ : Hom.opensRange (F.map k₁.snd.1) ≤ U\nh₂ ... | have : IsOpenImmersion α := by
have : IsOpenImmersion (α ≫ pullback.fst _ _) := by
simp only [pullback.lift_fst, α]; infer_instance
exact .of_comp _ (pullback.fst _ _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 305,
"column": 2
} | {
"line": 305,
"column": 35
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : GlueData C\ninst✝ : HasLimits C\ni j k : D.J\nU : Opens ↑↑(D.U i)\nX' : C\nf' :\n ((TopCat.Presheaf.pushforward C (D.f j k).base).obj (D.V (j, k)).presheaf).obj\n (op ((Opens.map (D.ι j).base).obj (⋯.functor.obj U))) ⟶\n X'\n⊢ D.opensImagePreimageMap... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 274,
"column": 8
} | {
"line": 274,
"column": 45
} | [
{
"pp": "ι : Type u\nf : ι → Scheme\nX : Scheme\nα : (i : ι) → f i ⟶ X\ninst✝ : ∀ (i : ι), IsOpenImmersion (α i)\nhα : _root_.Pairwise (Disjoint on fun x ↦ Set.range ⇑(α x))\nix : ι\nx : ↥(f ix)\niy : ι\ny : ↥(f iy)\ne : (⇑(Sigma.desc α) ∘ ⇑(sigmaMk f)) ⟨ix, x⟩ = (⇑(Sigma.desc α) ∘ ⇑(sigmaMk f)) ⟨iy, y⟩\n⊢ (α i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 281,
"column": 6
} | {
"line": 281,
"column": 43
} | [
{
"pp": "case left.refine_3\nι : Type u\nf : ι → Scheme\nX : Scheme\nα : (i : ι) → f i ⟶ X\ninst✝ : ∀ (i : ι), IsOpenImmersion (α i)\nhα : _root_.Pairwise (Disjoint on fun x ↦ Set.range ⇑(α x))\ni : ι\n⊢ IsOpenMap fun a ↦ (⇑(Sigma.desc α) ∘ ⇑(sigmaMk f)) ⟨i, a⟩",
"usedConstants": [
"AlgebraicGeometry.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 15
} | [
{
"pp": "case refine_2\nX 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\ni j : 𝒰.I₀\n⊢ ((Precoverage.ZeroHypercover.pullback₁ s.fst 𝒰).f i ≫ s.fst) ≫ 𝟙 X =\n ((pullbackSymmetry s.fst (𝒰.f i)).hom ≫ pullback.map (𝒰.f i) s.fst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 316,
"column": 4
} | {
"line": 316,
"column": 67
} | [
{
"pp": "σ : Type v\ninst✝¹ : Small.{u, v} σ\nX : σ → Scheme\nS : Scheme\nf : (i : σ) → X i ⟶ S\ninst✝ : ∀ (i : σ), IsOpenImmersion (f i)\nhcov : ⨆ i, Scheme.Hom.opensRange (f i) = ⊤\nhdisj : _root_.Pairwise (Disjoint on fun x ↦ Scheme.Hom.opensRange (f x))\ni j : σ\nhij : i ≠ j\n⊢ (Disjoint on fun x ↦ Set.rang... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 322,
"column": 27
} | {
"line": 322,
"column": 89
} | [
{
"pp": "σ : Type v\ninst✝¹ : Small.{u, v} σ\nX : σ → Scheme\nS : Scheme\nf : (i : σ) → X i ⟶ S\ninst✝ : ∀ (i : σ), IsOpenImmersion (f i)\nhcov : ⨆ i, Scheme.Hom.opensRange (f i) = ⊤\nhdisj : _root_.Pairwise (Disjoint on fun x ↦ Scheme.Hom.opensRange (f x))\nthis✝ : IsOpenImmersion (Sigma.desc f)\nx : ↥S\nhx : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 381,
"column": 12
} | {
"line": 382,
"column": 50
} | [
{
"pp": "case op.op.snd.e_a\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\nthis :\n D.t' j k i ≫ pullback.fst (D.f k i) (D.f k j) ≫ D.t k i ≫ D.f i k =\n (pullbackSymmetry (D.f j k) (D.f j i)).hom ≫ pullback.fst (D.f j i) (D.f j k) ≫ D.... | simp_rw [Category.assoc, Functor.op_obj, comp_base, Opens.map_comp_obj,
TopCat.Presheaf.pushforward_obj_map] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 292,
"column": 4
} | {
"line": 292,
"column": 94
} | [
{
"pp": "case hf.e_a.h₀.e_a\nX 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\ni j : (Precoverage.ZeroHypercover.pullback₁ s.fst 𝒰).I₀\n⊢ pullback.snd s.fst (𝒰.f j) =\n (pullbackSymmetry s.fst (𝒰.f j)).hom ≫\n pullback.map... | rw [← Iso.inv_comp_eq, pullbackSymmetry_inv_comp_snd, pullback.lift_fst, Category.comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Limits | {
"line": 427,
"column": 4
} | {
"line": 427,
"column": 88
} | [
{
"pp": "case convert_2\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\ninst✝¹ : IsOpenImmersion f\ninst✝ : IsOpenImmersion g\nhf : IsCompl (Scheme.Hom.opensRange f) (Scheme.Hom.opensRange g)\nc' : Cofan fun j ↦ WalkingPair.casesOn j X Y := Cofan.mk S fun j ↦ WalkingPair.casesOn j f g\ni : BinaryCofan.mk f g ≅ c' := Cof... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 430,
"column": 23
} | {
"line": 430,
"column": 39
} | [
{
"pp": "X Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\ninst✝¹ : IsOpenImmersion f\ninst✝ : IsOpenImmersion g\nhf : IsCompl (Scheme.Hom.opensRange f) (Scheme.Hom.opensRange g)\nc' : Cofan fun j ↦ WalkingPair.casesOn j X Y := Cofan.mk S fun j ↦ WalkingPair.casesOn j f g\ni✝ : BinaryCofan.mk f g ≅ c' := Cofan.ext (Iso.ref... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 431,
"column": 23
} | {
"line": 431,
"column": 39
} | [
{
"pp": "X Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\ninst✝¹ : IsOpenImmersion f\ninst✝ : IsOpenImmersion g\nhf : IsCompl (Scheme.Hom.opensRange f) (Scheme.Hom.opensRange g)\nc' : Cofan fun j ↦ WalkingPair.casesOn j X Y := Cofan.mk S fun j ↦ WalkingPair.casesOn j f g\ni✝ : BinaryCofan.mk f g ≅ c' := Cofan.ext (Iso.ref... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 302,
"column": 2
} | {
"line": 302,
"column": 87
} | [
{
"pp": "case h\nX 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\nb : (MultispanShape.prod (Cover.gluedCover (Precoverage.ZeroHypercover.pullback₁ s.fst 𝒰)).J).R\n⊢ Multicoequalizer.π (Cover.gluedCover (Precoverage.ZeroHypercover.p... | simp_rw [Cover.fromGlued, Multicoequalizer.π_desc_assoc, gluedLift, ← Category.assoc] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 311,
"column": 2
} | {
"line": 311,
"column": 87
} | [
{
"pp": "case h\nX 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\nb : (MultispanShape.prod (Cover.gluedCover (Precoverage.ZeroHypercover.pullback₁ s.fst 𝒰)).J).R\n⊢ Multicoequalizer.π (Cover.gluedCover (Precoverage.ZeroHypercover.p... | simp_rw [Cover.fromGlued, Multicoequalizer.π_desc_assoc, gluedLift, ← Category.assoc] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.AlgebraicGeometry.Limits | {
"line": 467,
"column": 6
} | {
"line": 467,
"column": 82
} | [
{
"pp": "case refine_2\nι : Type u\nf : ι → Scheme\nσ : Type v\ng : σ → Scheme\nX✝ Y✝ : Scheme\nX Y : Scheme\nc : BinaryCofan X Y\nhc : IsColimit c\n⊢ {Z : Scheme} → (f : Z ⟶ X ⨿ Y) → IsColimit (BinaryCofan.mk (pullback.fst f coprod.inl) (pullback.fst f coprod.inr))",
"usedConstants": [
"CategoryTheor... | refine fun {Z} f ↦ (nonempty_isColimit_binaryCofanMk_of_isCompl _ _ ?_).some | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.Limits | {
"line": 466,
"column": 4
} | {
"line": 470,
"column": 88
} | [
{
"pp": "case refine_2\nι : Type u\nf : ι → Scheme\nσ : Type v\ng : σ → Scheme\nX✝ Y✝ : Scheme\nX Y : Scheme\nc : BinaryCofan X Y\nhc : IsColimit c\n⊢ {Z : Scheme} →\n (f : Z ⟶ (BinaryCofan.mk coprod.inl coprod.inr).pt) →\n IsColimit\n (BinaryCofan.mk\n (PullbackCone.mk (pullback.fst f (... | · dsimp
refine fun {Z} f ↦ (nonempty_isColimit_binaryCofanMk_of_isCompl _ _ ?_).some
rw [Scheme.Hom.opensRange_pullbackFst, Scheme.Hom.opensRange_pullbackFst]
convert! (isCompl_range_inl_inr X Y).map (CompleteLatticeHom.setPreimage f)
simp [isCompl_iff, disjoint_iff, codisjoint_iff, ← Topologica... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 776,
"column": 4
} | {
"line": 776,
"column": 43
} | [
{
"pp": "J : Type w\ninst✝⁴ : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝³ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝² : (F ⋙ forget).IsLocallyDirected\ninst✝¹ : Quiver.IsThin J\ninst✝ : Small.{u, w} J\ns : Cocone (F ⋙ forgetToLocallyRingedSpace)\nj : J\n⊢ failed to pretty print expression (use 's... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 619,
"column": 17
} | {
"line": 619,
"column": 28
} | [
{
"pp": "case H.mpr\nι : Type u\nf : ι → Scheme\nσ : Type v\ng : σ → Scheme\nX Y : Scheme\nR✝ S : Type u\ninst✝² : CommRing R✝\ninst✝¹ : CommRing S\ni : ι\nR : ι → Type (max u_1 u)\ninst✝ : (i : ι) → CommRing (R i)\nthis : Algebra ((i : ι) → R i) (R i) := (Pi.evalRingHom R i).toAlgebra\nx y : (a : ι) → R a\ne :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 377,
"column": 6
} | {
"line": 377,
"column": 17
} | [
{
"pp": "case hom_inv_id.h₀\nX 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\ni : 𝒰.I₀\n⊢ (pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯ ≫\n pullback.lift ((gluing 𝒰 f g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 168,
"column": 84
} | {
"line": 168,
"column": 95
} | [
{
"pp": "P : MorphismProperty Scheme\ninst✝¹ : IsZariskiLocalAtTarget P\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : P.RespectsRight IsOpenImmersion\nι : Type u_1\nU : ι → Y.Opens\nH : Set.range ⇑f ⊆ ↑(⨆ i, U i)\nhf : ∀ (i : ι), P (f ∣_ U i)\n⊢ Set.range ⇑f ⊆ Set.range ⇑(⨆ i, U i).ι",
"usedConstants": [
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 169,
"column": 52
} | {
"line": 169,
"column": 63
} | [
{
"pp": "P : MorphismProperty Scheme\ninst✝¹ : IsZariskiLocalAtTarget P\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : P.RespectsRight IsOpenImmersion\nι : Type u_1\nU : ι → Y.Opens\nH : Set.range ⇑f ⊆ ↑(⨆ i, U i)\nhf : ∀ (i : ι), P (f ∣_ U i)\ng : X ⟶ ↑(⨆ i, U i) := IsOpenImmersion.lift (⨆ i, U i).ι f ⋯\n⊢ Set.range ⇑f ⊆ S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 707,
"column": 4
} | {
"line": 707,
"column": 15
} | [
{
"pp": "case left\nU X Y : Scheme\nf : U ⟶ X\ng : U ⟶ Y\ninst✝¹ : IsOpenImmersion f\ninst✝ : IsOpenImmersion g\n⊢ Function.Injective\n ⇑(ConcreteCategory.hom ((span f g ⋙ Scheme.forget).map (WidePushoutShape.Hom.init WalkingPair.left)))",
"usedConstants": [
"CategoryTheory.Limits.WalkingSpan",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 708,
"column": 4
} | {
"line": 708,
"column": 15
} | [
{
"pp": "case right\nU X Y : Scheme\nf : U ⟶ X\ng : U ⟶ Y\ninst✝¹ : IsOpenImmersion f\ninst✝ : IsOpenImmersion g\n⊢ Function.Injective\n ⇑(ConcreteCategory.hom ((span f g ⋙ Scheme.forget).map (WidePushoutShape.Hom.init WalkingPair.right)))",
"usedConstants": [
"CategoryTheory.Limits.WalkingSpan",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 13
} | [
{
"pp": "P : MorphismProperty Scheme\ninst✝ : IsZariskiLocalAtTarget P\nX Y : Scheme\nf : X ⟶ Y\nU : ↥Y → Y.Opens\nhxU : ∀ (x : ↥Y), x ∈ U x\nhU : ∀ (x : ↥Y), P (f ∣_ U x)\nx : ↥Y\nx✝ : x ∈ ⊤\n⊢ x ∈ iSup U",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrier... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 48,
"column": 42
} | {
"line": 48,
"column": 53
} | [
{
"pp": "X Y Z : Scheme\nf✝ : X ⟶ Y\ng : Y ⟶ Z\nα✝ β✝ : Type u_1\ninst✝¹ : TopologicalSpace α✝\ninst✝ : TopologicalSpace β✝\nf : α✝ → β✝\nι : Type u_1\nU : ι → Opens β✝\nH : IsOpenCover U\nx✝ : Continuous[inst✝¹, inst✝] f\nhf : ∀ (i : ι), Function.Injective ((U i).carrier.restrictPreimage f)\nx₁ x₂ : α✝\ne : f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 99,
"column": 39
} | {
"line": 99,
"column": 50
} | [
{
"pp": "X Y Z : Scheme\nf✝ : X ⟶ Y\ng : Y ⟶ Z\nthis : (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ Function.Surjective).RespectsIso\nα β : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nι : Type u_1\nU : ι → Opens β\nH : IsOpenCover U\nx✝ : Continuous[inst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 33
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : Surjective f\n⊢ Set.range ⇑f = Set.univ",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.PresheafedSpace.carrier",
"CategoryTheory.ConcreteCategory.hom",
"CommRingCat",
"TopCat.instCategory",
"ContinuousMap",
"Set.univ",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 119,
"column": 2
} | {
"line": 123,
"column": 51
} | [
{
"pp": "X : Scheme\nι : Type v\ninst✝ : Small.{u, v} ι\nY : ι → Scheme\nf : (i : ι) → Y i ⟶ X\nH : ⋃ i, Set.range ⇑(f i) = Set.univ\n⊢ Surjective (Limits.Sigma.desc f)",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"_private.Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap.0.... | refine ⟨fun x ↦ ?_⟩
simp_rw [Set.eq_univ_iff_forall, Set.mem_iUnion] at H
obtain ⟨i, x, rfl⟩ := H x
use Limits.Sigma.ι Y i x
rw [← Scheme.Hom.comp_apply, Limits.Sigma.ι_desc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 119,
"column": 2
} | {
"line": 123,
"column": 51
} | [
{
"pp": "X : Scheme\nι : Type v\ninst✝ : Small.{u, v} ι\nY : ι → Scheme\nf : (i : ι) → Y i ⟶ X\nH : ⋃ i, Set.range ⇑(f i) = Set.univ\n⊢ Surjective (Limits.Sigma.desc f)",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"_private.Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap.0.... | refine ⟨fun x ↦ ?_⟩
simp_rw [Set.eq_univ_iff_forall, Set.mem_iUnion] at H
obtain ⟨i, x, rfl⟩ := H x
use Limits.Sigma.ι Y i x
rw [← Scheme.Hom.comp_apply, Limits.Sigma.ι_desc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 30
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : Dense ↑U\n⊢ DenseRange ⇑U.ι",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"congrArg",
"CategoryTheory.ConcreteCategory.hom",
"CommRing... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 269,
"column": 48
} | {
"line": 269,
"column": 59
} | [
{
"pp": "X : Scheme\nU V : X.Opens\nhU : Dense ↑U\nhU' : U ≤ V\n⊢ IsDominant (X.homOfLE hU' ≫ V.ι)",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme.homOfLE",
"AlgebraicGeometry.Scheme",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"Alg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 464,
"column": 2
} | {
"line": 464,
"column": 34
} | [
{
"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✝\nX Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\ni : Z.affineCover.I₀\n⊢ HasPullback ((Precoverage.ZeroHypercover.pullback₁ f Z.affineCover).f i ≫ f) g",
"usedConstant... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 636,
"column": 54
} | {
"line": 636,
"column": 65
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝ : HasAffineProperty P Q\nhP' : Q.IsStableUnderBaseChange\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nx✝ : HasPullback f g\nH : P g\ni : (Precoverage.ZeroHypercover.pullback₁ f S.affineCover).I₀\n⊢ ((Precoverage.ZeroHypercover.pullback₁ f S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Away.Lemmas | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 27
} | [
{
"pp": "case h\nR : Type u_1\ninst✝³ : CommRing R\ns : Set R\nhsone : Ideal.span s = ⊤\nRₜ : ↑s → Type u_2\ninst✝² : (t : ↑s) → CommRing (Rₜ t)\ninst✝¹ : (t : ↑s) → Algebra R (Rₜ t)\ninst✝ : ∀ (t : ↑s), Away (↑t) (Rₜ t)\np : (t : ↑s) → Set (Rₜ t)\nhtone : ∀ (r : ↑s), Ideal.span (p r) = ⊤\na : R\nha : a ∈ s\nth... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 216,
"column": 6
} | {
"line": 216,
"column": 17
} | [
{
"pp": "case of_sSup_eq_top.x\nP : MorphismProperty Scheme\nhP₂ : ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), IsOpenCover U → (∀ (i : ι), P (f ∣_ U i)) → P f\nX Y : Scheme\nf : X ⟶ Y\nι : Type u\nU : ι → Y.Opens\nH : ∀ (i : ι), P.universally (f ∣_ U i)\nX' Y' : Scheme\ni₁ : X' ⟶ X\ni₂ : Y' ⟶ Y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 598,
"column": 58
} | {
"line": 598,
"column": 69
} | [
{
"pp": "X Y Z W : Scheme\nfWX : W ⟶ X\nfWY : W ⟶ Y\nfXZ : X ⟶ Z\nfYZ : Y ⟶ Z\n𝒰 : X.OpenCover\nH :\n ∀ (i : 𝒰.toPreZeroHypercover.1),\n IsPullback (Cover.pullbackHom 𝒰 fWX i) ((Precoverage.ZeroHypercover.pullback₁ fWX 𝒰).f i ≫ fWY) (𝒰.f i ≫ fXZ) fYZ\ni : (Precoverage.ZeroHypercover.pullback₁ fWX 𝒰).I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 603,
"column": 6
} | {
"line": 603,
"column": 72
} | [
{
"pp": "X Y Z W : Scheme\nfWX : W ⟶ X\nfWY : W ⟶ Y\nfXZ : X ⟶ Z\nfYZ : Y ⟶ Z\n𝒰 : X.OpenCover\nH :\n ∀ (i : 𝒰.toPreZeroHypercover.1),\n IsPullback (Cover.pullbackHom 𝒰 fWX i) ((Precoverage.ZeroHypercover.pullback₁ fWX 𝒰).f i ≫ fWY) (𝒰.f i ≫ fXZ) fYZ\nh : fWX ≫ fXZ = fWY ≫ fYZ\ni : (openCoverOfLeft 𝒰 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 45
} | [
{
"pp": "P : {α β : Type u} → [TopologicalSpace α] → [TopologicalSpace β] → (α → β) → Prop\nhP : ∀ {α β : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α ≃ₜ β), P ⇑f\nX Y : Scheme\ne : X ⟶ Y\nhe : IsIso e\n⊢ topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) e",
... | exact hP (TopCat.homeoOfIso (asIso e.base)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 606,
"column": 49
} | {
"line": 606,
"column": 68
} | [
{
"pp": "X Y Z W : Scheme\nfWX : W ⟶ X\nfWY : W ⟶ Y\nfXZ : X ⟶ Z\nfYZ : Y ⟶ Z\n𝒰 : X.OpenCover\nH :\n ∀ (i : 𝒰.toPreZeroHypercover.1),\n IsPullback (Cover.pullbackHom 𝒰 fWX i) ((Precoverage.ZeroHypercover.pullback₁ fWX 𝒰).f i ≫ fWY) (𝒰.f i ≫ fXZ) fYZ\nh : fWX ≫ fXZ = fWY ≫ fYZ\ni : (openCoverOfLeft 𝒰 ... | by ext <;> simp [f] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 607,
"column": 6
} | {
"line": 607,
"column": 17
} | [
{
"pp": "X Y Z W : Scheme\nfWX : W ⟶ X\nfWY : W ⟶ Y\nfXZ : X ⟶ Z\nfYZ : Y ⟶ Z\n𝒰 : X.OpenCover\nH :\n ∀ (i : 𝒰.toPreZeroHypercover.1),\n IsPullback (Cover.pullbackHom 𝒰 fWX i) ((Precoverage.ZeroHypercover.pullback₁ fWX 𝒰).f i ≫ fWY) (𝒰.f i ≫ fXZ) fYZ\nh : fWX ≫ fXZ = fWY ≫ fYZ\ni : (openCoverOfLeft 𝒰 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 311,
"column": 2
} | {
"line": 311,
"column": 16
} | [
{
"pp": "P : {α β : Type u} → [TopologicalSpace α] → [TopologicalSpace β] → (α → β) → Prop\ninst✝ : (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P).RespectsIso\nhP :\n ∀ {α β : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α → β) {ι : Type u} (U : ι → Opens β),\... | introv hf hs H | Mathlib.Tactic.evalIntrov | Mathlib.Tactic.introv |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 396,
"column": 4
} | {
"line": 396,
"column": 42
} | [
{
"pp": "case of_sSup_eq_top\nP : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhP : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P\nthis : (stalkwise fun {R S} [CommRing R] [CommRing S] ↦ P).RespectsIso := stalkwise_respectsIso hP\nX Y : Scheme\nf : X ⟶ Y\nι : Ty... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 39
} | [
{
"pp": "case h.e'_3.h₂\nX✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : IsIso f\nU : Set ↥Y\na✝ : IsOpen U\nhU' : IsCompact U\n⊢ Function.RightInverse (⇑f) (TopCat.Hom.hom (inv f.base)).toFun",
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.carrier",
"CommRingCat",
"... | exact IsIso.hom_inv_id_apply f.base | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 39
} | [
{
"pp": "case h.e'_3.h₂\nX✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : IsIso f\nU : Set ↥Y\na✝ : IsOpen U\nhU' : IsCompact U\n⊢ Function.RightInverse (⇑f) (TopCat.Hom.hom (inv f.base)).toFun",
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.carrier",
"CommRingCat",
"... | exact IsIso.hom_inv_id_apply f.base | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 39
} | [
{
"pp": "case h.e'_3.h₂\nX✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : IsIso f\nU : Set ↥Y\na✝ : IsOpen U\nhU' : IsCompact U\n⊢ Function.RightInverse (⇑f) (TopCat.Hom.hom (inv f.base)).toFun",
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.carrier",
"CommRingCat",
"... | exact IsIso.hom_inv_id_apply f.base | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 165,
"column": 6
} | {
"line": 165,
"column": 17
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nh₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P\nh₂ : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P\nh₃ : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ P\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RingHom.Locally | {
"line": 297,
"column": 6
} | {
"line": 297,
"column": 30
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhPi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P\nhPb : IsStableUnderBaseChange fun {R S} [CommRing R] [CommRing S] ↦ P\nR S T : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : ... | locally_iff_span_eq_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 63
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nX Y : Scheme\nf : X ⟶ Y\nhPa : StableUnderCompositionWithLocalizationAwayTarget fun {R S} [CommRing R] [CommRing S] ↦ P\nhPl : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P\nx : ↥X\nU₁ U₂ : ↑Y.affin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 240,
"column": 31
} | {
"line": 240,
"column": 42
} | [
{
"pp": "X : Scheme\nP : X.Opens → Prop\nh₁ : P ⊥\nh₂ : ∀ (S : X.Opens), IsCompact S.carrier → ∀ (U : ↑X.affineOpens), P S → P (S ⊔ ↑U)\ns : Set ↑X.affineOpens\nhs : s.Finite\nhS : IsCompact ↑(⨆ i ∈ s, ↑i)\n⊢ P (⨆ i ∈ ∅, ↑i)",
"usedConstants": [
"Eq.mpr",
"False",
"AlgebraicGeometry.Sheafe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 250,
"column": 15
} | {
"line": 250,
"column": 39
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nx f : ↑Γ(X, U)\nH : (x |_ X.basicOpen f) ⋯ = (CommRingCat.Hom.hom (X.presheaf.map (homOfLE ⋯).op)) 0\nn : ℕ\ne : f ^ n * x = f ^ n * 0\n⊢ f ^ n * x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 260,
"column": 10
} | {
"line": 260,
"column": 21
} | [
{
"pp": "case h\nX : Scheme\nU : X.Opens\nhU : IsCompact U.carrier\nx f : ↑Γ(X, U)\nH : (x |_ X.basicOpen f) ⋯ = 0\ns : Set ↑X.affineOpens\nhs : s.Finite\ne : U.carrier = ⋃ i ∈ s, ↑↑i\n⊢ ↑U = ↑(⨆ i, ↑↑i)",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalProperties.Reduced | {
"line": 32,
"column": 4
} | {
"line": 32,
"column": 15
} | [
{
"pp": "case eq_zero.zero\nR : Type u_1\nhR : CommRing R\nM : Submonoid R\nS : Type u_1\nhS : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\na✝ : IsReduced R\nx : S\ne : x ^ 0 = 0\n⊢ x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 352,
"column": 2
} | {
"line": 352,
"column": 17
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝² : HasRingHomProperty P Q\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsAffine X\ninst✝ : IsAffine Y\n⊢ (∀ (i : (Scheme.coverOfIsIso (𝟙 X)).I₀),\n Q (CommRingCat.Hom.hom (Scheme.Hom.appT... | simp +instances | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 367,
"column": 2
} | {
"line": 367,
"column": 39
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝¹ : HasRingHomProperty P Q\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : IsAffine Y\nι : Type u_1\nU : ι → ↑X.affineOpens\nhU : ⨆ i, ↑(U i) = ⊤\nH : ∀ (i : ι), Q (CommRingCat.Hom.hom (Scheme.Hom.a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 377,
"column": 2
} | {
"line": 377,
"column": 17
} | [
{
"pp": "case H\nP : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝ : HasRingHomProperty P Q\nX Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) [inst : IsAffine Y] (𝒰 : X.OpenCover),\n sourceAffineLocally (fun {R S} [Com... | intro X Y f _ 𝒰 | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Ideal.Height | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nI J : Ideal R\ninst✝¹ : I.IsPrime\ninst✝ : J.IsPrime\nh : I ≤ J\n⊢ I.primeHeight ≤ J.primeHeight",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nI J : Ideal R\ninst✝¹ : I.IsPrime\ninst✝ : J.IsPrime\nh : I < J\n⊢ I.primeHeight + 1 ≤ J.primeHeight",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nI J : Ideal R\ninst✝² : I.IsPrime\ninst✝¹ : J.IsPrime\nh : I < J\ninst✝ : J.FiniteHeight\n⊢ I.primeHeight < J.primeHeight",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"ENat",
"congr",
"LT.lt",
"instLTENat",
"_pri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 370,
"column": 4
} | {
"line": 370,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nS : Submonoid R\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : IsLocalization S A\nJ : Ideal A\ninst✝ : J.IsPrime\n⊢ Order.krullDim ↑(Set.Iic { asIdeal := J, isPrime := inst✝ }) =\n ↑(Order.height { asIdeal := comap (algebraMap R A) J, isPrime :... | Order.height_eq_krullDim_Iic | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 387,
"column": 2
} | {
"line": 387,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nS : Submonoid R\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : IsLocalization S A\nJ : Ideal A\ninst✝ : J.IsPrime\n⊢ (comap (algebraMap R A) J).primeHeight = J.primeHeight",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 484,
"column": 4
} | {
"line": 486,
"column": 61
} | [
{
"pp": "case a.refine_2\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\np : LTSeries (PrimeSpectrum R)\n⊢ (RelSeries.last p).asIdeal.height ≤ ⨆ I, ⨆ (_ : I ≠ ⊤), I.height",
"usedConstants": [
"Preorder.toLT",
"instCompleteLinearOrderENat",
"Semiring.toModule",
"le_rfl",
... | · apply le_trans (b := ⨆ (_ : (p.last).asIdeal ≠ ⊤), p.last.asIdeal.height)
· exact le_iSup_of_le p.last.isPrime.ne_top' le_rfl
· exact le_iSup (fun I => ⨆ _, I.height) p.last.asIdeal | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Ideal.Height | {
"line": 518,
"column": 4
} | {
"line": 519,
"column": 60
} | [
{
"pp": "case a.inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nI : Ideal R\nI_top : I ≠ ⊤\n⊢ ∃ i', ⨆ (_ : I ≠ ⊤), I.height ≤ ⨆ (_ : i'.IsMaximal), i'.height",
"usedConstants": [
"instCompleteLinearOrderENat",
"Semiring.toModule",
"CommSemiring.toSemiring",
"iSup",
... | · obtain ⟨M, hM, hIM⟩ := exists_le_maximal I I_top
exact ⟨M, iSup_mono' (fun hI ↦ ⟨hM, height_mono hIM⟩)⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 567,
"column": 12
} | {
"line": 568,
"column": 78
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝¹ : HasRingHomProperty P Q\nhQ : RingHom.StableUnderCompositionWithLocalizationAwaySource fun {R S} [CommRing R] [CommRing S] ↦ Q\nX Y : Scheme\ninst✝ : IsAffine Y\nU : Y.Opens\nf : X ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 13
} | [
{
"pp": "X : Scheme\nI : Type u_1\nU : I → X.Opens\nhU : IsOpenCover U\nhU₁ : ∀ (i : I), IsAffineOpen (U i)\nhU₂ : ∀ (i j : I), IsCompact (↑(U i) ∩ ↑(U j))\nthis✝ : AffineTargetMorphismProperty.IsLocal fun X x x_1 x_2 ↦ CompactSpace ↥X :=\n HasAffineProperty.isLocal_affineProperty @QuasiCompact\nthis : ∀ (i : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 13
} | [
{
"pp": "case isCompact_univ\nX Y Z : Scheme\nf✝ : X ⟶ Y\ninst✝¹ : CompactSpace ↥X\ninst✝ : QuasiSeparatedSpace ↥Y\nf g : X ⟶ Y\n⊢ IsCompact Set.univ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 258,
"column": 2
} | {
"line": 258,
"column": 26
} | [
{
"pp": "case h\nX : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nf : ↑Γ(X, U)\nx : ↑Γ(X, X.basicOpen f)\nthis :\n ∀ (z : ↑Γ(X, X.basicOpen f)),\n ∃ x, z * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑x.2 = (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) x.1\ny : ↑Γ(X, U)\nn : ℕ\nd :\n x * (algebraMap ↑Γ(X, U) ↑Γ(X... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 628,
"column": 44
} | {
"line": 628,
"column": 55
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nX Y : Scheme\nf : X ⟶ Y\nhQ : StableUnderCompositionWithLocalizationAwaySource fun {R S} [CommRing R] [CommRing S] ↦ Q\nhQi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q\ninst✝ : Ha... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Properties | {
"line": 342,
"column": 2
} | {
"line": 342,
"column": 13
} | [
{
"pp": "X : Scheme\ninst✝ : IsIntegral X\nU V : X.Opens\ni : U ⟶ V\nH : Nonempty ↥↑U\nx : ↑Γ(X, V)\n⊢ (↑U).Nonempty",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 340,
"column": 71
} | {
"line": 340,
"column": 82
} | [
{
"pp": "case h\nX : Scheme\nU✝ : X.Opens\nhU✝ : IsCompact U✝.carrier\nS : X.Opens\nhS : IsCompact S.carrier\nU : ↑X.affineOpens\nhU :\n IsQuasiSeparated S.carrier →\n ∀ (f : ↑Γ(X, S)) (x : ↑Γ(X, X.basicOpen f)),\n ∃ n y,\n (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) y =\n (C... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 701,
"column": 2
} | {
"line": 701,
"column": 12
} | [
{
"pp": "case a\nP : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝ : HasRingHomProperty P Q\nX Y : Scheme\nf : X ⟶ Y\nhQ : OfLocalizationPrime fun {R S} [CommRing R] [CommRing S] ↦ Q\nhQi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q\... | intro P hP | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.AlgebraicGeometry.Morphisms.OpenImmersion | {
"line": 64,
"column": 61
} | {
"line": 64,
"column": 72
} | [
{
"pp": "R S : CommRingCat\nf : R ⟶ S\nhf : Function.Surjective ⇑(CommRingCat.Hom.hom f)\ne : ↑R\nhe : IsIdempotentElem e\nhe' : RingHom.ker (CommRingCat.Hom.hom f) = Ideal.span {e}\nthis : Algebra ↑R ↑S := (CommRingCat.Hom.hom f).toAlgebra\n⊢ RingHom.ker (algebraMap ↑R ↑S) = Ideal.span {1 - (1 - e)}",
"use... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 15
} | [
{
"pp": "case h\nX : Scheme\nx : ↥X\n⊢ x ∈ ⊥ ↔ x ∈ ⋂ U, X.zeroLocus ↑(⊤ U)",
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"False",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.OpenImmersion | {
"line": 94,
"column": 8
} | {
"line": 94,
"column": 19
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nhf : Function.Injective ⇑f\nU : ↥X → Scheme\ni : (x : ↥X) → U x ⟶ X\nx✝ : ∀ (x : ↥X), IsOpenImmersion (i x)\nhxi : ∀ (x : ↥X), x ∈ Scheme.Hom.opensRange (i x)\nhi : ∀ (x : ↥X), IsOpenImmersion (i x ≫ f)\n⊢ { I₀ := ↥X, X := U, f := i }.presieve₀ ∈ Scheme.jointlySurjectivePrecove... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 399,
"column": 4
} | {
"line": 399,
"column": 88
} | [
{
"pp": "case toIsLocalizationMap.surj\nX : Scheme\nU : X.Opens\nhU : IsCompact U.carrier\nhU' : IsQuasiSeparated U.carrier\nf : ↑Γ(X, U)\nz : ↑Γ(X, X.basicOpen f)\nn : ℕ\ny : ↑Γ(X, U)\ne : (y |_ X.basicOpen f) ⋯ = (f |_ X.basicOpen f) ⋯ ^ n * z\n⊢ z * (algebraMap ↑Γ(X, U) ↑Γ(X, X.basicOpen f)) ↑(y, ⟨(fun x ↦ f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 407,
"column": 4
} | {
"line": 407,
"column": 28
} | [
{
"pp": "case toIsLocalizationMap.exists_of_eq\nX : Scheme\nU : X.Opens\nhU : IsCompact U.carrier\nhU' : IsQuasiSeparated U.carrier\nf x y z : ↑Γ(X, U)\nH : (CommRingCat.Hom.hom (X.presheaf.map (homOfLE ⋯).op)) z = 0\nn : ℕ\ne : f ^ n * z = 0\n⊢ ↑⟨(fun x ↦ f ^ x) n, ⋯⟩ * z = 0",
"usedConstants": [
"Ad... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 426,
"column": 4
} | {
"line": 426,
"column": 15
} | [
{
"pp": "X : Scheme\nU : Opens ↥X\nhU : IsCompact U.carrier\nhU' : IsQuasiSeparated U.carrier\nf s : ↑Γ(X, U)\nhf : (f |_ X.basicOpen s) ⋯ = 0\nh : ∃ n, s ^ n * f = s ^ n * 0\n⊢ ∃ n, s ^ n * f = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 13
} | [
{
"pp": "X : Scheme\nι : Type u_1\nI : ι → X.IdealSheafData\ninst✝ : Finite ι\n⊢ (⨅ i, I i).ideal = ⨅ i, (I i).ideal",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 13
} | [
{
"pp": "X Y : LocallyRingedSpace\nf : X ⟶ Y\nV : Opens ↑Y.toTopCat\nx : ↥((Opens.map f.base).obj V)\na : ↑(Y.presheaf.obj (op V))\n⊢ (ConcreteCategory.hom (residueFieldMap f ↑x))\n ((ConcreteCategory.hom (Y.evaluation ⟨(ConcreteCategory.hom f.base) ↑x, ⋯⟩)) a) =\n (ConcreteCategory.hom (X.evaluation x)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 304,
"column": 4
} | {
"line": 304,
"column": 85
} | [
{
"pp": "case zero\nX : Scheme\nI : X.IdealSheafData\nU V : ↑X.affineOpens\nx : ↥X\nhxV : x ∈ ↑↑V\nhxU : x ∈ ↑↑U\nhx : ∀ f ∈ I.ideal U, x ∉ X.basicOpen f\ns : ↑Γ(X, ↑V)\nhfU : s ∈ I.ideal V\nhxs : x ∈ X.basicOpen s\nf : ↑Γ(X, ↑U)\ng : ↑Γ(X, ↑V)\nhfg : X.basicOpen f = X.basicOpen g\nhxf : x ∈ X.basicOpen f\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 304,
"column": 4
} | {
"line": 304,
"column": 85
} | [
{
"pp": "case succ\nX : Scheme\nI : X.IdealSheafData\nU V : ↑X.affineOpens\nx : ↥X\nhxV : x ∈ ↑↑V\nhxU : x ∈ ↑↑U\nhx : ∀ f ∈ I.ideal U, x ∉ X.basicOpen f\ns : ↑Γ(X, ↑V)\nhfU : s ∈ I.ideal V\nhxs : x ∈ X.basicOpen s\nf : ↑Γ(X, ↑U)\ng : ↑Γ(X, ↑V)\nhfg : X.basicOpen f = X.basicOpen g\nhxf : x ∈ X.basicOpen f\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 342,
"column": 2
} | {
"line": 342,
"column": 37
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nx : ↥X\nU : ↑X.affineOpens\nh : x ∈ ↑U\n⊢ x ∈ I.support ↔ x ∈ X.zeroLocus ↑(I.ideal U)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 203,
"column": 29
} | {
"line": 203,
"column": 40
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nU V : ↑X.affineOpens\nF : pullback (I.glueDataObjι U) (X.homOfLE ⋯) ⟶ ↑(↑U ⊓ ↑V) := pullback.snd (I.glueDataObjι U) (X.homOfLE ⋯)\nx✝¹ : ↑Γ(X, ↑V)\nhx : x✝¹ ∈ I.ideal V\nx : ↥(pullback (I.glueDataObjι U) (X.homOfLE ⋯))\nx✝ : x ∈ ⊤\n⊢ x ∈ (F ≫ (↑U ⊓ ↑V).ι) ⁻¹ᵁ ↑V",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 206,
"column": 73
} | {
"line": 206,
"column": 84
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nU V : ↑X.affineOpens\nF : pullback (I.glueDataObjι U) (X.homOfLE ⋯) ⟶ ↑(↑U ⊓ ↑V) := pullback.snd (I.glueDataObjι U) (X.homOfLE ⋯)\nx✝¹ : ↑Γ(X, ↑V)\nhx : x✝¹ ∈ I.ideal V\nx : ↥↑(↑U ⊓ ↑V)\nx✝ : x ∈ ⊤\n⊢ x ∈ (↑U ⊓ ↑V).ι ⁻¹ᵁ ↑V",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Stalk | {
"line": 324,
"column": 30
} | {
"line": 324,
"column": 41
} | [
{
"pp": "X Y : Scheme\nf✝¹ : X ⟶ Y\nU✝ V : X.Opens\nhU✝ : IsAffineOpen U✝\nhV : IsAffineOpen V\nR : CommRingCat\ninst✝¹ : IsLocalRing ↑R\nf✝ : Spec R ⟶ X\nx : ↥X\nf : X.presheaf.stalk x ⟶ R\ninst✝ : IsLocalHom (CommRingCat.Hom.hom f)\nU : X.Opens\nhU : (Spec.map f ≫ X.fromSpecStalk x) (closedPoint ↑R) ∈ U\n⊢ x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 255,
"column": 58
} | {
"line": 255,
"column": 69
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nU V W U₀ : ↑X.affineOpens\nhU₀ : ↑U ⊓ ↑W ≤ ↑U₀\n⊢ Set.range ⇑(↑U ⊓ ↑W).ι ⊆ ↑↑U₀",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"TopologicalSpac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 572,
"column": 10
} | {
"line": 572,
"column": 56
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nZ : Closeds ↥X\nU : ↑X.affineOpens\nf : ↑Γ(X, ↑U)\nF : Γ(X, ↑U) ⟶ Γ(X, X.basicOpen f) := X.presheaf.map (homOfLE ⋯).op\nx : ↑Γ(X, ↑U)\nhx : x ∈ (fun U ↦ PrimeSpectrum.vanishingIdeal (⇑(IsAffineOpen.fromSpec ⋯) ⁻¹' ↑Z)) U\nthis :\n ∀ (p : ↥(Spec Γ(X, ↑(X.affineBasicOpe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 644,
"column": 21
} | {
"line": 644,
"column": 44
} | [
{
"pp": "X : Scheme\n⊢ vanishingIdeal ⊥.support = X.nilradical",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme.IdealSheafData.support",
"AlgebraicGeometry.Scheme.IdealSheafData.instOrderBot",
"congrArg",
"OrderBot.toBot",
"PartialOrder.toPreorder",
"Preorde... | vanishingIdeal_support, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 340,
"column": 17
} | {
"line": 340,
"column": 33
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nU V : ↑X.affineOpens\nh : U ≤ V\nthis : IsIso (X.homOfLE ⋯)\n⊢ X.homOfLE ⋯ = inv (X.homOfLE ⋯)",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Scheme.homOfLE",
"Algebraic... | ← hom_comp_eq_id | Lean.Elab.Tactic.evalRewriteSeq | null |
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