module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 1044,
"column": 6
} | {
"line": 1044,
"column": 53
} | {
"line": 1045,
"column": 6
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : LocallyRingedSpace\nf : X ⟶ Z\ng : Y ⟶ Z\nH : IsOpenImmersion f\ns : PullbackCone f g\nm : s.pt ⟶ (pullbackConeOfLeft f g).pt\na✝ : m ≫ (pullbackConeOfLeft f g).fst = s.fst\nh₂ : m ≫ (pullbackConeOfLeft f g).snd = s.snd\n⊢ m =\n {\n ... | [
"case refine_2\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : LocallyRingedSpace\nf : X ⟶ Z\ng : Y ⟶ Z\nH : IsOpenImmersion f\ns : PullbackCone f g\nm : s.pt ⟶ (pullbackConeOfLeft f g).pt\na✝ : m ≫ (pullbackConeOfLeft f g).fst = s.fst\nh₂ : m ≫ (pullbackConeOfLeft f g).snd = s.snd\n⊢ m ≫ (pullbackConeOfLeft f g).s... | rw [← cancel_mono (pullbackConeOfLeft f g).snd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Sheaves.CommRingCat | {
"line": 354,
"column": 4
} | {
"line": 354,
"column": 97
} | {
"line": 355,
"column": 4
} | [
{
"pp": "case mp\nX : TopCat\nF : Sheaf CommRingCat X\nU V W : Opens ↑X\ne : W ≤ U ⊔ V\nx :\n ↑(CommRingCat.of\n ↥(((CommRingCat.Hom.hom (F.obj.map (homOfLE ⋯).op)).comp\n (RingHom.fst ↑(F.obj.obj (op U)) ↑(F.obj.obj (op V)))).eqLocus\n ((CommRingCat.Hom.hom (F.obj.map (homOfLE ⋯).op... | [
"case mp\nX : TopCat\nF : Sheaf CommRingCat X\nU V W : Opens ↑X\ne : W ≤ U ⊔ V\nx :\n ↑(CommRingCat.of\n ↥(((CommRingCat.Hom.hom (F.obj.map (homOfLE ⋯).op)).comp\n (RingHom.fst ↑(F.obj.obj (op U)) ↑(F.obj.obj (op V)))).eqLocus\n ((CommRingCat.Hom.hom (F.obj.map (homOfLE ⋯).op)).comp\n ... | rw [← TopCat.Sheaf.objSupIsoProdEqLocus_inv_fst, ← TopCat.Sheaf.objSupIsoProdEqLocus_inv_snd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 152,
"column": 2
} | {
"line": 153,
"column": 43
} | {
"line": 154,
"column": 2
} | [
{
"pp": "X : LocallyRingedSpace\nr : ↑(Γ.obj (op X))\nf : (structureSheaf ↑(Γ.obj (op X))).obj.obj (op (basicOpen r)) ⟶ X.presheaf.obj (op (X.toΓSpecMapBasicOpen r))\nloc_inst : IsLocalization.Away r ↑((structureSheaf ↑(Γ.obj (op X))).obj.obj (op (basicOpen r)))\n⊢ ConcreteCategory.hom\n (CommRingCat.ofH... | [
"X : LocallyRingedSpace\nr : ↑(Γ.obj (op X))\nf : (structureSheaf ↑(Γ.obj (op X))).obj.obj (op (basicOpen r)) ⟶ X.presheaf.obj (op (X.toΓSpecMapBasicOpen r))\nloc_inst : IsLocalization.Away r ↑((structureSheaf ↑(Γ.obj (op X))).obj.obj (op (basicOpen r)))\n⊢ ConcreteCategory.hom\n (CommRingCat.ofHom\n ... | rw [← @IsLocalization.Away.lift_comp _ _ _ _ _ _ _ r loc_inst _
(X.isUnit_res_toΓSpecMapBasicOpen r)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 233,
"column": 4
} | {
"line": 233,
"column": 16
} | {
"line": 234,
"column": 4
} | [
{
"pp": "X : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X))\nx : ↑X.toTopCat\np : PrimeSpectrum ↑(Γ.obj (op X)) := X.toΓSpecFun x\nS : CommRingCat := (structureSheaf ↑(Γ.obj (op X))).presheaf.stalk p\nt : ↑S\nht : IsUnit ((CommRingCat.Hom.hom (PresheafedSpace.Hom.stalkMap X.toΓSpecSheafedSpace.hom x)) t)\nr : ↑(Γ.obj... | [
"X : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X))\nx : ↑X.toTopCat\np : PrimeSpectrum ↑(Γ.obj (op X)) := X.toΓSpecFun x\nS : CommRingCat := (structureSheaf ↑(Γ.obj (op X))).presheaf.stalk p\nt : ↑S\nht : IsUnit ((CommRingCat.Hom.hom (PresheafedSpace.Hom.stalkMap X.toΓSpecSheafedSpace.hom x)) t)\nr : ↑(Γ.obj (op X))\ns ... | rw [map_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 63
} | {
"line": 37,
"column": 2
} | [
{
"pp": "case refine_3\nC : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasEqualizer f g\ninst✝ : HasBinaryProduct Y Y\n⊢ ∀ (s : PullbackCone (prod.lift f g) (prod.lift (𝟙 Y) (𝟙 Y))), equalizer.lift s.fst ⋯ ≫ equalizer.ι f g ≫ f = s.snd",
"ppTerm": "?refine_3",
"assigned": true,... | [
"case refine_4\nC : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasEqualizer f g\ninst✝ : HasBinaryProduct Y Y\n⊢ ∀ (s : PullbackCone (prod.lift f g) (prod.lift (𝟙 Y) (𝟙 Y))) (m : s.pt ⟶ equalizer f g),\n m ≫ equalizer.ι f g = s.fst → m ≫ equalizer.ι f g ≫ f = s.snd → m = equalizer.lift ... | · exact fun s ↦ by simpa using congr($s.condition ≫ prod.fst) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 84
} | {
"line": 183,
"column": 2
} | [
{
"pp": "I : Type w\ninst✝¹ : Category.{v, w} I\nD : I ⥤ Schemeᵒᵖ\ninst✝ : ∀ (i : I), IsAffine (unop (D.obj i))\n⊢ PreservesColimit D Scheme.Γ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme",
"Opposite",
"CategoryTheory.Functor.leftOp_obj",
... | [
"I : Type w\ninst✝¹ : Category.{v, w} I\nD : I ⥤ Schemeᵒᵖ\ninst✝ : ∀ (i : I), IsAffine (unop (D.obj i))\nthis : ∀ (i : Iᵒᵖ), IsAffine (D.leftOp.obj i)\n⊢ PreservesColimit D Scheme.Γ"
] | have (i : _) : IsAffine (D.leftOp.obj i) := Functor.leftOp_obj D _ ▸ inferInstance | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.GlueData | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 55
} | {
"line": 115,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nD : GlueData C\ni j k : D.J\n⊢ D.t' i j k ≫ (pullbackSymmetry (D.f j k) (D.f j i)).hom ≫ D.t' j i k ≫ (pullbackSymmetry (D.f i k) (D.f i j)).hom =\n 𝟙 (pullback (D.f i j) (D.f i k))",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
... | [
"C : Type u₁\ninst✝ : Category.{v, u₁} C\nD : GlueData C\ni j k : D.J\n⊢ (D.t' i j k ≫ (pullbackSymmetry (D.f j k) (D.f j i)).hom ≫ D.t' j i k ≫ (pullbackSymmetry (D.f i k) (D.f i j)).hom) ≫\n pullback.fst (D.f i j) (D.f i k) =\n 𝟙 (pullback (D.f i j) (D.f i k)) ≫ pullback.fst (D.f i j) (D.f i k)"
] | rw [← cancel_mono (pullback.fst (D.f i j) (D.f i k))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.GlueData | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 57
} | {
"line": 130,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nD : GlueData C\ni j k : D.J\n⊢ inv (D.t' i j k) =\n (pullbackSymmetry (D.f j k) (D.f j i)).hom ≫ D.t' j i k ≫ (pullbackSymmetry (D.f i k) (D.f i j)).hom",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.pullbac... | [
"C : Type u₁\ninst✝ : Category.{v, u₁} C\nD : GlueData C\ni j k : D.J\n⊢ inv (D.t' i j k) ≫ pullback.fst (D.f i j) (D.f i k) =\n ((pullbackSymmetry (D.f j k) (D.f j i)).hom ≫ D.t' j i k ≫ (pullbackSymmetry (D.f i k) (D.f i j)).hom) ≫\n pullback.fst (D.f i j) (D.f i k)"
] | rw [← cancel_mono (pullback.fst (D.f i j) (D.f i k))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.GlueData | {
"line": 197,
"column": 2
} | {
"line": 197,
"column": 46
} | {
"line": 198,
"column": 2
} | [
{
"pp": "D : GlueData (Type v)\nx : D.glued\n⊢ ∃ i y, (ConcreteCategory.hom (Sigma.ι D.diagram.right i ≫ Multicoequalizer.sigmaπ D.diagram)) y = x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.Types.hasColimitsOfSize",
"CategoryTheory.GlueData.diagram",... | [
"D : GlueData (Type v)\nx' : D.sigmaOpens\n⊢ ∃ i y,\n (ConcreteCategory.hom (Sigma.ι D.diagram.right i ≫ Multicoequalizer.sigmaπ D.diagram)) y =\n (ConcreteCategory.hom D.π) x'"
] | rcases D.types_π_surjective x with ⟨x', rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Gluing | {
"line": 124,
"column": 26
} | {
"line": 124,
"column": 53
} | {
"line": 124,
"column": 53
} | [
{
"pp": "D : GlueData\na b : (i : D.J) × ↑(D.U i)\nx : ↑(D.V (a.fst, b.fst))\ne₁ : (ConcreteCategory.hom (D.f a.fst b.fst)) x = a.snd\ne₂ : (ConcreteCategory.hom (D.f b.fst a.fst)) ((ConcreteCategory.hom (D.t a.fst b.fst)) x) = b.snd\n⊢ (ConcreteCategory.hom (D.f a.fst b.fst))\n ((ConcreteCategory.hom (D.t... | [] | by rw [← e₁, D.t_inv_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 16
} | {
"line": 57,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nJ : Type w\ninst✝² : Category.{w', w} J\ninst✝¹ : Small.{v, w} J\nF : J ⥤ SheafedSpace C\ninst✝ : HasLimits C\nX Y : SheafedSpace C\nf g : X ⟶ Y\n⊢ Epi (coequalizer.π ((forget C).map f) ((forget C).map g) ≫ (PreservesCoequalizer.iso (forget C) f g).hom)",
"pp... | [] | apply epi_comp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 766,
"column": 45
} | {
"line": 769,
"column": 5
} | {
"line": 771,
"column": 0
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nx : ↥U\n⊢ hU.fromSpec (hU.primeIdealOf x) = ↑x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.IsAffineOpen.isoSpec",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTo... | [] | by
dsimp only [IsAffineOpen.fromSpec, Subtype.coe_mk, IsAffineOpen.primeIdealOf]
rw [← Scheme.Hom.comp_apply, Iso.hom_inv_id_assoc]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.GlueData | {
"line": 468,
"column": 6
} | {
"line": 468,
"column": 86
} | {
"line": 469,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData' C\ni j : D.J\nhij : ¬i = j\n⊢ ((if hij : i = j then\n (pullbackSymmetry (D.f' i j) (D.f' i i)).hom ≫\n pullback.map (D.f' i i) (D.f' i j) (D.f' j i) (D.f' j i) (eqToHom ⋯) (eqToHom ⋯) (eqT... | [] | ext <;> simp [hij, Ne.symm hij, fst_eq_snd_of_mono_eq, pullback.condition_assoc] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.Gluing | {
"line": 235,
"column": 67
} | {
"line": 244,
"column": 54
} | {
"line": 246,
"column": 0
} | [
{
"pp": "D : GlueData\ni j : D.J\nU : Set ↑(D.U i)\n⊢ ⇑(ConcreteCategory.hom (D.ι j)) ⁻¹' ⇑(ConcreteCategory.hom (D.ι i)) '' U =\n ⇑(ConcreteCategory.hom (D.f j i)) '' ⇑(ConcreteCategory.hom (D.t j i ≫ D.f i j)) ⁻¹' U",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"CategoryTheory... | [] | by
have : D.f _ _ ⁻¹' 𝖣.ι j ⁻¹' 𝖣.ι i '' U = (D.t j i ≫ D.f _ _) ⁻¹' U := by
ext x
conv_rhs => rw [← Set.preimage_image_eq U (D.ι_injective _)]
simp
rw [← this, Set.image_preimage_eq_inter_range]
symm
apply Set.inter_eq_self_of_subset_left
rw [← D.preimage_range i j]
exact Set.preimage_mono (S... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits | {
"line": 204,
"column": 32
} | {
"line": 204,
"column": 60
} | {
"line": 204,
"column": 60
} | [
{
"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⊢ X.toRingedSpace.basicOpen\n ((ConcreteCategory.hom ((f.toHom ≫ (coequalizer.π (Hom.toShHom f) (Hom... | [
"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⊢ X.toRingedSpace.basicOpen\n ((ConcreteCategory.hom ((f.toHom ≫ (coequalizer.π (Hom.toShHom f) (Hom.toShHom g))... | ← PresheafedSpace.comp_c_app | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 254,
"column": 63
} | {
"line": 254,
"column": 89
} | {
"line": 254,
"column": 89
} | [
{
"pp": "D : GlueData\nU : Set ↥D.glued\n⊢ IsOpen (⇑(TopCat.homeoOfIso D.isoCarrier.symm) ⁻¹' U) ↔ ∀ (i : D.J), IsOpen (⇑(D.ι i) ⁻¹' U)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.GlueData.ι",
"CategoryTheory.GlueData.diagram",
"Eq.mpr",
... | [
"D : GlueData\nU : Set ↥D.glued\n⊢ (∀ (i : D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.J),\n IsOpen\n (⇑(ConcreteCategory.hom\n (D.toLocallyRingedSpaceGlueData.toSheafedSpaceGlueData.toPresheafedSpaceGlueData.toTopGlueData.ι i)) ⁻¹'\n ... | TopCat.GlueData.isOpen_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1265,
"column": 2
} | {
"line": 1265,
"column": 99
} | {
"line": 1267,
"column": 0
} | [
{
"pp": "X : Scheme\nA : CommRingCat\nf : X ⟶ Spec A\nφ : A ⟶ Γ(X, ⊤) := ⋯\nφ' : specTargetImage f ⟶ Scheme.Γ.obj (op X) := ⋯\n⊢ Function.Injective ⇑(Scheme.ΓSpecIso (specTargetImage f)).hom.hom'",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": [
"CommRingCat.forgetReflectIsos",
"... | [] | exact ((ConcreteCategory.isIso_iff_bijective (Scheme.ΓSpecIso _).hom).mp inferInstance).injective | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 444,
"column": 2
} | {
"line": 444,
"column": 21
} | {
"line": 445,
"column": 2
} | [
{
"pp": "case k\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\n⊢ (b : (MultispanShape.prod (ulift 𝒰).gluedCover.J).R) → (ulift 𝒰).gluedCover.diagram.right b ⟶ Y",
... | [
"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\n⊢ ∀ (a : (MultispanShape.prod (ulift 𝒰).gluedCover.J).L),\n (ulift 𝒰).gluedCover.diagram.fst a ≫ f (idx 𝒰 ((Mul... | · exact fun i ↦ f _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Limits | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 81
} | {
"line": 216,
"column": 2
} | [
{
"pp": "σ : Type v\ng : σ → Scheme\ninst✝ : Small.{u, v} σ\ni j : σ\nZ : Scheme\na : Z ⟶ g i\nb : Z ⟶ g j\nh : CommSq a b (Sigma.ι g i) (Sigma.ι g j)\nhij : i ≠ j\nz : ↥Z\n⊢ False",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCar... | [
"case x\nσ : Type v\ng : σ → Scheme\ninst✝ : Small.{u, v} σ\ni j : σ\nZ : Scheme\na : Z ⟶ g i\nb : Z ⟶ g j\nh : CommSq a b (Sigma.ι g i) (Sigma.ι g j)\nhij : i ≠ j\nz : ↥Z\n⊢ ↥(∐ g)",
"case a\nσ : Type v\ng : σ → Scheme\ninst✝ : Small.{u, v} σ\ni j : σ\nZ : Scheme\na : Z ⟶ g i\nb : Z ⟶ g j\nh : CommSq a b (Sigma.... | fapply eq_bot_iff.mp <| disjoint_iff.mp <| disjoint_opensRange_sigmaι g i j hij | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_tacticFapply__1 | Batteries.Tactic.tacticFapply_ |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 664,
"column": 75
} | {
"line": 669,
"column": 71
} | {
"line": 669,
"column": 71
} | [
{
"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\ni j k : Shrink.{u, w} J\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError tru... | [] | by
rintro _ ⟨x, rfl⟩
obtain ⟨l, fi, fj, fk, α, z, hα, hα₁, hα₂, rfl⟩ := exists_of_pullback_V_V F x
rw [← Scheme.Hom.comp_apply, reassoc_of% hα₁, homOfLE_tAux F ↓i ↓j fi fj,
Iso.hom_inv_id_assoc, Scheme.Opens.range_ι, SetLike.mem_coe]
exact TopologicalSpace.Opens.mem_iSup.mpr ⟨⟨l, fj, fk⟩... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Limits | {
"line": 472,
"column": 6
} | {
"line": 472,
"column": 81
} | {
"line": 473,
"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\nZ : Scheme\nf : Z ⟶ X ⨿ Y\n⊢ IsCompl (f ⁻¹ᵁ Scheme.Hom.opensRange coprod.inl) (f ⁻¹ᵁ Scheme.Hom.opensRange coprod.inr)",
"ppTerm": "?refine_2",
"assigned"... | [
"ι : 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 (f ⁻¹ᵁ Scheme.Hom.opensRange coprod.inl) (f ⁻¹ᵁ Scheme.Hom.opensRange coprod.inr) ↔\n IsCompl ((CompleteLatticeHom.setPreimage ⇑f) (Set.range ⇑coprod... | convert! (isCompl_range_inl_inr X Y).map (CompleteLatticeHom.setPreimage f) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.AlgebraicGeometry.Limits | {
"line": 620,
"column": 4
} | {
"line": 620,
"column": 41
} | {
"line": 621,
"column": 4
} | [
{
"pp": "case he\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) := ⋯\n⊢ IsIdempotentElem (Function.update 0 i 1)",
"ppTerm... | [
"case H\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) := ⋯\n⊢ ∀ (x y : (a : ι) → R a),\n (algebraMap ((a : ι) → R a) (R i)) x ... | · ext j; by_cases h : j = i <;> aesop | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 699,
"column": 2
} | {
"line": 699,
"column": 44
} | {
"line": 700,
"column": 2
} | [
{
"pp": "X Y S X' Y' S' : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nf' : X' ⟶ S'\ng' : Y' ⟶ S'\ni₁ : X ⟶ X'\ni₂ : Y ⟶ Y'\ni₃ : S ⟶ S'\ne₁ : f ≫ i₃ = i₁ ≫ f'\ne₂ : g ≫ i₃ = i₂ ≫ g'\ninst✝² : IsOpenImmersion i₁\ninst✝¹ : IsOpenImmersion i₂\ninst✝ : Mono i₃\n⊢ IsOpenImmersion (pullback.map f g f' g' i₁ i₂ i₃ e₁ e₂)",
"ppT... | [
"X Y S X' Y' S' : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nf' : X' ⟶ S'\ng' : Y' ⟶ S'\ni₁ : X ⟶ X'\ni₂ : Y ⟶ Y'\ni₃ : S ⟶ S'\ne₁ : f ≫ i₃ = i₁ ≫ f'\ne₂ : g ≫ i₃ = i₂ ≫ g'\ninst✝² : IsOpenImmersion i₁\ninst✝¹ : IsOpenImmersion i₂\ninst✝ : Mono i₃\n⊢ IsOpenImmersion\n ((pullbackFstFstIso f g f' g' i₁ i₂ i₃ e₁ e₂).inv ≫\n ... | rw [pullback_map_eq_pullbackFstFstIso_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 84,
"column": 31
} | {
"line": 101,
"column": 82
} | {
"line": 103,
"column": 0
} | [
{
"pp": "X Y : Scheme\ninst✝ : IsAffine Y\nf : X ⟶ Y\n⊢ AffineTargetMorphismProperty.diagonal (fun X x x_1 x_2 ↦ CompactSpace ↥X) f ↔ QuasiSeparatedSpace ↥X",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"AlgebraicGeometry.Shea... | [] | by
delta AffineTargetMorphismProperty.diagonal
rw [quasiSeparatedSpace_iff_forall_affineOpens]
constructor
· intro H U V
let g : pullback U.1.ι V.1.ι ⟶ X := pullback.fst _ _ ≫ U.1.ι
have e := g.isOpenEmbedding.isEmbedding.toHomeomorph
rw [IsOpenImmersion.range_pullback_to_base_of_left, Scheme.Opens.... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 10
} | {
"line": 260,
"column": 2
} | [
{
"pp": "X : 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, X.basi... | [
"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, X.basicOpe... | use n, y | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Ideal.Height | {
"line": 467,
"column": 8
} | {
"line": 467,
"column": 71
} | {
"line": 469,
"column": 0
} | [
{
"pp": "case neg.a\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nI : Ideal R\nr : ℕ\nih : ↑r ≤ I.height → ∃ J ≤ I, Submodule.spanRank J ≤ ↑r ∧ ↑r ≤ J.height\nhr : ↑(r + 1) ≤ I.height\nJ : Ideal R\nh₁ : J ≤ I\nh₂ : Submodule.spanRank J ≤ ↑r\nh₃ : ↑r ≤ J.height\nS : Set (Ideal R) := ⋯\nhS : S.F... | [] | apply hp.le <| Ideal.mem_sup_right <| mem_span_singleton_self x | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Ideal.Height | {
"line": 467,
"column": 8
} | {
"line": 467,
"column": 71
} | {
"line": 469,
"column": 0
} | [
{
"pp": "case neg.a\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nI : Ideal R\nr : ℕ\nih : ↑r ≤ I.height → ∃ J ≤ I, Submodule.spanRank J ≤ ↑r ∧ ↑r ≤ J.height\nhr : ↑(r + 1) ≤ I.height\nJ : Ideal R\nh₁ : J ≤ I\nh₂ : Submodule.spanRank J ≤ ↑r\nh₃ : ↑r ≤ J.height\nS : Set (Ideal R) := ⋯\nhS : S.F... | [] | apply hp.le <| Ideal.mem_sup_right <| mem_span_singleton_self x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Height | {
"line": 467,
"column": 8
} | {
"line": 467,
"column": 71
} | {
"line": 469,
"column": 0
} | [
{
"pp": "case neg.a\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nI : Ideal R\nr : ℕ\nih : ↑r ≤ I.height → ∃ J ≤ I, Submodule.spanRank J ≤ ↑r ∧ ↑r ≤ J.height\nhr : ↑(r + 1) ≤ I.height\nJ : Ideal R\nh₁ : J ≤ I\nh₂ : Submodule.spanRank J ≤ ↑r\nh₃ : ↑r ≤ J.height\nS : Set (Ideal R) := ⋯\nhS : S.F... | [] | apply hp.le <| Ideal.mem_sup_right <| mem_span_singleton_self x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 608,
"column": 4
} | {
"line": 608,
"column": 84
} | {
"line": 610,
"column": 0
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝ : HasRingHomProperty P Q\nhQ : RingHom.StableUnderCompositionWithLocalizationAwaySource fun {R S} [CommRing R] [CommRing S] ↦ Q\nX Y Z : Scheme\ni : Y ⟶ Z\nhi : IsOpenImmersion i\nf :... | [] | exact respects_isOpenImmersion_aux hQ _ (by rwa [P.cancel_right_of_respectsIso]) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 450,
"column": 6
} | {
"line": 450,
"column": 33
} | {
"line": 451,
"column": 2
} | [
{
"pp": "case inst.refine_2\nX Y Z : Scheme\nf✝ : X ⟶ Y\ninst✝¹ : CompactSpace ↥X\ninst✝ : QuasiSeparatedSpace ↥X\nf : ↑Γ(X, ⊤)\n⊢ IsQuasiSeparated ⊤.carrier",
"ppTerm": "?inst.refine_2✝",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",... | [] | exact isQuasiSeparated_univ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 450,
"column": 6
} | {
"line": 450,
"column": 33
} | {
"line": 451,
"column": 2
} | [
{
"pp": "case inst.refine_2\nX Y Z : Scheme\nf✝ : X ⟶ Y\ninst✝¹ : CompactSpace ↥X\ninst✝ : QuasiSeparatedSpace ↥X\nf : ↑Γ(X, ⊤)\n⊢ IsQuasiSeparated ⊤.carrier",
"ppTerm": "?inst.refine_2✝",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",... | [] | exact isQuasiSeparated_univ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 450,
"column": 6
} | {
"line": 450,
"column": 33
} | {
"line": 451,
"column": 2
} | [
{
"pp": "case inst.refine_2\nX Y Z : Scheme\nf✝ : X ⟶ Y\ninst✝¹ : CompactSpace ↥X\ninst✝ : QuasiSeparatedSpace ↥X\nf : ↑Γ(X, ⊤)\n⊢ IsQuasiSeparated ⊤.carrier",
"ppTerm": "?inst.refine_2✝",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",... | [] | exact isQuasiSeparated_univ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.IsIso | {
"line": 29,
"column": 88
} | {
"line": 32,
"column": 5
} | {
"line": 34,
"column": 0
} | [
{
"pp": "⊢ isomorphisms Scheme = IsOpenImmersion ⊓ @Surjective",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"AlgebraicGeometry.Scheme",
"CategoryTheory.IsIso",
"CategoryTheory.MorphismProperty.isomorphisms.iff",
... | [] | by
ext
rw [isomorphisms.iff, isIso_iff_isOpenImmersion_and_surjective]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.OpenImmersion | {
"line": 82,
"column": 4
} | {
"line": 85,
"column": 18
} | {
"line": 87,
"column": 0
} | [
{
"pp": "case refine_2\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : X.OpenCover\nhf : Function.Injective ⇑f\nh𝒰 : ∀ (i : 𝒰.I₀), IsOpenImmersion (𝒰.f i ≫ f)\n⊢ ∀ (x : ↥X), IsIso (Scheme.Hom.stalkMap f x)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Algebr... | [] | intro x
obtain ⟨i, x, rfl⟩ := 𝒰.exists_eq x
rw [← (IsIso.comp_inv_eq _).mpr (Scheme.Hom.stalkMap_comp (𝒰.f i) f x)]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.OpenImmersion | {
"line": 82,
"column": 4
} | {
"line": 85,
"column": 18
} | {
"line": 87,
"column": 0
} | [
{
"pp": "case refine_2\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : X.OpenCover\nhf : Function.Injective ⇑f\nh𝒰 : ∀ (i : 𝒰.I₀), IsOpenImmersion (𝒰.f i ≫ f)\n⊢ ∀ (x : ↥X), IsIso (Scheme.Hom.stalkMap f x)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Algebr... | [] | intro x
obtain ⟨i, x, rfl⟩ := 𝒰.exists_eq x
rw [← (IsIso.comp_inv_eq _).mpr (Scheme.Hom.stalkMap_comp (𝒰.f i) f x)]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Stalk | {
"line": 51,
"column": 2
} | {
"line": 52,
"column": 84
} | {
"line": 53,
"column": 2
} | [
{
"pp": "X : Scheme\nU V : X.Opens\nhU : IsAffineOpen U\nhV : IsAffineOpen V\nx : ↥X\nhxU : x ∈ U\nhxV : x ∈ V\n⊢ hU.fromSpecStalk hxU = hV.fromSpecStalk hxV",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTopologica... | [
"X : Scheme\nU V : X.Opens\nhU : IsAffineOpen U\nhV : IsAffineOpen V\nx : ↥X\nhxU : x ∈ U\nhxV : x ∈ V\nU' : Opens ↥X\nh₁ : U' ∈ X.affineOpens\nh₂ : x ∈ U'\nh₃ : U' ≤ U ⊓ V\n⊢ hU.fromSpecStalk hxU = hV.fromSpecStalk hxV"
] | obtain ⟨U', h₁, h₂, h₃ : U' ≤ U ⊓ V⟩ :=
Opens.isBasis_iff_nbhd.mp X.isBasis_affineOpens (show x ∈ U ⊓ V from ⟨hxU, hxV⟩) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 247,
"column": 72
} | {
"line": 255,
"column": 75
} | {
"line": 255,
"column": 75
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nU V W U₀ : ↑X.affineOpens\nhU₀ : ↑U ⊓ ↑W ≤ ↑U₀\n⊢ Set.range\n ⇑(pullback.fst (pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯)) ≫\n pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯) ≫ I.glueDataObjι U ≫ (↑U).ι) ⊆\n... | [] | by
simp only [Scheme.Opens.range_ι, TopologicalSpace.Opens.coe_inf, Set.subset_inter_iff]
constructor
· rw [pullback.condition_assoc (f := I.glueDataObjι U), X.homOfLE_ι,
← Category.assoc, Scheme.Hom.comp_base, TopCat.coe_comp]
exact (Set.range_comp_subset_range _ _).trans (by simp)
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 124,
"column": 49
} | {
"line": 124,
"column": 75
} | {
"line": 124,
"column": 76
} | [
{
"pp": "case refine_2\nX : Scheme\ns : Set ↑Γ(X, ⊤)\nhs : Ideal.span s = ⊤\nhs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)\nthis✝ : QuasiSeparatedSpace ↥X\nthis : CompactSpace ↥X\n⊢ Ideal.span (Set.range Subtype.val) = ⊤",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case refine_2\nX : Scheme\ns : Set ↑Γ(X, ⊤)\nhs : Ideal.span s = ⊤\nhs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)\nthis✝ : QuasiSeparatedSpace ↥X\nthis : CompactSpace ↥X\n⊢ Ideal.span {x | x ∈ s} = ⊤"
] | Subtype.range_coe_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.AffineAnd | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 98
} | {
"line": 63,
"column": 0
} | [
{
"pp": "case refine_2\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhP : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q\nX Y Z : Scheme\ne : Y ≅ Z\nf : X ⟶ Y\nhZ : IsAffine Y\nhf : affineAnd (fun {R S} [CommRing R] [CommRing S] ↦ Q) f\n⊢ (affineAnd fun {R S}... | [] | simpa [AffineTargetMorphismProperty.toProperty, IsAffine.of_isIso e.inv, hP.cancel_left_isIso] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.RingHom.EssFiniteType | {
"line": 25,
"column": 2
} | {
"line": 25,
"column": 29
} | {
"line": 26,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.EssFiniteType\nhg : g.EssFiniteType\n⊢ (g.comp f).EssFiniteType",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.EssFiniteType\nhg : g.EssFiniteType\nalgInst✝² : Algebra R S := f.toAlgebra\nalgInst✝¹ : Algebra S T := g.toAlgebra\nalgInst✝ : Algebra R T := (g.comp f).toAlgebra\nscalarTowerIn... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.RingHom.EssFiniteType | {
"line": 30,
"column": 2
} | {
"line": 30,
"column": 29
} | {
"line": 31,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.EssFiniteType\n⊢ (g.comp f).EssFiniteType ↔ g.EssFiniteType",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
"C... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.EssFiniteType\nalgInst✝² : Algebra R S := f.toAlgebra\nalgInst✝¹ : Algebra S T := g.toAlgebra\nalgInst✝ : Algebra R T := (g.comp f).toAlgebra\nscalarTowerInst✝ : IsScalarTower R ... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.RingHom.EssFiniteType | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 29
} | {
"line": 36,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nh : (g.comp f).EssFiniteType\n⊢ g.EssFiniteType",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
"CommSemiring.toSemir... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nh : (g.comp f).EssFiniteType\nalgInst✝² : Algebra R S := f.toAlgebra\nalgInst✝¹ : Algebra S T := g.toAlgebra\nalgInst✝ : Algebra R T := (g.comp f).toAlgebra\nscalarTowerInst✝ : IsScalar... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.Spectrum.Prime.Noetherian | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 89
} | {
"line": 65,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsArtinianRing R\np : Ideal R\ninst✝ : p.IsPrime\nr : R\nhr : (toPiLocalization R) r = Pi.single { asIdeal := p, isPrime := ⋯ } 1\nthis : (algebraMap R (Localization p.primeCompl)) r = 1\n⊢ r ∉ p",
"ppTerm": "?refine_1",
"assigned": tru... | [
"case refine_1\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsArtinianRing R\np : Ideal R\ninst✝ : p.IsPrime\nr : R\nhr : (toPiLocalization R) r = Pi.single { asIdeal := p, isPrime := ⋯ } 1\nthis : (algebraMap R (Localization p.primeCompl)) r = 1\n⊢ 1 ∉ IsLocalRing.maximalIdeal (Localization.AtPrime p)"
] | rw [← IsLocalization.AtPrime.to_map_mem_maximal_iff (Localization.AtPrime p) p, this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.JacobsonSpace | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 16
} | {
"line": 139,
"column": 2
} | [
{
"pp": "X : Type u_2\nY : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\ninst✝ : JacobsonSpace Y\nhf : IsOpenEmbedding f\n⊢ ∀ (Z : Set X), Z.Nonempty → IsLocallyClosed Z → (Z ∩ f ⁻¹' closedPoints Y).Nonempty",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
... | [
"X : Type u_2\nY : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\ninst✝ : JacobsonSpace Y\nhf : IsOpenEmbedding f\nZ : Set X\nhZ : Z.Nonempty\nhZ' : IsLocallyClosed Z\n⊢ (Z ∩ f ⁻¹' closedPoints Y).Nonempty"
] | intro Z hZ hZ' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.JacobsonSpace | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 16
} | {
"line": 147,
"column": 2
} | [
{
"pp": "X : Type u_2\nY : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\ninst✝ : JacobsonSpace Y\nhf : IsClosedEmbedding f\n⊢ ∀ (Z : Set X), Z.Nonempty → IsLocallyClosed Z → (Z ∩ f ⁻¹' closedPoints Y).Nonempty",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [... | [
"X : Type u_2\nY : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\ninst✝ : JacobsonSpace Y\nhf : IsClosedEmbedding f\nZ : Set X\nhZ : Z.Nonempty\nhZ' : IsLocallyClosed Z\n⊢ (Z ∩ f ⁻¹' closedPoints Y).Nonempty"
] | intro Z hZ hZ' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.JacobsonSpace | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 16
} | {
"line": 176,
"column": 2
} | [
{
"pp": "X : Type u_2\ninst✝ : TopologicalSpace X\nι : Type u_1\nU : ι → Opens X\nhU : IsOpenCover U\nH : ∀ (i : ι), JacobsonSpace ↥(U i)\n⊢ ∀ (Z : Set X), Z.Nonempty → IsLocallyClosed Z → (Z ∩ closedPoints X).Nonempty",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"IsLocallyClosed"... | [
"X : Type u_2\ninst✝ : TopologicalSpace X\nι : Type u_1\nU : ι → Opens X\nhU : IsOpenCover U\nH : ∀ (i : ι), JacobsonSpace ↥(U i)\nZ : Set X\nhZ : Z.Nonempty\nhZ' : IsLocallyClosed Z\n⊢ (Z ∩ closedPoints X).Nonempty"
] | intro Z hZ hZ' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Finiteness.FinitePresentationLocal | {
"line": 91,
"column": 6
} | {
"line": 91,
"column": 80
} | {
"line": 92,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\ns✝ : Set S\nhs✝ : Ideal.span s✝ = ⊤\ns : Finset S\nh₁ : ↑s ⊆ s✝\nhs : Ideal.span ↑s = ⊤\nh : ∀ (i : ↥s), FinitePresentation R (Localization.Away ↑i)\nhfintype : FiniteType R S\nn : ℕ\nf : MvPolynomial (Fin n) R →... | [] | simp only [Finset.coe_sort_coe, smul_eq_mul, mul_comm, sub_self, zero_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Morphisms.FinitePresentation | {
"line": 137,
"column": 6
} | {
"line": 137,
"column": 32
} | {
"line": 137,
"column": 33
} | [
{
"pp": "case e'_3\nX Y : Scheme\nf : X ⟶ Y\nhf : LocallyOfFinitePresentation f\ninst✝¹ : QuasiCompact f\ns : Set ↥X\nhs : IsLocallyConstructible s\nthis :\n ∀ {X Y : Scheme} (f : X ⟶ Y) [hf : LocallyOfFinitePresentation f] [QuasiCompact f] {s : Set ↥X},\n IsLocallyConstructible s → (∃ R, Y = Spec R) → IsLo... | [
"case e'_3\nX Y : Scheme\nf : X ⟶ Y\nhf : LocallyOfFinitePresentation f\ninst✝¹ : QuasiCompact f\ns : Set ↥X\nhs : IsLocallyConstructible s\nthis :\n ∀ {X Y : Scheme} (f : X ⟶ Y) [hf : LocallyOfFinitePresentation f] [QuasiCompact f] {s : Set ↥X},\n IsLocallyConstructible s → (∃ R, Y = Spec R) → IsLocallyConstru... | Subtype.range_coe_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 102,
"column": 48
} | {
"line": 102,
"column": 70
} | {
"line": 102,
"column": 70
} | [
{
"pp": "R : Type u_2\nA : Type u_1\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nf : A\ns : Set A\nx : PrimeSpectrum R\nq : Ideal ((A ⧸ Ideal.span s) ⊗[R] x.asIdeal.ResidueField)\nhq : q.IsPrime\nhfq : (Ideal.Quotient.mk (Ideal.span s)) f ⊗ₜ[R] 1 ∉ q\nthis : ∀ a ∈ s, (Ideal.Quotient.mk (Ideal... | [] | simpa [Set.subset_def] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 102,
"column": 48
} | {
"line": 102,
"column": 70
} | {
"line": 102,
"column": 70
} | [
{
"pp": "R : Type u_2\nA : Type u_1\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nf : A\ns : Set A\nx : PrimeSpectrum R\nq : Ideal ((A ⧸ Ideal.span s) ⊗[R] x.asIdeal.ResidueField)\nhq : q.IsPrime\nhfq : (Ideal.Quotient.mk (Ideal.span s)) f ⊗ₜ[R] 1 ∉ q\nthis : ∀ a ∈ s, (Ideal.Quotient.mk (Ideal... | [] | simpa [Set.subset_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 102,
"column": 48
} | {
"line": 102,
"column": 70
} | {
"line": 102,
"column": 70
} | [
{
"pp": "R : Type u_2\nA : Type u_1\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nf : A\ns : Set A\nx : PrimeSpectrum R\nq : Ideal ((A ⧸ Ideal.span s) ⊗[R] x.asIdeal.ResidueField)\nhq : q.IsPrime\nhfq : (Ideal.Quotient.mk (Ideal.span s)) f ⊗ₜ[R] 1 ∉ q\nthis : ∀ a ∈ s, (Ideal.Quotient.mk (Ideal... | [] | simpa [Set.subset_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 26
} | {
"line": 164,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nS : Set (Set (PrimeSpectrum R[X]))\nhS : S ⊆ Set.range fun r ↦ ↑(basicOpen r)\nhU : IsOpen (⋃₀ S)\nt : Set (PrimeSpectrum R[X])\nht : t ∈ S\n⊢ IsOpen (comap C '' t)",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Prime... | [
"R : Type u_1\ninst✝ : CommRing R\nS : Set (Set (PrimeSpectrum R[X]))\nhS : S ⊆ Set.range fun r ↦ ↑(basicOpen r)\nhU : IsOpen (⋃₀ S)\nr : R[X]\nht : (fun r ↦ ↑(basicOpen r)) r ∈ S\n⊢ IsOpen (comap C '' (fun r ↦ ↑(basicOpen r)) r)"
] | obtain ⟨r, rfl⟩ := hS ht | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 26
} | {
"line": 227,
"column": 2
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\nσ : Type u_1\nS : Set (Set (PrimeSpectrum (MvPolynomial σ R)))\nhS : S ⊆ Set.range fun r ↦ ↑(basicOpen r)\nhU : IsOpen (⋃₀ S)\nt : Set (PrimeSpectrum (MvPolynomial σ R))\nht : t ∈ S\n⊢ IsOpen (comap C '' t)",
"ppTerm": "?m.85",
"assigned": true,
"usedConsta... | [
"R : Type u_2\ninst✝ : CommRing R\nσ : Type u_1\nS : Set (Set (PrimeSpectrum (MvPolynomial σ R)))\nhS : S ⊆ Set.range fun r ↦ ↑(basicOpen r)\nhU : IsOpen (⋃₀ S)\nr : MvPolynomial σ R\nht : (fun r ↦ ↑(basicOpen r)) r ∈ S\n⊢ IsOpen (comap C '' (fun r ↦ ↑(basicOpen r)) r)"
] | obtain ⟨r, rfl⟩ := hS ht | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Artinian | {
"line": 70,
"column": 8
} | {
"line": 70,
"column": 43
} | {
"line": 70,
"column": 44
} | [
{
"pp": "X : Scheme\ninst✝ : IsLocallyNoetherian X\nh : topologicalKrullDim ↥X ≤ 0\nU : ↑X.affineOpens\nx✝ : IsNoetherianRing ↑Γ(X, ↑U)\n⊢ IsArtinianRing ↑Γ(X, ↑U)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Opposite",
"IsArtinianRing",
"CommRingCat.c... | [
"X : Scheme\ninst✝ : IsLocallyNoetherian X\nh : topologicalKrullDim ↥X ≤ 0\nU : ↑X.affineOpens\nx✝ : IsNoetherianRing ↑Γ(X, ↑U)\n⊢ Ring.KrullDimLE 0 ↑Γ(X, ↑U)"
] | isArtinianRing_iff_krullDimLE_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.HopkinsLevitzki | {
"line": 186,
"column": 6
} | {
"line": 186,
"column": 41
} | {
"line": 187,
"column": 4
} | [
{
"pp": "R : Type u_3\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\n⊢ IsArtinianRing R ↔ IsNilpotent (IsLocalRing.maximalIdeal R)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"IsArtinianRing",
"IsScalarTo... | [
"R : Type u_3\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\n⊢ Ring.KrullDimLE 0 R ↔ IsNilpotent (IsLocalRing.maximalIdeal R)"
] | isArtinianRing_iff_krullDimLE_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 64
} | {
"line": 131,
"column": 2
} | [
{
"pp": "X : Scheme\nK : Type u\ninst✝¹ : Field K\nf : X ⟶ Spec (CommRingCat.of K)\ninst✝ : UniversallyClosed f\n𝒰 : X.OpenCover := X.affineCover\nU : 𝒰.I₀ → X.Opens := fun i ↦ Scheme.Hom.opensRange (𝒰.f i)\nT : Scheme := Spec (CommRingCat.of (MvPolynomial 𝒰.I₀ K))\nq : T ⟶ Spec (CommRingCat.of K) := Spec.m... | [
"X : Scheme\nK : Type u\ninst✝¹ : Field K\nf : X ⟶ Spec (CommRingCat.of K)\ninst✝ : UniversallyClosed f\n𝒰 : X.OpenCover := X.affineCover\nU : 𝒰.I₀ → X.Opens := fun i ↦ Scheme.Hom.opensRange (𝒰.f i)\nT : Scheme := Spec (CommRingCat.of (MvPolynomial 𝒰.I₀ K))\nq : T ⟶ Spec (CommRingCat.of K) := Spec.map (CommRing... | have ht (i : 𝒰.I₀) : t ∈ Ti i := show ψ (.X i) ≠ 0 by simp [ψ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 447,
"column": 4
} | {
"line": 448,
"column": 86
} | {
"line": 449,
"column": 4
} | [
{
"pp": "n✝ : Type u\nS : Scheme\nn : Type u\nR : CommRingCat\nh : IsIntegralHom (𝔸(n; Spec R) ↘ Spec R)\nh✝ : Nonempty ↥(Spec R)\n⊢ IsEmpty n",
"ppTerm": "?m.150",
"assigned": true,
"usedConstants": [
"Nontrivial",
"AlgebraicGeometry.Spec",
"CommRingCat.carrier",
"Algebraic... | [
"n✝ : Type u\nS : Scheme\nn : Type u\nR : CommRingCat\nh : IsIntegralHom (𝔸(n; Spec R) ↘ Spec R)\nh✝ : Nonempty ↥(Spec R)\nthis : Nontrivial ↑R\n⊢ IsEmpty n"
] | have : Nontrivial R := (subsingleton_or_nontrivial R).resolve_left fun H ↦
not_isEmpty_of_nonempty (Spec R) (inferInstanceAs (IsEmpty (PrimeSpectrum R))) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 57
} | {
"line": 240,
"column": 0
} | [
{
"pp": "case H\nX Y : Scheme\nx : ↥X\ninst✝ : X.IsGermInjectiveAt x\nf g : X ⟶ Y\ne : X.fromSpecStalk x ≫ f = X.fromSpecStalk x ≫ g\n⊢ Spec.map (Scheme.Hom.stalkMap f x) ≫ Y.fromSpecStalk (f x) =\n Spec.map (Y.presheaf.stalkSpecializes ⋯ ≫ Scheme.Hom.stalkMap g x) ≫ Y.fromSpecStalk (f x)",
"ppTerm": "?H... | [] | simpa [Scheme.SpecMap_stalkSpecializes_fromSpecStalk] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 416,
"column": 4
} | {
"line": 419,
"column": 61
} | {
"line": 421,
"column": 0
} | [
{
"pp": "case mpr\nX Y S : Scheme\ninst✝¹ : X.Over S\ninst✝ : Y.Over S\nf : X ⤏ Y\n⊢ f.compHom (Y ↘ S) = Hom.toRationalMap (X ↘ S) → IsOver S f",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"AlgebraicGeometry.SheafedSpace.instTo... | [] | intro e
obtain ⟨f, rfl⟩ := PartialMap.toRationalMap_surjective f
obtain ⟨U, hU, hUl, hUr, e⟩ := PartialMap.toRationalMap_eq_iff.mp e
exact ⟨⟨f.restrict U hU hUl, by simpa using! e, by simp⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 416,
"column": 4
} | {
"line": 419,
"column": 61
} | {
"line": 421,
"column": 0
} | [
{
"pp": "case mpr\nX Y S : Scheme\ninst✝¹ : X.Over S\ninst✝ : Y.Over S\nf : X ⤏ Y\n⊢ f.compHom (Y ↘ S) = Hom.toRationalMap (X ↘ S) → IsOver S f",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"AlgebraicGeometry.SheafedSpace.instTo... | [] | intro e
obtain ⟨f, rfl⟩ := PartialMap.toRationalMap_surjective f
obtain ⟨U, hU, hUl, hUr, e⟩ := PartialMap.toRationalMap_eq_iff.mp e
exact ⟨⟨f.restrict U hU hUl, by simpa using! e, by simp⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Cover.Directed | {
"line": 128,
"column": 6
} | {
"line": 129,
"column": 71
} | {
"line": 130,
"column": 4
} | [
{
"pp": "P : MorphismProperty Scheme\nX : Scheme\n𝒰 : Cover (precoverage P) X\ninst✝¹ : Category.{v_1, u_1} 𝒰.I₀\ninst✝ : 𝒰.LocallyDirected\ni j k : 𝒰.I₀\nfi : i ⟶ k\nfj : j ⟶ k\nxi : (𝒰.functorOfLocallyDirected ⋙ forget).obj i\nxj : (𝒰.functorOfLocallyDirected ⋙ forget).obj j\nhxij : (𝒰.trans fi) xi = (... | [] | rw [← 𝒰.trans_map fi, ← 𝒰.trans_map fj, Hom.comp_base, Hom.comp_base,
ConcreteCategory.comp_apply, hxij, ConcreteCategory.comp_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.Minpoly.Finite | {
"line": 31,
"column": 2
} | {
"line": 32,
"column": 32
} | {
"line": 33,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\ninst✝ : Module.Finite A B\nx : B\n⊢ (minpoly A x).natDegree ≤ ⊤.spanFinrank",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Submodule",
"Algebra.lmul",
"instSMulOfMul",
... | [
"A : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\ninst✝ : Module.Finite A B\nx : B\nf : A[X]\nf_monic : f.Monic\nf_deg : f.natDegree = ⊤.spanFinrank\nf_aeval : (Polynomial.aeval ((Algebra.lmul A B) x)) f = 0\n⊢ (minpoly A x).natDegree ≤ ⊤.spanFinrank"
] | rcases LinearMap.exists_monic_and_natDegree_eq_and_aeval_eq_zero _ (Algebra.lmul A _ x) with
⟨f, f_monic, f_deg, f_aeval⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 446,
"column": 45
} | {
"line": 449,
"column": 74
} | {
"line": 451,
"column": 0
} | [
{
"pp": "X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\ninst✝ : Flat iX\nhUS : IsAffineOpen US\nhUX : IsAffineOpen UX\nhUT : IsCompact ... | [] | by
suffices Mono (pushoutSection H.flip hUSX hUST (hUY.trans (inf_comm _ _))) by
rw [← mono_comp_iff_of_isIso (pushoutSymmetry _ _).hom]; convert! this; cat_disch
exact mono_pushoutSection_of_isCompact_of_flat_right _ _ _ _ hUS hUX hUT | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Normal.Basic | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 61
} | {
"line": 87,
"column": 2
} | [
{
"pp": "case inr.refine_1\nF : Type u_1\ninst✝² : Field F\nE : Type u_3\ninst✝¹ : Field E\ninst✝ : Algebra F E\np : F[X]\nhFEp : IsSplittingField F E p\nhp : p ≠ 0\nx : E\nthis✝ : FiniteDimensional F E\nhx : IsIntegral F x\nL : Type u_1 := (p * minpoly F x).SplittingField\nhL1 : (Polynomial.map (algebraMap F (... | [] | · rwa [Polynomial.map_map, ← IsScalarTower.algebraMap_eq] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 288,
"column": 2
} | {
"line": 429,
"column": 11
} | {
"line": 431,
"column": 0
} | [
{
"pp": "R₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\n⊢ Statement R₀ (Localization.Away c) n\n {\n val :=\n Polynomial.C (IsLocalization.Away.invSelf ... | [] | set q₁ := IsScalarTower.toAlgHom R₀ R (Away c)
set q₂ := Ideal.Quotient.mkₐ R₀ (.span {c})
have q₂_surjective : Surjective q₂ := Ideal.Quotient.mk_surjective
set e₁ : InductionObj (Away c) n :=
⟨Polynomial.C (IsLocalization.Away.invSelf (S := Away c) c) • mapRingHom q₁ ∘ e⟩
set e₂ : InductionObj (R ⧸ Ideal.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 288,
"column": 2
} | {
"line": 429,
"column": 11
} | {
"line": 431,
"column": 0
} | [
{
"pp": "R₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\n⊢ Statement R₀ (Localization.Away c) n\n {\n val :=\n Polynomial.C (IsLocalization.Away.invSelf ... | [] | set q₁ := IsScalarTower.toAlgHom R₀ R (Away c)
set q₂ := Ideal.Quotient.mkₐ R₀ (.span {c})
have q₂_surjective : Surjective q₂ := Ideal.Quotient.mk_surjective
set e₁ : InductionObj (Away c) n :=
⟨Polynomial.C (IsLocalization.Away.invSelf (S := Away c) c) • mapRingHom q₁ ∘ e⟩
set e₂ : InductionObj (R ⧸ Ideal.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 569,
"column": 86
} | {
"line": 570,
"column": 36
} | {
"line": 572,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\n⊢ ¬M.Dep X ↔ M.Indep X",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.Dep",
"Matroid.Dep.eq_1",
"Classical.not_not",
"congrArg",
"Matroid.E",
"Iff.rfl",
"and_if... | [] | by
rw [Dep, and_iff_left hX, not_not] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 761,
"column": 4
} | {
"line": 761,
"column": 78
} | {
"line": 762,
"column": 4
} | [
{
"pp": "α : Type u_1\nM✝ : Matroid α\nB B' I✝ J D X : Set α\ne f : α\nM : Matroid α\ninst✝ : M.RankFinite\nI : Set α\nhI : ∀ J ⊆ I, J.Finite → M.Indep J\n⊢ M.Indep I",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Set.finite_or_infinite",
"False.elim",
"Set.Finite",
... | [
"α : Type u_1\nM✝ : Matroid α\nB B' I✝ J D X : Set α\ne f : α\nM : Matroid α\ninst✝ : M.RankFinite\nI : Set α\nhI : ∀ J ⊆ I, J.Finite → M.Indep J\nh : I.Infinite\n⊢ False"
] | refine I.finite_or_infinite.elim (hI _ Subset.rfl) (fun h ↦ False.elim ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 963,
"column": 2
} | {
"line": 963,
"column": 27
} | {
"line": 965,
"column": 0
} | [
{
"pp": "α : Type u_1\nX Y : Set α\nM : Matroid α\nhXY : X ⊆ Y\nhY : Y ⊆ M.E\nI : Set α\nhI : M.IsBasis I X\nJ : Set α\nhJ : M.IsBasis J Y\nhIJ : I ⊆ J\n⊢ ∃ I J, M.IsBasis I X ∧ M.IsBasis J Y ∧ I ⊆ J",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Exists",
"HasSubset.Subset",
... | [] | exact ⟨_, _, hI, hJ, hIJ⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 427,
"column": 4
} | {
"line": 427,
"column": 67
} | {
"line": 428,
"column": 4
} | [
{
"pp": "case inr\nI : Type u\ninst✝⁷ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝⁶ : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝⁵ : IsCofiltered I\ninst✝⁴ : ∀ (i : I), IsAffine (D.obj i)\ninst✝³ : IsAffine c.pt\ni j : I\nR : CommRingCat\nt : D ⟶ (Functor.const I).obj (Spec R)\ninst✝² : ... | [
"case inr\nI : Type u\ninst✝⁷ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝⁶ : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝⁵ : IsCofiltered I\ninst✝⁴ : ∀ (i : I), IsAffine (D.obj i)\ninst✝³ : IsAffine c.pt\ni j : I\nR : CommRingCat\nt : D ⟶ (Functor.const I).obj (Spec R)\ninst✝² : IsAffine (Sp... | have inst (i) : IsIso (e.app i) := by dsimp [e]; infer_instance | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 359,
"column": 8
} | {
"line": 359,
"column": 65
} | {
"line": 360,
"column": 8
} | [
{
"pp": "α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug : ∀ ⦃I J : Set α⦄, Indep I → Indep J → I.encard < J.encard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nindep_bdd : ∃ n, ∀ (I : Set α), Indep I → I.encard ≤ ↑n\nsubset_gr... | [
"α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug : ∀ ⦃I J : Set α⦄, Indep I → Indep J → I.encard < J.encard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nindep_bdd : ∃ n, ∀ (I : Set α), Indep I → I.encard ≤ ↑n\nsubset_ground : ∀ (I ... | obtain ⟨e, heB, heI, hi⟩ := indep_aug hI hBmax.prop hcard | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 367,
"column": 8
} | {
"line": 367,
"column": 47
} | {
"line": 368,
"column": 6
} | [
{
"pp": "α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug : ∀ ⦃I J : Set α⦄, Indep I → Indep J → I.encard < J.encard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nsubset_ground : ∀ (I : Set α), Indep I → I ⊆ E\nI B : Set α\nhI : ... | [] | exact finite_of_encard_le_coe (hn _ hI) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Map | {
"line": 217,
"column": 2
} | {
"line": 220,
"column": 62
} | {
"line": 222,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nN : Matroid β\nI X : Set α\n⊢ (N.comap f).IsBasis' I X ↔ N.IsBasis' (f '' I) (f '' X) ∧ InjOn f I ∧ I ⊆ X",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.Matroid.Map.0.Matroid.comap_isBasi... | [] | simp only [isBasis'_iff_isBasis_inter_ground, comap_ground_eq, comap_isBasis_iff,
image_inter_preimage, subset_inter_iff, ← and_assoc, and_iff_left_iff_imp,
and_imp]
exact fun h _ _ ↦ (image_subset_iff.1 h.indep.subset_ground) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Map | {
"line": 217,
"column": 2
} | {
"line": 220,
"column": 62
} | {
"line": 222,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nN : Matroid β\nI X : Set α\n⊢ (N.comap f).IsBasis' I X ↔ N.IsBasis' (f '' I) (f '' X) ∧ InjOn f I ∧ I ⊆ X",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.Matroid.Map.0.Matroid.comap_isBasi... | [] | simp only [isBasis'_iff_isBasis_inter_ground, comap_ground_eq, comap_isBasis_iff,
image_inter_preimage, subset_inter_iff, ← and_assoc, and_iff_left_iff_imp,
and_imp]
exact fun h _ _ ↦ (image_subset_iff.1 h.indep.subset_ground) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Map | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 41
} | {
"line": 253,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nN : Matroid β\nf : α → β\n⊢ N.comapOn (f ⁻¹' N.E) f = N.comap f",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.restrict_eq_self_iff",
"Matroid.comapOn.eq_1",
"Matroid.comapOn",
"congrArg",
"Matroid.... | [] | rw [comapOn, restrict_eq_self_iff]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Map | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 41
} | {
"line": 253,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nN : Matroid β\nf : α → β\n⊢ N.comapOn (f ⁻¹' N.E) f = N.comap f",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.restrict_eq_self_iff",
"Matroid.comapOn.eq_1",
"Matroid.comapOn",
"congrArg",
"Matroid.... | [] | rw [comapOn, restrict_eq_self_iff]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Map | {
"line": 435,
"column": 8
} | {
"line": 435,
"column": 33
} | {
"line": 435,
"column": 33
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nf : α → β\nM : Matroid α\nhf : InjOn f M.E\nI : Set α\nhI : I ⊆ M.E\nX : Set α\nhX : X ⊆ M.E\nh : (M.map f hf).IsBasis (f '' I) (f '' X)\n⊢ ∃ I₀ X₀, M.IsBasis I₀ X₀ ∧ f '' I = f '' I₀ ∧ f '' X = f '' X₀",
"ppTerm": "?refine_1",
"assigned": true,
"u... | [
"case refine_1\nα : Type u_1\nβ : Type u_2\nf : α → β\nM : Matroid α\nhf : InjOn f M.E\nI : Set α\nhI : I ⊆ M.E\nX : Set α\nhX : X ⊆ M.E\nh : M.IsBasis I X\n⊢ ∃ I₀ X₀, M.IsBasis I₀ X₀ ∧ f '' I = f '' I₀ ∧ f '' X = f '' X₀"
] | map_isBasis_iff _ _ hI hX | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Map | {
"line": 673,
"column": 63
} | {
"line": 676,
"column": 88
} | {
"line": 678,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Set α\nM N : Matroid α\nhM : M.E = E\nhN : N.E = E\nh : M.restrictSubtype E = N.restrictSubtype E\n⊢ M = N",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Iff.rfl",
"Membership.mem",
"Matroid.... | [] | by
subst hM
refine ext_indep (by rw [hN]) (fun I hI ↦ ?_)
rwa [← restrictSubtype_indep_iff_of_subset hI, h, restrictSubtype_indep_iff_of_subset] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 592,
"column": 2
} | {
"line": 592,
"column": 84
} | {
"line": 593,
"column": 2
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nX I J : Set α\nhI : M.IsBasis I X\nhJI : J ⊆ I\nhJ : X ⊆ M.closure J\n⊢ J = I",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Set.instInter",
"Inter.inter",
"Matroid.Indep.closure_inter_eq... | [
"α : Type u_2\nM : Matroid α\nX I J : Set α\nhI : M.IsBasis I X\nhJI : J ⊆ I\nhJ : X ⊆ M.closure J\n⊢ I ⊆ M.closure J"
] | rw [← hI.indep.closure_inter_eq_self_of_subset hJI, inter_eq_self_of_subset_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 626,
"column": 46
} | {
"line": 626,
"column": 68
} | {
"line": 626,
"column": 69
} | [
{
"pp": "k₀ k : ℕ\nD : ℕ → ℕ\nn : ℕ\n⊢ numBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) (n + 1) ^ numBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) (n + 1) *\n degBound k₀ (fun t ↦ Nat.casesOn t k D) (n + 1) =\n (k₀ * k) ^ (k₀ * k) * degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) (n + 1)",
"ppTerm": "?m.133",
... | [
"k₀ k : ℕ\nD : ℕ → ℕ\nn : ℕ\n⊢ numBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) (n + 1) ^ numBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) (n + 1) *\n ((k₀ * k) ^ (k₀ * k) * degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) n) =\n (k₀ * k) ^ (k₀ * k) * degBound (k₀ * k) ((k₀ * k) ^ (k₀ * k) • D) (n + 1)"
] | degBound_casesOn_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 708,
"column": 6
} | {
"line": 708,
"column": 39
} | {
"line": 708,
"column": 40
} | [
{
"pp": "case refine_2.inr\nα : Type u_2\nM : Matroid α\nI J : Set α\nhI : M.Indep I\nhJ : M.Indep J\nh : ∀ e ∈ J \\ I, e ∉ M.closure (I ∪ J \\ {e})\nK : Set α\nhKIJ : M.IsBasis K (I ∪ J)\nhK : I ⊆ K\nhssu : K ⊂ I ∪ J\ne : α\nheI : e ∈ I ∪ J\nheK : e ∉ K\nheJI : e ∈ J \\ I\n⊢ e ∈ M.closure (I ∪ J \\ {e})",
... | [
"case refine_2.inr\nα : Type u_2\nM : Matroid α\nI J : Set α\nhI : M.Indep I\nhJ : M.Indep J\nh : ∀ e ∈ J \\ I, e ∉ M.closure (I ∪ J \\ {e})\nK : Set α\nhKIJ : M.IsBasis K (I ∪ J)\nhK : I ⊆ K\nhssu : K ⊂ I ∪ J\ne : α\nheI : e ∈ I ∪ J\nheK : e ∉ K\nheJI : e ∈ J \\ I\n⊢ e ∈ M.closure (I \\ {e} ∪ J \\ {e})"
] | ← sdiff_singleton_eq_self heJI.2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 454,
"column": 4
} | {
"line": 462,
"column": 14
} | {
"line": 463,
"column": 2
} | [
{
"pp": "case refine_5\nα : Type u_1\nM : Matroid α\nι : Type u_2\nC₀ : Set α\nhC₀ : M.IsCircuit C₀\nx : ι → α\nC : ι → Set α\nz : α\nhC : ∀ (i : ι), M.IsCircuit (C i)\nh_mem_C₀ : ∀ (i : ι), x i ∈ C₀\nh_mem : ∀ (i : ι), x i ∈ C i\nh_unique : ∀ ⦃i i' : ι⦄, x i ∈ C i' → i = i'\nhzC₀ : z ∈ C₀\nhzC : ∀ (i : ι), z ∉... | [] | obtain ⟨C', hC'ss, hC', hzC'⟩ := hwin
refine ⟨C', hC'ss.trans ?_, hC', hzC'⟩
refine union_subset (sdiff_subset_sdiff_left subset_union_left)
(iUnion_subset fun i ↦ subset_sdiff.2
⟨sdiff_subset.trans (subset_union_of_subset_right (subset_iUnion ..) _), ?_⟩)
rw [disjoint_iff_forall_ne]
rintr... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 454,
"column": 4
} | {
"line": 462,
"column": 14
} | {
"line": 463,
"column": 2
} | [
{
"pp": "case refine_5\nα : Type u_1\nM : Matroid α\nι : Type u_2\nC₀ : Set α\nhC₀ : M.IsCircuit C₀\nx : ι → α\nC : ι → Set α\nz : α\nhC : ∀ (i : ι), M.IsCircuit (C i)\nh_mem_C₀ : ∀ (i : ι), x i ∈ C₀\nh_mem : ∀ (i : ι), x i ∈ C i\nh_unique : ∀ ⦃i i' : ι⦄, x i ∈ C i' → i = i'\nhzC₀ : z ∈ C₀\nhzC : ∀ (i : ι), z ∉... | [] | obtain ⟨C', hC'ss, hC', hzC'⟩ := hwin
refine ⟨C', hC'ss.trans ?_, hC', hzC'⟩
refine union_subset (sdiff_subset_sdiff_left subset_union_left)
(iUnion_subset fun i ↦ subset_sdiff.2
⟨sdiff_subset.trans (subset_union_of_subset_right (subset_iUnion ..) _), ?_⟩)
rw [disjoint_iff_forall_ne]
rintr... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 887,
"column": 85
} | {
"line": 888,
"column": 47
} | {
"line": 890,
"column": 0
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nS : Set α\nhS : M.Spanning S\n⊢ ∃ B, M.IsBase B ∧ B ⊆ S",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"congrArg",
"Matroid.spanning_iff_exists_isBase_subset",
"Exists",
"Matroid.IsBase",
"Eq.mp",
"HasSubset.Subset",... | [] | by
rwa [spanning_iff_exists_isBase_subset] at hS | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 699,
"column": 4
} | {
"line": 699,
"column": 38
} | {
"line": 700,
"column": 4
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\nn✝ n : ℕ\nIH :\n ∀ {M : Submodule ℤ R},\n 1 ∈ M →\n ∀ (k : ℕ) (d : Multiset (Fin n)) (S : ConstructibleSetData (MvPolynomial (Fin n) R)),\n (∀ C ∈ S, C.n ≤ k) →\n (∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin n) M ⊓ Submodule.restrictScalars... | [
"R : Type u_2\ninst✝ : CommRing R\nn✝ n : ℕ\nIH :\n ∀ {M : Submodule ℤ R},\n 1 ∈ M →\n ∀ (k : ℕ) (d : Multiset (Fin n)) (S : ConstructibleSetData (MvPolynomial (Fin n) R)),\n (∀ C ∈ S, C.n ≤ k) →\n (∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin n) M ⊓ Submodule.restrictScalars ℤ (degreesL... | apply Finset.sup_le fun x hxS ↦ ?_ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 1033,
"column": 4
} | {
"line": 1034,
"column": 79
} | {
"line": 1036,
"column": 2
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nM : Matroid α\nf : α → β\nhf : InjOn f M.E\nX : Set β\naux : ∀ ⦃I : Set α⦄, M.Indep I → (M.map f hf).closure (f '' I) = f '' M.closure I\nI : Set α\nhI : M.IsBasis I (f ⁻¹' X ∩ M.E)\n⊢ (M.map f hf).closure X = f '' M.closure (f ⁻¹' X)",
"ppTerm": "?m.41",
"assigned":... | [] | rw [← closure_inter_ground, map_ground, ← M.closure_inter_ground, ← hI.closure_eq_closure,
← aux hI.indep, ← image_preimage_inter, ← (hI.map hf).closure_eq_closure] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 302,
"column": 2
} | {
"line": 302,
"column": 49
} | {
"line": 304,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\ne : α\n⊢ M.IsNonloop e ↔ e ∈ M.E \\ M.loops",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.IsLoop.eq_1",
"congrArg",
"Matroid.E",
"Iff.rfl",
"Membership.mem",
"id",
"Matroid.IsNonloop",... | [] | rw [isNonloop_iff, IsLoop, and_comm, mem_sdiff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 302,
"column": 2
} | {
"line": 302,
"column": 49
} | {
"line": 304,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\ne : α\n⊢ M.IsNonloop e ↔ e ∈ M.E \\ M.loops",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.IsLoop.eq_1",
"congrArg",
"Matroid.E",
"Iff.rfl",
"Membership.mem",
"id",
"Matroid.IsNonloop",... | [] | rw [isNonloop_iff, IsLoop, and_comm, mem_sdiff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 302,
"column": 2
} | {
"line": 302,
"column": 49
} | {
"line": 304,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\ne : α\n⊢ M.IsNonloop e ↔ e ∈ M.E \\ M.loops",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.IsLoop.eq_1",
"congrArg",
"Matroid.E",
"Iff.rfl",
"Membership.mem",
"id",
"Matroid.IsNonloop",... | [] | rw [isNonloop_iff, IsLoop, and_comm, mem_sdiff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 668,
"column": 66
} | {
"line": 670,
"column": 42
} | {
"line": 672,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nC K : Set α\nhC : M.IsCircuit C\nhK : M.IsCocircuit K\n⊢ Disjoint C K ∨ (C ∩ K).Nontrivial",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"CompleteBooleanAlgebra.toCompleteD... | [] | by
rw [or_iff_not_imp_left, disjoint_iff_inter_eq_empty, ← ne_eq, ← nonempty_iff_ne_empty]
exact hC.isCocircuit_inter_nontrivial hK | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 681,
"column": 39
} | {
"line": 681,
"column": 52
} | {
"line": 681,
"column": 52
} | [
{
"pp": "α : Type u_1\nM : Matroid α\ninst✝ : M✶.RankPos\n⊢ M✶.RankPos",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"inst✝"
],
"usedGoals": []
}
] | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 258,
"column": 18
} | {
"line": 258,
"column": 40
} | {
"line": 258,
"column": 40
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX I : Set α\nhI : M.IsBasis I (X ∩ M.E)\nh : I = ∅\n⊢ X ∩ M.E ⊆ M.loops",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Matroid.empty_isBasis_iff._simp_1",
"congrArg",
"Matroid.E",
"Eq.mp",
"HasSubset.Subset",
"Set.... | [] | by simpa [h] using! hI | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 364,
"column": 28
} | {
"line": 364,
"column": 41
} | {
"line": 366,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\ninst✝ : M.RankInfinite\n⊢ M.RankInfinite",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"inst✝"
],
"usedGoals": []
}
] | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 631,
"column": 2
} | {
"line": 631,
"column": 47
} | {
"line": 632,
"column": 2
} | [
{
"pp": "α : Type u_1\nX Y : Set α\nhXY : X ⊆ Y\n⊢ (freeOn Y).eRk X = X.encard",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Set.encard",
"Exists",
"Exists.casesOn",
"ENat",
"Matroid.eRk",
"Matroid.exists_isBasis",
"Matroid.IsBasis",
"Eq",
... | [
"α : Type u_1\nX Y : Set α\nhXY : X ⊆ Y\nI : Set α\nhI : (freeOn Y).IsBasis I X\n⊢ (freeOn Y).eRk X = X.encard"
] | obtain ⟨I, hI⟩ := (freeOn Y).exists_isBasis X | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 648,
"column": 4
} | {
"line": 648,
"column": 37
} | {
"line": 648,
"column": 38
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ (B ∩ X).encard + (M.E \\ B).encard = (M.E \\ B ∩ (M.E \\ X)).encard + X.encard",
"ppTerm": "?m.122",
"assign... | [
"α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\nhd : M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)\n⊢ (B ∩ X ∪ M.E \\ B).encard = (M.E \\ B ∩ (M.E \\ X)).encard + X.encard"
] | ← encard_union_eq (by tauto_set), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Rank.Cardinal | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 83
} | {
"line": 127,
"column": 2
} | [
{
"pp": "α : Type u\nM : Matroid α\nX Y : Set α\nhXY : X ⊆ Y\nI : Set α\nhIX : M.IsBasis' I X\n⊢ #↑I ≤ M.cRk Y",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Cardinal",
"Matroid.IsBasis'",
"Cardinal.mk",
"Exists",
"HasSubset.Subset.trans",
"Set.Elem",
... | [
"α : Type u\nM : Matroid α\nX Y : Set α\nhXY : X ⊆ Y\nI : Set α\nhIX : M.IsBasis' I X\nJ : Set α\nhJ : M.IsBasis' J Y\nhIJ : I ⊆ J\n⊢ #↑I ≤ M.cRk Y"
] | obtain ⟨J, hJ, hIJ⟩ := hIX.indep.subset_isBasis'_of_subset (hIX.subset.trans hXY) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 886,
"column": 98
} | {
"line": 888,
"column": 93
} | {
"line": 890,
"column": 0
} | [
{
"pp": "α : Type u_1\nM N : Matroid α\nhNM : N ≤r M\ninst✝ : N.Loopless\n⊢ N ≤r M.removeLoops",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Matroid.IsRestriction.of_subset",
"Matroid.IsRestriction.exists_eq_restrict",
"Matroid.E",
"setOf",
"Membership.mem",... | [] | by
obtain ⟨R, hR, rfl⟩ := hNM.exists_eq_restrict
exact IsRestriction.of_subset M fun e heR ↦ ((M ↾ R).isNonloop_of_loopless heR).of_restrict | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.AlgebraicIndependent.Transcendental | {
"line": 188,
"column": 12
} | {
"line": 188,
"column": 96
} | {
"line": 189,
"column": 2
} | [
{
"pp": "case e'_1\nι : Type u_1\nR : Type u_3\nA : Type v\nx : ι → A\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : Set ι\n⊢ AlgebraicIndependent R (Sum.elim (fun i ↦ x ↑i) fun i ↦ x ↑i) ↔ AlgebraicIndependent R x",
"ppTerm": "?e'_1",
"assigned": true,
"usedConstants": [
... | [
"case e'_1\nι : Type u_1\nR : Type u_3\nA : Type v\nx : ι → A\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : Set ι\n⊢ x ∘ ⇑((Equiv.sumComm ↑sᶜ ↑s).trans (Equiv.Set.sumCompl s)) = Sum.elim (fun i ↦ x ↑i) fun i ↦ x ↑i"
] | apply algebraicIndependent_equiv' ((Equiv.sumComm ..).trans (Equiv.Set.sumCompl ..)) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.AlgebraicIndependent.Transcendental | {
"line": 188,
"column": 12
} | {
"line": 188,
"column": 96
} | {
"line": 189,
"column": 2
} | [
{
"pp": "case e'_1\nι : Type u_1\nR : Type u_3\nA : Type v\nx : ι → A\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : Set ι\n⊢ AlgebraicIndependent R (Sum.elim (fun i ↦ x ↑i) fun i ↦ x ↑i) ↔ AlgebraicIndependent R x",
"ppTerm": "?e'_1",
"assigned": true,
"usedConstants": [
... | [
"case e'_1\nι : Type u_1\nR : Type u_3\nA : Type v\nx : ι → A\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : Set ι\n⊢ x ∘ ⇑((Equiv.sumComm ↑sᶜ ↑s).trans (Equiv.Set.sumCompl s)) = Sum.elim (fun i ↦ x ↑i) fun i ↦ x ↑i"
] | apply algebraicIndependent_equiv' ((Equiv.sumComm ..).trans (Equiv.Set.sumCompl ..)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AlgebraicIndependent.Transcendental | {
"line": 188,
"column": 12
} | {
"line": 188,
"column": 96
} | {
"line": 189,
"column": 2
} | [
{
"pp": "case e'_1\nι : Type u_1\nR : Type u_3\nA : Type v\nx : ι → A\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : Set ι\n⊢ AlgebraicIndependent R (Sum.elim (fun i ↦ x ↑i) fun i ↦ x ↑i) ↔ AlgebraicIndependent R x",
"ppTerm": "?e'_1",
"assigned": true,
"usedConstants": [
... | [
"case e'_1\nι : Type u_1\nR : Type u_3\nA : Type v\nx : ι → A\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : Set ι\n⊢ x ∘ ⇑((Equiv.sumComm ↑sᶜ ↑s).trans (Equiv.Set.sumCompl s)) = Sum.elim (fun i ↦ x ↑i) fun i ↦ x ↑i"
] | apply algebraicIndependent_equiv' ((Equiv.sumComm ..).trans (Equiv.Set.sumCompl ..)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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