module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 47
} | {
"line": 148,
"column": 4
} | [
{
"pp": "case inst.inst\nX Y : LocallyRingedSpace\nf g : X ⟶ Y\nU : Opens ↑↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace\nthis✝ :\n coequalizer.π f.toHom g.toHom ≫\n (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace (Hom.toShHom f) (Hom.toShHom g)).hom =\n (coequalizer.π... | [
"case inst.inst\nX Y : LocallyRingedSpace\nf g : X ⟶ Y\nU : Opens ↑↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace\nthis✝ :\n coequalizer.π f.toHom g.toHom ≫\n (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace (Hom.toShHom f) (Hom.toShHom g)).hom =\n (coequalizer.π (Hom.toShHo... | · apply CommRingCat.equalizer_ι_isLocalHom' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits | {
"line": 185,
"column": 2
} | {
"line": 186,
"column": 61
} | {
"line": 187,
"column": 2
} | [
{
"pp": "X Y : LocallyRingedSpace\nf g : X ⟶ Y\nU : Opens ↑↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace\ns : ↑((coequalizer (Hom.toShHom f) (Hom.toShHom g)).presheaf.obj (op U))\n⊢ ⇑(ConcreteCategory.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base) ⁻¹'\n ⇑(ConcreteCategory.... | [
"case e\nX Y : LocallyRingedSpace\nf g : X ⟶ Y\nU : Opens ↑↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace\ns : ↑((coequalizer (Hom.toShHom f) (Hom.toShHom g)).presheaf.obj (op U))\n⊢ ↾⇑(ConcreteCategory.hom f.base) ≫ ↾⇑(ConcreteCategory.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base)... | fapply Types.coequalizer_preimage_image_eq_of_preimage_eq (↾f.base)
(↾g.base) (↾(coequalizer.π f.toShHom g.toShHom).hom.base) | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_tacticFapply__1 | Batteries.Tactic.tacticFapply_ |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits | {
"line": 210,
"column": 4
} | {
"line": 210,
"column": 50
} | {
"line": 212,
"column": 0
} | [
{
"pp": "case H\nX Y : LocallyRingedSpace\nf g : X ⟶ Y\nU : Opens ↑↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace\ns : ↑((coequalizer (Hom.toShHom f) (Hom.toShHom g)).presheaf.obj (op U))\n⊢ unop ((Opens.map (Hom.toShHom g ≫ coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base).op.obj (op U... | [] | rw [coequalizer.condition f.toShHom g.toShHom] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Gluing | {
"line": 407,
"column": 4
} | {
"line": 407,
"column": 42
} | {
"line": 408,
"column": 4
} | [
{
"pp": "case right.right\nα : Type u\ninst✝ : TopologicalSpace α\nJ : Type u\nU : J → Opens α\ns : Set ↑(ofOpenSubsets U).glued\nhs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s)\ni : (ofOpenSubsets U).J\nx : ↑(of α)\nhx' : x ∈ U i\nhx : (ConcreteCategory.hom ((of... | [
"case right.right\nα : Type u\ninst✝ : TopologicalSpace α\nJ : Type u\nU : J → Opens α\ns : Set ↑(ofOpenSubsets U).glued\nhs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s)\ni : (ofOpenSubsets U).J\nx : ↑(of α)\nhx' : x ∈ U i\nhx : (ConcreteCategory.hom ((ofOpenSubsets ... | refine ⟨Set.mem_image_of_mem _ hx, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1055,
"column": 43
} | {
"line": 1055,
"column": 77
} | {
"line": 1055,
"column": 78
} | [
{
"pp": "R S : CommRingCat\nX : Scheme\nφ : R ⟶ S\nhφ : Function.Injective ⇑(ConcreteCategory.hom φ)\nf g : Spec R ⟶ X\nU : X.Opens\nhU : IsAffineOpen U\nhUf : f ⁻¹ᵁ U = ⊤\nhUg : g ⁻¹ᵁ U = ⊤\nH : Spec.map φ ≫ f = Spec.map φ ≫ g\nthis : Mono φ\n⊢ Set.range ⇑f ⊆ Set.range ⇑U.ι",
"ppTerm": "?m.78",
"assign... | [
"R S : CommRingCat\nX : Scheme\nφ : R ⟶ S\nhφ : Function.Injective ⇑(ConcreteCategory.hom φ)\nf g : Spec R ⟶ X\nU : X.Opens\nhU : IsAffineOpen U\nhUf : f ⁻¹ᵁ U = ⊤\nhUg : g ⁻¹ᵁ U = ⊤\nH : Spec.map φ ≫ f = Spec.map φ ≫ g\nthis : Mono φ\n⊢ ∀ (y : ↥(Spec R)), f y ∈ U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1056,
"column": 41
} | {
"line": 1056,
"column": 75
} | {
"line": 1056,
"column": 76
} | [
{
"pp": "R S : CommRingCat\nX : Scheme\nφ : R ⟶ S\nhφ : Function.Injective ⇑(ConcreteCategory.hom φ)\nf g : Spec R ⟶ X\nU : X.Opens\nhU : IsAffineOpen U\nhUf : f ⁻¹ᵁ U = ⊤\nhUg : g ⁻¹ᵁ U = ⊤\nH : Spec.map φ ≫ f = Spec.map φ ≫ g\nthis : Mono φ\n⊢ Set.range ⇑g ⊆ Set.range ⇑U.ι",
"ppTerm": "?m.104",
"assig... | [
"R S : CommRingCat\nX : Scheme\nφ : R ⟶ S\nhφ : Function.Injective ⇑(ConcreteCategory.hom φ)\nf g : Spec R ⟶ X\nU : X.Opens\nhU : IsAffineOpen U\nhUf : f ⁻¹ᵁ U = ⊤\nhUg : g ⁻¹ᵁ U = ⊤\nH : Spec.map φ ≫ f = Spec.map φ ≫ g\nthis : Mono φ\n⊢ ∀ (y : ↥(Spec R)), g y ∈ U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1106,
"column": 4
} | {
"line": 1106,
"column": 84
} | {
"line": 1107,
"column": 4
} | [
{
"pp": "case refine_1\nX : Scheme\ninst✝ : IsAffine X\ns : Set ↥X\nhs : IsClosed s\nZ : Set ↥(Spec Γ(X, ⊤)) := X.toΓSpecFun '' s\nhZ : IsClosed Z\n⊢ ∃ I, s = X.zeroLocus ↑I",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrie... | [
"case refine_1\nX : Scheme\ninst✝ : IsAffine X\ns : Set ↥X\nhs : IsClosed s\nZ : Set ↥(Spec Γ(X, ⊤)) := X.toΓSpecFun '' s\nhZ : IsClosed Z\nI : Ideal ↑Γ(X, ⊤)\nhI : Z = PrimeSpectrum.zeroLocus ↑I\n⊢ ∃ I, s = X.zeroLocus ↑I"
] | obtain ⟨I, (hI : Z = _)⟩ := (PrimeSpectrum.isClosed_iff_zeroLocus_ideal _).mp hZ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Monoidal.Cartesian.Over | {
"line": 37,
"column": 42
} | {
"line": 37,
"column": 53
} | {
"line": 37,
"column": 54
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasPullbacks C\nX : C\nY : Over X\nm : Y ⟶ mk (𝟙 X)\n⊢ Hom.left m = Hom.left (homMk Y.hom ⋯)",
"ppTerm": "?m.96",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTh... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasPullbacks C\nX : C\nY : Over X\nm : Y ⟶ mk (𝟙 X)\n⊢ Hom.left m = Y.hom"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1156,
"column": 6
} | {
"line": 1156,
"column": 23
} | {
"line": 1156,
"column": 24
} | [
{
"pp": "case pos\nX : Scheme\nU : X.Opens\nI J : Ideal ↑Γ(X, U)\nthis : U.carrier ↓∩ X.zeroLocus ↑(I ⊓ J) = U.carrier ↓∩ (X.zeroLocus ↑I ∪ X.zeroLocus ↑J)\nx : ↥X\nhxU : x ∈ U\n⊢ x ∈ X.zeroLocus ↑(I ⊓ J) ↔ x ∈ X.zeroLocus ↑I ∪ X.zeroLocus ↑J",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [... | [
"case pos\nX : Scheme\nU : X.Opens\nI J : Ideal ↑Γ(X, U)\nthis : U.carrier ↓∩ X.zeroLocus ↑(I ⊓ J) = U.carrier ↓∩ (X.zeroLocus ↑I ∪ X.zeroLocus ↑J)\nx : ↥X\nhxU : x ∈ U\n⊢ (∀ f ∈ I, f ∈ J → x ∉ X.basicOpen f) ↔ (∀ f ∈ I, x ∉ X.basicOpen f) ∨ ∀ f ∈ J, x ∉ X.basicOpen f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1186,
"column": 2
} | {
"line": 1186,
"column": 13
} | {
"line": 1186,
"column": 14
} | [
{
"pp": "X : Scheme\nU : X.Opens\nι : Type u_1\nI : ι → Ideal ↑Γ(X, U)\ninst✝ : Finite ι\n⊢ X.zeroLocus ↑(⨅ i, I i) = (⋃ i, X.zeroLocus ↑(I i)) ∪ (↑U)ᶜ",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"AlgebraicGeometry.SheafedSpace.instTopologicalSp... | [
"X : Scheme\nU : X.Opens\nι : Type u_1\nI : ι → Ideal ↑Γ(X, U)\ninst✝ : Finite ι\n⊢ X.zeroLocus (⋂ i, ↑(I i)) = (⋃ i, X.zeroLocus ↑(I i)) ∪ (↑U)ᶜ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1192,
"column": 2
} | {
"line": 1192,
"column": 13
} | {
"line": 1192,
"column": 14
} | [
{
"pp": "X : Scheme\nU : X.Opens\nι : Type u_1\nI : ι → Ideal ↑Γ(X, U)\ninst✝¹ : Finite ι\ninst✝ : Nonempty ι\n⊢ X.zeroLocus ↑(⨅ i, I i) = ⋃ i, X.zeroLocus ↑(I i)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"iInf",
"Semiring.toModule",
... | [
"X : Scheme\nU : X.Opens\nι : Type u_1\nI : ι → Ideal ↑Γ(X, U)\ninst✝¹ : Finite ι\ninst✝ : Nonempty ι\n⊢ X.zeroLocus (⋂ i, ↑(I i)) = ⋃ i, X.zeroLocus ↑(I i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Constructions.ZeroObjects | {
"line": 40,
"column": 24
} | {
"line": 40,
"column": 35
} | {
"line": 40,
"column": 36
} | [
{
"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 = s.snd",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals... | [
"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 = s.snd"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Constructions.ZeroObjects | {
"line": 66,
"column": 24
} | {
"line": 66,
"column": 35
} | {
"line": 66,
"column": 36
} | [
{
"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 = s.fst",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals... | [
"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 = s.fst"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Constructions.ZeroObjects | {
"line": 92,
"column": 24
} | {
"line": 92,
"column": 35
} | {
"line": 92,
"column": 36
} | [
{
"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 = s.inr",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoa... | [
"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 = s.inr"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Constructions.ZeroObjects | {
"line": 118,
"column": 24
} | {
"line": 118,
"column": 35
} | {
"line": 118,
"column": 36
} | [
{
"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 = s.inl",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoa... | [
"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 = s.inl"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 86,
"column": 34
} | {
"line": 86,
"column": 91
} | {
"line": 87,
"column": 4
} | [
{
"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.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 164,
"column": 7
} | {
"line": 164,
"column": 39
} | {
"line": 164,
"column": 40
} | [
{
"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.... | [
"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.f i k)).c.ap... | (π₁ i, j, k).c.naturality_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 448,
"column": 2
} | {
"line": 448,
"column": 45
} | {
"line": 448,
"column": 46
} | [
{
"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... | [
"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 (𝒰.f (idx 𝒰 i)) (𝒰.f (idx 𝒰 j)) ≫ f (idx 𝒰 i) =\n pullback.snd ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 481,
"column": 29
} | {
"line": 481,
"column": 40
} | {
"line": 481,
"column": 41
} | [
{
"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)",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"CategoryTheory.PreZeroH... | [
"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 ∈ U x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 548,
"column": 10
} | {
"line": 548,
"column": 32
} | {
"line": 548,
"column": 33
} | [
{
"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... | [
"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 = ↑((pullback.fst (V F i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 296,
"column": 4
} | {
"line": 296,
"column": 94
} | {
"line": 297,
"column": 2
} | [
{
"pp": "case 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 (𝒰.f j) ... | [] | 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.Gluing | {
"line": 565,
"column": 4
} | {
"line": 565,
"column": 15
} | {
"line": 565,
"column": 16
} | [
{
"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)\nk₂ : (k_1 : J) × (k_1 ⟶ i) × (k_1 ⟶ k)\nl : J\nhli : l ⟶ k₁.... | [
"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)\nk₂ : (k_1 : J) × (k_1 ⟶ i) × (k_1 ⟶ k)\nl : J\nhli : l ⟶ k₁.fst\nhlk : l... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 307,
"column": 2
} | {
"line": 307,
"column": 87
} | {
"line": 308,
"column": 2
} | [
{
"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\nb : (MultispanShape.prod (Cover.gluedCover (Precoverage.ZeroHypercover.pullback₁ s.fst 𝒰)).J).R\n⊢ Multicoequalizer.π (Cover.gluedCover (Precoverage.ZeroHypercover.pullback₁... | [
"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\nb : (MultispanShape.prod (Cover.gluedCover (Precoverage.ZeroHypercover.pullback₁ s.fst 𝒰)).J).R\n⊢ ((Precoverage.ZeroHypercover.pullback₁ s.fst 𝒰).f b ≫\n Cover.glueMorphisms (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.Gluing | {
"line": 589,
"column": 6
} | {
"line": 589,
"column": 17
} | {
"line": 589,
"column": 18
} | [
{
"pp": "J : 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 : Hom.op... | [
"J : 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 : Hom.opensRange (F.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 317,
"column": 2
} | {
"line": 317,
"column": 87
} | {
"line": 318,
"column": 2
} | [
{
"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\nb : (MultispanShape.prod (Cover.gluedCover (Precoverage.ZeroHypercover.pullback₁ s.fst 𝒰)).J).R\n⊢ Multicoequalizer.π (Cover.gluedCover (Precoverage.ZeroHypercover.pullback₁... | [
"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\nb : (MultispanShape.prod (Cover.gluedCover (Precoverage.ZeroHypercover.pullback₁ s.fst 𝒰)).J).R\n⊢ ((Precoverage.ZeroHypercover.pullback₁ s.fst 𝒰).f b ≫\n Cover.glueMorphisms (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": 275,
"column": 8
} | {
"line": 275,
"column": 45
} | {
"line": 275,
"column": 46
} | [
{
"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... | [
"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 ix) x = (α iy... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 597,
"column": 2
} | {
"line": 600,
"column": 39
} | {
"line": 601,
"column": 2
} | [
{
"pp": "J : 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 : Hom.op... | [
"J : 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 : Hom.opensRange (F.... | 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.AlgebraicGeometry.Limits | {
"line": 282,
"column": 6
} | {
"line": 282,
"column": 43
} | {
"line": 282,
"column": 44
} | [
{
"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⟩",
"ppTerm": "?left.refine_3",
"assigned": ... | [
"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 ↦ (α i) a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 67
} | {
"line": 318,
"column": 68
} | [
{
"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... | [
"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⊢ Set.range ⇑(f i) ∩ Set.range ⇑(f j) = ∅"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 324,
"column": 27
} | {
"line": 324,
"column": 89
} | {
"line": 324,
"column": 90
} | [
{
"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 : ... | [
"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 : x ∈ ⊤\nthis ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 429,
"column": 4
} | {
"line": 429,
"column": 88
} | {
"line": 429,
"column": 89
} | [
{
"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... | [
"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' := Cofan.ext (Iso.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 432,
"column": 23
} | {
"line": 432,
"column": 39
} | {
"line": 432,
"column": 40
} | [
{
"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... | [
"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.refl (BinaryCof... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 433,
"column": 23
} | {
"line": 433,
"column": 39
} | {
"line": 433,
"column": 40
} | [
{
"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... | [
"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.refl (BinaryCof... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 384,
"column": 6
} | {
"line": 384,
"column": 17
} | {
"line": 384,
"column": 18
} | [
{
"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... | [
"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 Multicoequalizer.π (gluing 𝒰 f g).diagram i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 35
} | {
"line": 308,
"column": 36
} | [
{
"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... | [
"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 i j U ≫ (D.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 470,
"column": 6
} | {
"line": 470,
"column": 82
} | {
"line": 471,
"column": 6
} | [
{
"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))",
"ppTerm": "?refine_2",
"assigned": ... | [
"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\nZ : Scheme\nf : Z ⟶ X ⨿ Y\n⊢ IsCompl (Scheme.Hom.opensRange (pullback.fst f coprod.inl)) (Scheme.Hom.opensRange (pullback.fst f coprod.inr))"
] | refine fun {Z} f ↦ (nonempty_isColimit_binaryCofanMk_of_isCompl _ _ ?_).some | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.Limits | {
"line": 469,
"column": 4
} | {
"line": 473,
"column": 88
} | {
"line": 475,
"column": 0
} | [
{
"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.Pullbacks | {
"line": 473,
"column": 2
} | {
"line": 473,
"column": 34
} | {
"line": 474,
"column": 4
} | [
{
"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",
"ppTerm": "?m... | [
"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 (pullback.snd f (Z.affineCover.f i) ≫ Z.affineCover.f i) g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 385,
"column": 12
} | {
"line": 386,
"column": 50
} | {
"line": 387,
"column": 12
} | [
{
"pp": "case 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.t j i ≫ D.... | [
"case 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.t j i ≫ D.f i j\n⊢ (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.Morphisms.Basic | {
"line": 170,
"column": 84
} | {
"line": 170,
"column": 95
} | {
"line": 170,
"column": 96
} | [
{
"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",
"ppTerm": "?m.35",
"assigned"... | [
"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 ⊆ ⋃ i, ↑(U i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 171,
"column": 52
} | {
"line": 171,
"column": 63
} | {
"line": 171,
"column": 64
} | [
{
"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... | [
"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 ⊆ ⋃ i, ↑(U i)"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 13
} | {
"line": 191,
"column": 14
} | [
{
"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",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeom... | [
"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⊢ ∃ i, x ∈ U i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 612,
"column": 58
} | {
"line": 612,
"column": 69
} | {
"line": 612,
"column": 70
} | [
{
"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... | [
"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).I₀\n⊢ pullbac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 617,
"column": 6
} | {
"line": 617,
"column": 72
} | {
"line": 617,
"column": 73
} | [
{
"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 ... | [
"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 fXZ fYZ).toP... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 620,
"column": 49
} | {
"line": 620,
"column": 68
} | {
"line": 620,
"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": 621,
"column": 6
} | {
"line": 621,
"column": 17
} | {
"line": 621,
"column": 18
} | [
{
"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 ... | [
"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 fXZ fYZ).toP... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 784,
"column": 4
} | {
"line": 784,
"column": 43
} | {
"line": 784,
"column": 44
} | [
{
"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... | [
"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 'set_option pp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 48,
"column": 42
} | {
"line": 48,
"column": 53
} | {
"line": 48,
"column": 54
} | [
{
"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 ... | [
"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 x₁ = f x₂\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 99,
"column": 39
} | {
"line": 99,
"column": 50
} | {
"line": 99,
"column": 51
} | [
{
"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... | [
"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✝¹, inst✝] f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 33
} | {
"line": 105,
"column": 34
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : Surjective f\n⊢ Set.range ⇑f = Set.univ",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.PresheafedSpace.carrier",
"CategoryTheory.ConcreteCategory.hom",
"CommRingCat",
"TopCat.instCategory"... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝ : Surjective f\n⊢ Function.Surjective ⇑f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 119,
"column": 2
} | {
"line": 123,
"column": 51
} | {
"line": 125,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"_private.Mathlib... | [] | 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
} | {
"line": 125,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"_private.Mathlib... | [] | 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
} | {
"line": 265,
"column": 31
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : Dense ↑U\n⊢ DenseRange ⇑U.ι",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"congrArg",
"Category... | [
"X : Scheme\nU : X.Opens\nhU : Dense ↑U\n⊢ Dense ↑U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 269,
"column": 48
} | {
"line": 269,
"column": 59
} | {
"line": 269,
"column": 60
} | [
{
"pp": "X : Scheme\nU V : X.Opens\nhU : Dense ↑U\nhU' : U ≤ V\n⊢ IsDominant (X.homOfLE hU' ≫ V.ι)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme.homOfLE",
"AlgebraicGeometry.Scheme",
"CategoryTheory.CategoryStruct.toQuiver",
... | [
"X : Scheme\nU V : X.Opens\nhU : Dense ↑U\nhU' : U ≤ V\n⊢ IsDominant U.ι"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Away.Lemmas | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 27
} | {
"line": 64,
"column": 28
} | [
{
"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... | [
"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\nthis : IsLocal... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Localization.Away.Lemmas | {
"line": 85,
"column": 12
} | {
"line": 85,
"column": 23
} | {
"line": 85,
"column": 24
} | [
{
"pp": "case zero\nR : Type u_2\nS : Type u_3\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nx : R\ninst✝ : IsLocalization.Away x S\nI J : Ideal R\nhle : map (algebraMap R S) I ≤ map (algebraMap R S) J\nhxJ : ∀ (y : R), x * y ∈ J → y ∈ J\ny : R\nhy : y ∈ I\nh : x ^ 0 * y ∈ J\n⊢ y ∈ J"... | [
"case zero\nR : Type u_2\nS : Type u_3\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nx : R\ninst✝ : IsLocalization.Away x S\nI J : Ideal R\nhle : map (algebraMap R S) I ≤ map (algebraMap R S) J\nhxJ : ∀ (y : R), x * y ∈ J → y ∈ J\ny : R\nhy : y ∈ I\nh : x ^ 0 * y ∈ J\n⊢ y ∈ J"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 220,
"column": 6
} | {
"line": 220,
"column": 17
} | {
"line": 220,
"column": 18
} | [
{
"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... | [
"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\nf' : X' ⟶ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 277,
"column": 2
} | {
"line": 277,
"column": 45
} | {
"line": 279,
"column": 0
} | [
{
"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.RingTheory.RingHom.Locally | {
"line": 297,
"column": 6
} | {
"line": 297,
"column": 30
} | {
"line": 297,
"column": 31
} | [
{
"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 : ... | [
"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 : Algebra R S\... | locally_iff_span_eq_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 16
} | {
"line": 317,
"column": 2
} | [
{
"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),\... | [
"P : {α β : Type u} → [TopologicalSpace α] → [TopologicalSpace β] → (α → β) → Prop\ninst✝² : (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),\n IsOpen... | introv hf hs H | Mathlib.Tactic.evalIntrov | Mathlib.Tactic.introv |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 17
} | {
"line": 166,
"column": 18
} | [
{
"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\... | [
"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\nX Y : Schem... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 401,
"column": 4
} | {
"line": 401,
"column": 42
} | {
"line": 401,
"column": 43
} | [
{
"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... | [
"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 : Type u\nU : ι ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 39
} | {
"line": 69,
"column": 0
} | [
{
"pp": "case 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",
"ppTerm": "?e'_3.h₂",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Presheaf... | [] | 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
} | {
"line": 69,
"column": 0
} | [
{
"pp": "case 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",
"ppTerm": "?e'_3.h₂",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Presheaf... | [] | 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
} | {
"line": 69,
"column": 0
} | [
{
"pp": "case 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",
"ppTerm": "?e'_3.h₂",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Presheaf... | [] | exact IsIso.hom_inv_id_apply f.base | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 63
} | {
"line": 207,
"column": 64
} | [
{
"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... | [
"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.affineOpens\nV₁ V... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 241,
"column": 31
} | {
"line": 241,
"column": 42
} | {
"line": 241,
"column": 43
} | [
{
"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)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"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 ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 251,
"column": 15
} | {
"line": 251,
"column": 39
} | {
"line": 251,
"column": 40
} | [
{
"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",
"ppTerm": "?m.107",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedG... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 261,
"column": 10
} | {
"line": 261,
"column": 21
} | {
"line": 261,
"column": 22
} | [
{
"pp": "X : 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)",
"ppTerm": "?m.92",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Algebr... | [
"X : 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, ⋃ (x : i ∈ X.affineOpens), ⋃ (_ : ⟨i, ⋯⟩ ∈ s), ↑i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 17
} | {
"line": 357,
"column": 0
} | [
{
"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": 370,
"column": 2
} | {
"line": 370,
"column": 39
} | {
"line": 370,
"column": 40
} | [
{
"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... | [
"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.appLE f ⊤ ↑(U... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 381,
"column": 2
} | {
"line": 381,
"column": 17
} | {
"line": 382,
"column": 2
} | [
{
"pp": "P : 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} [CommRing R]... | [
"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✝ Z : Scheme\nf✝ : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : IsAffine Y\n𝒰 : X.OpenCover\n⊢ sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ ... | intro X Y f _ 𝒰 | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.LocalProperties.Reduced | {
"line": 32,
"column": 4
} | {
"line": 32,
"column": 15
} | {
"line": 32,
"column": 16
} | [
{
"pp": "case 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",
"ppTerm": "?zero",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"case 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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 186,
"column": 2
} | {
"line": 186,
"column": 13
} | {
"line": 186,
"column": 14
} | [
{
"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 : ... | [
"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 : (X.openCover... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 13
} | {
"line": 207,
"column": 14
} | [
{
"pp": "X Y Z : Scheme\nf✝ : X ⟶ Y\ninst✝¹ : CompactSpace ↥X\ninst✝ : QuasiSeparatedSpace ↥Y\nf g : X ⟶ Y\n⊢ IsCompact Set.univ",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X Y Z : Scheme\nf✝ : X ⟶ Y\ninst✝¹ : CompactSpace ↥X\ninst✝ : QuasiSeparatedSpace ↥Y\nf g : X ⟶ Y\n⊢ IsCompact Set.univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 43
} | {
"line": 113,
"column": 44
} | [
{
"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",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 43
} | {
"line": 123,
"column": 44
} | [
{
"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",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 45
} | {
"line": 164,
"column": 46
} | [
{
"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",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"ENat",
"congr",
... | [
"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.height < J.height"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 384,
"column": 4
} | {
"line": 384,
"column": 32
} | {
"line": 384,
"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 :... | [
"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.krullDim ↑(Set.Iic { asIdeal := comap (algebraMap R A) J, isPrime := ... | Order.height_eq_krullDim_Iic | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 401,
"column": 2
} | {
"line": 401,
"column": 43
} | {
"line": 401,
"column": 44
} | [
{
"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",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 498,
"column": 4
} | {
"line": 500,
"column": 61
} | {
"line": 502,
"column": 0
} | [
{
"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",
"ppTerm": "?a.refine_2✝",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"instCompleteLinearOrder... | [] | · 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": 532,
"column": 4
} | {
"line": 533,
"column": 60
} | {
"line": 535,
"column": 0
} | [
{
"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",
"ppTerm": "?a.inr✝",
"assigned": true,
"usedConstants": [
"instCompleteLinearOrderENat",
"Semiring.toModule",
... | [] | · 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.QuasiSeparated | {
"line": 342,
"column": 71
} | {
"line": 342,
"column": 82
} | {
"line": 342,
"column": 83
} | [
{
"pp": "X : 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 (ConcreteC... | [
"X : 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 (ConcreteCategory.hom ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 632,
"column": 44
} | {
"line": 632,
"column": 55
} | {
"line": 632,
"column": 56
} | [
{
"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... | [
"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✝ : HasRingHomProp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 15
} | {
"line": 135,
"column": 16
} | [
{
"pp": "X : Scheme\nx : ↥X\n⊢ x ∈ ⊥ ↔ x ∈ ⋂ U, X.zeroLocus ↑(⊤ U)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"False",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Semiring.toModule",
"Op... | [
"X : Scheme\nx : ↥X\n⊢ ∃ x_1 ∈ X.affineOpens, x ∈ x_1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 13
} | {
"line": 203,
"column": 14
} | [
{
"pp": "X : Scheme\nι : Type u_1\nI : ι → X.IdealSheafData\ninst✝ : Finite ι\n⊢ (⨅ i, I i).ideal = ⨅ i, (I i).ideal",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Scheme\nι : Type u_1\nI : ι → X.IdealSheafData\ninst✝ : Finite ι\n⊢ (⨅ i, I i).ideal = ⨅ i, (I i).ideal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Properties | {
"line": 351,
"column": 2
} | {
"line": 351,
"column": 13
} | {
"line": 351,
"column": 14
} | [
{
"pp": "X : Scheme\ninst✝ : IsIntegral X\nU V : X.Opens\ni : U ⟶ V\nH : Nonempty ↥↑U\nx : ↑Γ(X, V)\n⊢ (↑U).Nonempty",
"ppTerm": "?m.72",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Scheme\ninst✝ : IsIntegral X\nU V : X.Opens\ni : U ⟶ V\nH : Nonempty ↥↑U\nx : ↑Γ(X, V)\n⊢ (↑U).Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 705,
"column": 2
} | {
"line": 705,
"column": 12
} | {
"line": 706,
"column": 2
} | [
{
"pp": "P : 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\nR S : C... | [
"P✝ : 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\nR S : CommRingCat... | intro P hP | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 306,
"column": 4
} | {
"line": 306,
"column": 85
} | {
"line": 306,
"column": 86
} | [
{
"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... | [
"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 : IsLocaliz... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 306,
"column": 4
} | {
"line": 306,
"column": 85
} | {
"line": 306,
"column": 86
} | [
{
"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... | [
"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 : IsLocaliz... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 37
} | {
"line": 344,
"column": 38
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nx : ↥X\nU : ↑X.affineOpens\nh : x ∈ ↑U\n⊢ x ∈ I.support ↔ x ∈ X.zeroLocus ↑(I.ideal U)",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Scheme\nI : X.IdealSheafData\nx : ↥X\nU : ↑X.affineOpens\nh : x ∈ ↑U\n⊢ x ∈ I.support ↔ x ∈ X.zeroLocus ↑(I.ideal U)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 428,
"column": 4
} | {
"line": 428,
"column": 15
} | {
"line": 428,
"column": 16
} | [
{
"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",
"ppTerm": "?m.118",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.OpenImmersion | {
"line": 94,
"column": 8
} | {
"line": 94,
"column": 19
} | {
"line": 94,
"column": 20
} | [
{
"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... | [
"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⊢ ∀ (x : ↥X), ∃ i_1 y, (i i_1) y = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 203,
"column": 29
} | {
"line": 203,
"column": 40
} | {
"line": 203,
"column": 41
} | [
{
"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",
... | [
"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⊢ ↑((TopCat.Hom.hom F.base) x) ∈ ↑V"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 647,
"column": 21
} | {
"line": 647,
"column": 44
} | {
"line": 647,
"column": 45
} | [
{
"pp": "X : Scheme\n⊢ vanishingIdeal ⊥.support = X.nilradical",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme.IdealSheafData.support",
"AlgebraicGeometry.Scheme.IdealSheafData.instOrderBot",
"congrArg",
"OrderBot.toBot",
... | [
"X : Scheme\n⊢ ⊥.radical = X.nilradical"
] | vanishingIdeal_support, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 206,
"column": 73
} | {
"line": 206,
"column": 84
} | {
"line": 206,
"column": 85
} | [
{
"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",
"ppTerm": "?m.394",
"assigned": tru... | [
"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 ∈ ↑V"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Stalk | {
"line": 324,
"column": 30
} | {
"line": 324,
"column": 41
} | {
"line": 324,
"column": 42
} | [
{
"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 ... | [
"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 ∈ U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 722,
"column": 29
} | {
"line": 722,
"column": 40
} | {
"line": 722,
"column": 41
} | [
{
"pp": "X Y : Scheme\nf : X.Hom Y\ninst✝ : QuasiCompact f\nU✝ U : ↑Y.affineOpens\ns : ↑Γ(Y, ↑U)\nthis : IsLocalization.Away s ↑Γ(Y, Y.basicOpen s)\nx : ↑Γ(Y, ↑U)\nn✝ : ℕ\nhx :\n IsLocalization.mk' (↑Γ(Y, Y.basicOpen s)) x ⟨(fun x ↦ s ^ x) n✝, ⋯⟩ ∈\n RingHom.ker (CommRingCat.Hom.hom (f.app ↑(Y.affineBasicOp... | [
"X Y : Scheme\nf : X.Hom Y\ninst✝ : QuasiCompact f\nU✝ U : ↑Y.affineOpens\ns : ↑Γ(Y, ↑U)\nthis : IsLocalization.Away s ↑Γ(Y, Y.basicOpen s)\nx : ↑Γ(Y, ↑U)\nn✝ : ℕ\nhx :\n IsLocalization.mk' (↑Γ(Y, Y.basicOpen s)) x ⟨(fun x ↦ s ^ x) n✝, ⋯⟩ ∈\n RingHom.ker (CommRingCat.Hom.hom (f.app ↑(Y.affineBasicOpen s)))\nH :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 738,
"column": 52
} | {
"line": 738,
"column": 63
} | {
"line": 738,
"column": 64
} | [
{
"pp": "X Y : Scheme\nf : X.Hom Y\ninst✝ : IsEmpty ↥X\nU : ↑Y.affineOpens\nx : ↑Γ(Y, ↑U)\nx✝ : x ∈ ⊤.ideal U\n⊢ x ∈ RingHom.ker (CommRingCat.Hom.hom (f.app ↑U))",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"AlgebraicGeometry.SheafedSpace.instTop... | [
"X Y : Scheme\nf : X.Hom Y\ninst✝ : IsEmpty ↥X\nU : ↑Y.affineOpens\nx : ↑Γ(Y, ↑U)\nx✝ : x ∈ ⊤.ideal U\n⊢ (CommRingCat.Hom.hom (f.app ↑U)) x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 771,
"column": 14
} | {
"line": 771,
"column": 29
} | {
"line": 771,
"column": 30
} | [
{
"pp": "X Y : Scheme\nf : X.Hom Y\nH : f.ker = ⊤\n⊢ IsEmpty ↥X",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X Y : Scheme\nf : X.Hom Y\nH : f.ker = ⊤\n⊢ IsEmpty ↥X"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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