module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 699,
"column": 4
} | {
"line": 699,
"column": 12
} | [
{
"pp": "case add\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Monoid G\ninst✝ : MulSemiringAction G k\ng h : SkewMonoidAlgebra k G\nf✝ g✝ : SkewMonoidAlgebra k G\na✝¹ : f✝ * g * h = f✝ * (g * h)\na✝ : g✝ * g * h = g✝ * (g * h)\n⊢ (f✝ + g✝) * g * h = (f✝ + g✝) * (g * h)",
"usedConstants": [
... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 882,
"column": 42
} | {
"line": 882,
"column": 50
} | [
{
"pp": "case pos\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : SMulZeroClass G k\nf g : SkewMonoidAlgebra k G\nx : G\ns : Finset (G × G)\nhs : ∀ {p : G × G}, p ∈ s ↔ p.1 * p.2 = x\nF : G × G → k := fun p ↦ if p.1 * p.2 = x then f.coeff p.1 * p.1 • g.coeff p.2 else 0\np : G × G\nhps ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 882,
"column": 42
} | {
"line": 882,
"column": 50
} | [
{
"pp": "case neg\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : SMulZeroClass G k\nf g : SkewMonoidAlgebra k G\nx : G\ns : Finset (G × G)\nhs : ∀ {p : G × G}, p ∈ s ↔ p.1 * p.2 = x\nF : G × G → k := fun p ↦ if p.1 * p.2 = x then f.coeff p.1 * p.1 • g.coeff p.2 else 0\np : G × G\nhps ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 907,
"column": 42
} | {
"line": 907,
"column": 50
} | [
{
"pp": "case pos\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : SMulZeroClass G k\nf g : SkewMonoidAlgebra k G\nx : G\nthis : ({p | p.1 * p.2 = x} ∩ Function.support fun p ↦ f.coeff p.1 * p.1 • g.coeff p.2).Finite\ns : Finset (G × G) := this.toFinset\nF : G × G → k := fun p ↦ if p.1 ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 907,
"column": 42
} | {
"line": 907,
"column": 50
} | [
{
"pp": "case neg\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : SMulZeroClass G k\nf g : SkewMonoidAlgebra k G\nx : G\nthis : ({p | p.1 * p.2 = x} ∩ Function.support fun p ↦ f.coeff p.1 * p.1 • g.coeff p.2).Finite\ns : Finset (G × G) := this.toFinset\nF : G × G → k := fun p ↦ if p.1 ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Star.LinearMap | {
"line": 251,
"column": 59
} | {
"line": 252,
"column": 100
} | [
{
"pp": "R : Type u_4\nm : Type u_5\nn : Type u_6\ninst✝³ : CommSemiring R\ninst✝² : StarRing R\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nA : Matrix n m R\n⊢ star (toConv (toLin' A)) = toConv (toLin' (A.map star))",
"usedConstants": [
"Pi.Function.module",
"WithConv.toConv",
"Algebra.to_... | by
simp [← LinearMap.toMatrix'.injective.eq_iff, LinearMap.toMatrix'_intrinsicStar, WithConv.ext_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Vertex.HVertexOperator | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 65
} | [
{
"pp": "Γ✝ : Type u_1\ninst✝¹⁴ : PartialOrder Γ✝\nR✝ : Type u_2\nV✝ : Type u_3\nW✝ : Type u_4\ninst✝¹³ : CommRing R✝\ninst✝¹² : AddCommGroup V✝\ninst✝¹¹ : Module R✝ V✝\ninst✝¹⁰ : AddCommGroup W✝\ninst✝⁹ : Module R✝ W✝\nΓ : Type u_5\nΓ' : Type u_6\ninst✝⁸ : PartialOrder Γ\ninst✝⁷ : PartialOrder Γ'\nR : Type u_7... | refine Set.IsPWO.mono (((of R).symm (B u)).isPWO_support') ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.RingedSpace.Basic | {
"line": 206,
"column": 31
} | {
"line": 206,
"column": 39
} | [
{
"pp": "case succ\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj (op U))\nn : ℕ\nhn : 0 < n + Nat.succ 0 → X.basicOpen (f ^ (n + Nat.succ 0)) = X.basicOpen f\nh : 0 < n + 1 + Nat.succ 0\n⊢ X.basicOpen (f ^ (n + 1) * f ^ Nat.succ 0) = X.basicOpen f",
"usedConstants": [
"IsRigh... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.RingedSpace.PresheafedSpace | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 38
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : PresheafedSpace C\nbase : ↑X ⟶ ↑Y\nc c' : Y.presheaf ⟶ (Presheaf.pushforward C base).obj X.presheaf\nh : c ≫ whiskerRight (𝟙 (Opens.map base).op) X.presheaf = c'\n⊢ { base := base, c := c } = { base := base, c := c' }",
"usedConstants": [
"C... | erw [whiskerRight_id', comp_id] at h | Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_tacticErw____1 | Lean.Parser.Tactic.tacticErw___ |
Mathlib.Geometry.RingedSpace.PresheafedSpace | {
"line": 242,
"column": 53
} | {
"line": 242,
"column": 63
} | [
{
"pp": "case w\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : PresheafedSpace C\nH : X ≅ Y\nU : Opens ↑↑Y\neq₁ : (H.hom ≫ H.inv).c.app (op ((Opens.map H.hom.base).obj U)) = eqToHom ⋯\neq₂ :\n Y.presheaf.map (eqToHom ⋯).op ≫ H.hom.c.app (op ((Opens.map ((forget C).map (𝟙 Y))).obj U)) =\n H.hom.c.app (... | comp_c_app | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Scheme | {
"line": 865,
"column": 2
} | {
"line": 866,
"column": 57
} | [
{
"pp": "R : CommRingCat\nf : ↑Γ(Spec R, ⊤)\n⊢ (Spec R).basicOpen f = PrimeSpectrum.basicOpen ((ConcreteCategory.hom (Scheme.ΓSpecIso R).hom) f)",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeo... | convert! basicOpen_eq_of_affine ((Scheme.ΓSpecIso R).hom f)
exact (Iso.hom_inv_id_apply (Scheme.ΓSpecIso R) f).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Scheme | {
"line": 865,
"column": 2
} | {
"line": 866,
"column": 57
} | [
{
"pp": "R : CommRingCat\nf : ↑Γ(Spec R, ⊤)\n⊢ (Spec R).basicOpen f = PrimeSpectrum.basicOpen ((ConcreteCategory.hom (Scheme.ΓSpecIso R).hom) f)",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeo... | convert! basicOpen_eq_of_affine ((Scheme.ΓSpecIso R).hom f)
exact (Iso.hom_inv_id_apply (Scheme.ΓSpecIso R) f).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 56,
"column": 4
} | {
"line": 62,
"column": 35
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : LocallyRingedSpace\nh : ∀ (x : ↑X.toTopCat), ∃ R f, x ∈ Set.range ⇑(ConcreteCategory.hom f.base) ∧ IsOpenImmersion f\n⊢ ∀ (x : ↑X.toTopCat), ∃ U R, Nonempty (X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R))",
"usedConstants": [
"Opposite",
... | intro x
obtain ⟨R, f, h₁, h₂⟩ := h x
refine ⟨⟨⟨_, h₂.base_open.isOpen_range⟩, h₁⟩, R, ⟨?_⟩⟩
apply LocallyRingedSpace.isoOfSheafedSpaceIso
refine SheafedSpace.forgetToPresheafedSpace.preimageIso ?_
apply PresheafedSpace.IsOpenImmersion.isoOfRangeEq (PresheafedSpace.ofRestrict _ _) f.1
exact Subty... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 56,
"column": 4
} | {
"line": 62,
"column": 35
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : LocallyRingedSpace\nh : ∀ (x : ↑X.toTopCat), ∃ R f, x ∈ Set.range ⇑(ConcreteCategory.hom f.base) ∧ IsOpenImmersion f\n⊢ ∀ (x : ↑X.toTopCat), ∃ U R, Nonempty (X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R))",
"usedConstants": [
"Opposite",
... | intro x
obtain ⟨R, f, h₁, h₂⟩ := h x
refine ⟨⟨⟨_, h₂.base_open.isOpen_range⟩, h₁⟩, R, ⟨?_⟩⟩
apply LocallyRingedSpace.isoOfSheafedSpaceIso
refine SheafedSpace.forgetToPresheafedSpace.preimageIso ?_
apply PresheafedSpace.IsOpenImmersion.isoOfRangeEq (PresheafedSpace.ofRestrict _ _) f.1
exact Subty... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 186,
"column": 2
} | {
"line": 186,
"column": 86
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nH : IsOpenImmersion f\nV : Y.Opens\nhV : V ≤ opensRange f\n⊢ IsIso (app f V)",
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"CategoryTheory.IsIso",
... | rw [show V = f ''ᵁ f ⁻¹ᵁ V from Opens.ext (Set.image_preimage_eq_of_subset hV).symm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 126,
"column": 6
} | {
"line": 126,
"column": 57
} | [
{
"pp": "case app.op.mk.e_unop.e_carrier\nC : Type u\ninst✝ : Category.{v, u} C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nH : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\ncarrier✝ : Set ↑↑(Y.restrict ⋯)\nis_open'✝ : IsOpen carrier✝\n⊢ ⇑(ConcreteCategory.hom f.base) ... | erw [Set.preimage_image_eq _ H.base_open.injective] | Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_tacticErw____1 | Lean.Parser.Tactic.tacticErw___ |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 128,
"column": 4
} | {
"line": 133,
"column": 13
} | [
{
"pp": "case naturality\nC : Type u\ninst✝ : Category.{v, u} C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nH : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\n⊢ autoParam\n (∀ {X_1 Y_1 : (Opens ↑↑(Y.restrict ⋯))ᵒᵖ} (f_1 : X_1 ⟶ Y_1),\n (Y.restrict ⋯).presheaf.ma... | · intro U V i
dsimp
simp only [NatTrans.naturality_assoc, TopCat.Presheaf.pushforward_obj_obj,
TopCat.Presheaf.pushforward_obj_map, Quiver.Hom.unop_op, Category.assoc]
rw [← X.presheaf.map_comp, ← X.presheaf.map_comp]
congr 1 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 140,
"column": 2
} | {
"line": 141,
"column": 29
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nH : IsOpenImmersion f\nx : (Opens ↑↑Y)ᵒᵖ\n⊢ (((isoRestrict f).hom ≫ Y.ofRestrict ⋯).c ≫ Functor.whiskerRight (eqToHom ⋯) X.presheaf).app x = f.c.app x",
"usedConstants": [
"CategoryTheory.Functor.op",
"Eq.mpr",
... | simp only [eqToHom_refl,
Functor.whiskerRight_id'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Cover.MorphismProperty | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 13
} | [
{
"pp": "case right\nK : Precoverage Scheme\nX✝ Y✝ Z : Scheme\n𝒰 : Cover K X✝\nf : X✝ ⟶ Z\ng✝ : Y✝ ⟶ Z\ninst✝ : ∀ (x : 𝒰.I₀), HasPullback (𝒰.f x ≫ f) g✝\nP Q : MorphismProperty Scheme\nX : Scheme\nR : Presieve X\nx✝ : R ∈ (precoverage P).coverings X\nx : ↥X\nhR : ∀ (x : forget.obj X), ∃ Y f, R f ∧ x ∈ Set.ra... | exact heq | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Cover.MorphismProperty | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 38
} | [
{
"pp": "K : Precoverage Scheme\nX Y Z : Scheme\n𝒰✝ : Cover K X\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : ∀ (x : 𝒰✝.I₀), HasPullback (𝒰✝.f x ≫ f) g\nP Q : MorphismProperty Scheme\ninst✝ : JointlySurjective K\n𝒰 : Cover K X\nh : ∀ (j : 𝒰.I₀), Q (𝒰.f j)\n⊢ 𝒰.presieve₀ ∈ (precoverage Q).coverings X",
"usedConstan... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Cover.MorphismProperty | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 38
} | [
{
"pp": "K : Precoverage Scheme\nX✝ Y Z : Scheme\n𝒰✝ : Cover K X✝\nf : X✝ ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : ∀ (x : 𝒰✝.I₀), HasPullback (𝒰✝.f x ≫ f) g\nP Q : MorphismProperty Scheme\ninst✝ : P.RespectsIso\nX : Scheme\n𝒰 : Cover (precoverage P) X\nJ : Type u_1\nobj : J → Scheme\nmap : (i : J) → obj i ⟶ X\ne₁ : J ≃ 𝒰.... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 683,
"column": 2
} | {
"line": 683,
"column": 10
} | [
{
"pp": "X Y U : Scheme\nf : Y ⟶ U\ng : U ⟶ X\nfg : Y ⟶ X\nH : fg = f ≫ g\nh : IsOpenImmersion g\nV : U.Opens\n⊢ f ⁻¹ᵁ V = fg ⁻¹ᵁ g ''ᵁ V",
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"AlgebraicGeometry.PresheafedSpace.Hom",
"CategoryTheory.Functor",
"AlgebraicGeo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 683,
"column": 2
} | {
"line": 683,
"column": 10
} | [
{
"pp": "X Y U : Scheme\nf : Y ⟶ U\ng : U ⟶ X\nfg : Y ⟶ X\nH : fg = f ≫ g\nh : IsOpenImmersion g\nV : U.Opens\n⊢ f ⁻¹ᵁ V = fg ⁻¹ᵁ g ''ᵁ V",
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"AlgebraicGeometry.PresheafedSpace.Hom",
"CategoryTheory.Functor",
"AlgebraicGeo... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 683,
"column": 2
} | {
"line": 683,
"column": 10
} | [
{
"pp": "X Y U : Scheme\nf : Y ⟶ U\ng : U ⟶ X\nfg : Y ⟶ X\nH : fg = f ≫ g\nh : IsOpenImmersion g\nV : U.Opens\n⊢ f ⁻¹ᵁ V = fg ⁻¹ᵁ g ''ᵁ V",
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"AlgebraicGeometry.PresheafedSpace.Hom",
"CategoryTheory.Functor",
"AlgebraicGeo... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 682,
"column": 34
} | {
"line": 683,
"column": 10
} | [
{
"pp": "X Y U : Scheme\nf : Y ⟶ U\ng : U ⟶ X\nfg : Y ⟶ X\nH : fg = f ≫ g\nh : IsOpenImmersion g\nV : U.Opens\n⊢ f ⁻¹ᵁ V = fg ⁻¹ᵁ g ''ᵁ V",
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"AlgebraicGeometry.PresheafedSpace.Hom",
"CategoryTheory.Functor",
"AlgebraicGeo... | by
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Cover.MorphismProperty | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 38
} | [
{
"pp": "K : Precoverage Scheme\nX✝ Y✝ Z : Scheme\n𝒰✝ : Cover K X✝\nf✝ : X✝ ⟶ Z\ng : Y✝ ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.I₀), HasPullback (𝒰✝.f x ≫ f✝) g\nP Q : MorphismProperty Scheme\nX Y : Scheme\n𝒰 : Cover (precoverage P) X\nf : Y ⟶ X\nhf : P f\n⊢ (𝒰.add f).presieve₀ ∈ (precoverage P).coverings X",
"usedCons... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Cover.MorphismProperty | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 38
} | [
{
"pp": "K : Precoverage Scheme\nX✝ Y Z : Scheme\n𝒰✝ : Cover K X✝\nf : X✝ ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.I₀), HasPullback (𝒰✝.f x ≫ f) g\nP Q : MorphismProperty Scheme\nX : Scheme\n𝒰 : AffineCover P X\n⊢ { I₀ := 𝒰.I₀, X := fun j ↦ Spec (𝒰.X j), f := 𝒰.f }.presieve₀ ∈ (precoverage P).coverings X",
... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Cover.MorphismProperty | {
"line": 226,
"column": 4
} | {
"line": 226,
"column": 38
} | [
{
"pp": "K : Precoverage Scheme\nX Y Z : Scheme\n𝒰✝ : Cover K X\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.I₀), HasPullback (𝒰✝.f x ≫ f) g\nP Q : MorphismProperty Scheme\n𝒰 : Cover (precoverage P) X\n⊢ { I₀ := ↥X, X := fun x ↦ 𝒰.X (𝒰.idx x), f := fun x ↦ 𝒰.f (𝒰.idx x) }.presieve₀ ∈ (precoverage P).coverin... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 737,
"column": 2
} | {
"line": 737,
"column": 18
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsOpenImmersion f\nU : Y.Opens\n⊢ (Y.presheaf.map (homOfLE ⋯).op ≫ (ΓIso f U).inv) ≫ (ΓIso f U).hom = Y.presheaf.map (homOfLE ⋯).op",
"usedConstants": [
"CategoryTheory.Category.assoc",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",... | simp [-ΓIso_inv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Cover.Open | {
"line": 57,
"column": 6
} | {
"line": 57,
"column": 40
} | [
{
"pp": "X✝ Y Z : Scheme\n𝒰 : X✝.OpenCover\nf : X✝ ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰.I₀), HasPullback (𝒰.f x ≫ f) g\nX : Scheme\nU : (x : ↑X.toTopCat) → OpenNhds x\nR : ↑X.toTopCat → CommRingCat\nh : ∀ (x : ↑X.toTopCat), Nonempty (X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op (R x)))\ne : (x : ↑X.toTopCat)... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 436,
"column": 2
} | {
"line": 443,
"column": 56
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y Z : PresheafedSpace C\nf : X ⟶ Z\nhf : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\ng : Y ⟶ Z\ns : PullbackCone f g\n⊢ IsLimit (pullbackConeOfLeft f g)",
"usedConstants": [
"Eq.mpr",
"Algebraic... | apply PullbackCone.isLimitAux'
intro s
use pullbackConeOfLeftLift f g s
use pullbackConeOfLeftLift_fst f g s
use pullbackConeOfLeftLift_snd f g s
intro m _ h₂
rw [← cancel_mono (pullbackConeOfLeft f g).snd]
exact h₂.trans (pullbackConeOfLeftLift_snd f g s).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 436,
"column": 2
} | {
"line": 443,
"column": 56
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y Z : PresheafedSpace C\nf : X ⟶ Z\nhf : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\ng : Y ⟶ Z\ns : PullbackCone f g\n⊢ IsLimit (pullbackConeOfLeft f g)",
"usedConstants": [
"Eq.mpr",
"Algebraic... | apply PullbackCone.isLimitAux'
intro s
use pullbackConeOfLeftLift f g s
use pullbackConeOfLeftLift_fst f g s
use pullbackConeOfLeftLift_snd f g s
intro m _ h₂
rw [← cancel_mono (pullbackConeOfLeft f g).snd]
exact h₂.trans (pullbackConeOfLeftLift_snd f g s).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Cover.Open | {
"line": 94,
"column": 8
} | {
"line": 94,
"column": 42
} | [
{
"pp": "X✝ Y Z : Scheme\n𝒰✝ : X✝.OpenCover\nf : X✝ ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.I₀), HasPullback (𝒰✝.f x ≫ f) g\nX : Scheme\n𝒰 : X.OpenCover\nH : CompactSpace ↥X\nthis : ∃ t, ⋃ x ∈ t, (fun x ↦ Set.range ⇑(𝒰.f (Cover.idx 𝒰 x))) x = ⊤\nt : Finset ↥X := this.choose\nh : ∀ (x : ↥X), ∃ y, x ∈ Set.range ⇑... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Cover.Open | {
"line": 270,
"column": 4
} | {
"line": 270,
"column": 38
} | [
{
"pp": "X Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰.I₀), HasPullback (𝒰.f x ≫ f) g\nR : CommRingCat\n⊢ { I₀ := ↑R, X := fun r ↦ Spec (CommRingCat.of (Localization.Away r)),\n f := fun r ↦ Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r))) }.presieve₀ ∈\n ... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 33
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nX : Scheme\nU : X.Opens\n⊢ IsIso (Hom.appLE U.ι U ⊤ ⋯)",
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Eq.ge",
"AlgebraicGeometry.... | simp only [ι, ofRestrict_appLE] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 185,
"column": 4
} | {
"line": 185,
"column": 38
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nX✝ : Scheme\nU✝ : X✝.Opens\ns : Type u_1\nX : Scheme\nU : s → X.Opens\nhU : IsOpenCover U\n⊢ { I₀ := s, X := fun i ↦ ↑(U i), f := fun i ↦ (U i).ι }.presieve₀ ∈ (precoverage IsOpenImmersion).coverings X",
"usedConstants": [
"CategoryTheory.PreZeroHyperc... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 29
} | [
{
"pp": "X : Scheme\nU V : X.Opens\ne : U ≤ V\nW : (↑V).Opens\ne₁ : Hom.app (X.homOfLE e ≫ V.ι) (V.ι ''ᵁ W) = Hom.app U.ι (V.ι ''ᵁ W) ≫ (↑U).presheaf.map (eqToHom ⋯).op\nthis : V.ι ⁻¹ᵁ V.ι ''ᵁ W = W\ne₂ :\n (↑V).presheaf.map (eqToIso this).hom.op ≫ Hom.app (X.homOfLE e) (V.ι ⁻¹ᵁ V.ι ''ᵁ W) =\n Hom.app (X.ho... | have e₃ := e₂.symm.trans e₁ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.GroupTheory.Submonoid.Inverses | {
"line": 79,
"column": 38
} | {
"line": 79,
"column": 41
} | [
{
"pp": "case h.e'_5\nM : Type u_1\ninst✝ : Monoid M\nS : Submonoid M\nx y : M\nz : ↥S\nh₁ : y * ↑z = 1\nh₂ : x * y = 1\n⊢ x * y * ↑z = ↑z",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Semigroup.toMul",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Membership.... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 322,
"column": 4
} | {
"line": 322,
"column": 38
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nX✝ : Scheme\nU✝ : X✝.Opens\nJ : Type u_1\nX : Scheme\nU : J → X.Opens\n⊢ { I₀ := J, X := fun i ↦ ↑(U i), f := fun j ↦ X.homOfLE ⋯ }.presieve₀ ∈\n (precoverage IsOpenImmersion).coverings ↑(⨆ i, U i)",
"usedConstants": [
"CategoryTheory.PreZeroHyperco... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.MorphismProperty.Local | {
"line": 197,
"column": 2
} | {
"line": 197,
"column": 66
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasEqualizers C\ninst✝² : HasPullbacks C\nX Y S : C\nf g : X ⟶ Y\ns : X ⟶ S\nt : Y ⟶ S\nhf : f ≫ t = s\nhg : g ≫ t = s\nJ : Precoverage C\n𝒰 : J.ZeroHypercover S\ninst✝¹ : J.IsStableUnderBaseChange\ninst✝ : (MorphismProperty.isomorphisms C).IsLocalAtTar... | suffices IsIso (equalizer.ι f g) from Limits.eq_of_epi_equalizer | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 776,
"column": 2
} | {
"line": 776,
"column": 52
} | [
{
"pp": "case h.e'_10.h\nR M A : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nx : ↑(PrimeSpectrum.Top R)\nm : ↑(ModuleCat.of R M)\n⊢ (ModuleCat.Hom.hom (toStalkₗ' R M x)) m =\n (↑(stalkIsoₗ R M x).toLinearEquiv.symm ∘ₗ LocalizedModule.mk... | refine .trans ?_ (localizationtoStalkₗ_mk ..).symm | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Gluing | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 12
} | [
{
"pp": "D : GlueData\nx y : ↑(∐ D.U)\nh :\n (ConcreteCategory.hom (colimit.ι (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) one)) x =\n (ConcreteCategory.hom (colimit.ι (parallelPair D.diagram.fstSigmaMap D.diagram.sndSigmaMap) one)) y\ndiagram : WalkingParallelPair ⥤ Type u := parallelPair D.d... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 65
} | [
{
"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⊢ IsOpen\n (⇑(ConcreteCategory.hom\n (coequalizer.π ((SheafedSpace.forget CommRingCat).map (Hom.toShH... | rw [PreservesCoequalizer.iso_hom, ι_comp_coequalizerComparison] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 856,
"column": 11
} | {
"line": 856,
"column": 27
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nI J : Ideal ↑Γ(X, U)\n⊢ I = J ↔\n ∀ (x : ↥X) (h : x ∈ U),\n Ideal.map (CommRingCat.Hom.hom (X.presheaf.germ U x h)) I =\n Ideal.map (CommRingCat.Hom.hom (X.presheaf.germ U x h)) J",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeome... | le_antisymm_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 885,
"column": 4
} | {
"line": 887,
"column": 32
} | [
{
"pp": "case refine_3.hf.h.a\nX Y : Scheme\nU✝ : X.Opens\nhU✝ : IsAffineOpen U✝\nf✝ : ↑Γ(X, U✝)\nf : X ⟶ Y\nx : ↥X\nU : Y.Opens\nhU : IsAffineOpen U\nV : X.Opens\nhV : IsAffineOpen V\nhVU : V ≤ f ⁻¹ᵁ U\nhx : x ∈ V\nthis✝² : Algebra ↑(Y.presheaf.obj (op U)) ↑(Y.presheaf.stalk ↑⟨f x, ⋯⟩) := Y.presheaf.algebra_se... | simp only [RingHom.algebraMap_toAlgebra, Scheme.Hom.germ_stalkMap_apply, Scheme.Hom.appLE,
homOfLE_leOfHom, CommRingCat.hom_comp, RingHom.coe_comp, Function.comp_apply,
X.presheaf.germ_res_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 267,
"column": 4
} | {
"line": 267,
"column": 38
} | [
{
"pp": "D✝ : GlueData\nD : GlueData\n⊢ { I₀ := D.J, X := D.U, f := D.ι }.presieve₀ ∈ (precoverage IsOpenImmersion).coverings D.glued",
"usedConstants": [
"AlgebraicGeometry.Scheme.GlueData.ι",
"CategoryTheory.PreZeroHypercover.mk",
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"Catego... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 129,
"column": 6
} | {
"line": 129,
"column": 81
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : GlueData C\ninst✝ : HasLimits C\ni : D.J\n⊢ IsOpenEmbedding ⇑(ConcreteCategory.hom (D.ι i).base)",
"usedConstants": [
"CategoryTheory.Limits.PreservesColimitsOfShape.preservesColimit",
"CategoryTheory.GlueData.diagram",
"Eq.mpr",
"... | ← show _ = (𝖣.ι i).base from 𝖣.ι_gluedIso_inv (PresheafedSpace.forget _) _, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 477,
"column": 6
} | {
"line": 477,
"column": 40
} | [
{
"pp": "X Y : Scheme\nf g : X ⟶ Y\nU : ↥X → X.Opens\nhxU : ∀ (x : ↥X), x ∈ U x\nhU : ∀ (x : ↥X), (U x).ι ≫ f = (U x).ι ≫ g\n⊢ { I₀ := ↥X, X := fun i ↦ ↑(U i), f := fun i ↦ (U i).ι }.presieve₀ ∈ (precoverage IsOpenImmersion).coverings X",
"usedConstants": [
"CategoryTheory.PreZeroHypercover.mk",
... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 206,
"column": 49
} | {
"line": 206,
"column": 59
} | [
{
"pp": "case h\nC : Type u\ninst✝ : Category.{v, u} C\nD : GlueData C\ni j k : D.J\nU : Opens ↑↑(pullback (D.f i j) (D.f i k))\n⊢ (D.t k i).c.app (op ((opensFunctor (pullback.snd (D.f i j) (D.f i k))).obj U)) ≫\n (D.V (k, i)).presheaf.map (eqToHom ⋯) =\n (((pullback.fst (D.f k i) (D.f k j) ≫ D.t k i).c... | comp_c_app | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 32
} | [
{
"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 =\n pullback.fst (pullback.fst (𝒰.f i ≫ f) g ≫ 𝒰.f i) (𝒰.... | rw [← cancel_mono (𝒰.f i)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 30
} | [
{
"pp": "case h₁\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.snd (pullback.fst (𝒰.f i ≫ f) g ≫ 𝒰.f i) (𝒰.f i) =\n pullback.snd (pullback.fst (𝒰.f i ≫ f) g ≫ 𝒰.f i) (𝒰.f i)",
"usedConstants": [
... | rw [← cancel_mono (𝒰.f i)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 593,
"column": 4
} | {
"line": 594,
"column": 54
} | [
{
"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... | have : IsOpenImmersion (α ≫ pullback.fst _ _) := by
simp only [pullback.lift_fst, α]; infer_instance | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 601,
"column": 4
} | {
"line": 601,
"column": 12
} | [
{
"pp": "case a\nJ : Type w\ninst✝³ : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝² : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝¹ : (F ⋙ forget).IsLocallyDirected\ninst✝ : Quiver.IsThin J\ni j : J\nk₁ k₂ : (k : J) × (k ⟶ i) × (k ⟶ j)\nU : (F.obj i).Opens\nh₁ : Hom.opensRange (F.map k₁.snd.1) ≤ U\nh₂ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 738,
"column": 4
} | {
"line": 738,
"column": 64
} | [
{
"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\ni j : (glueData F).J\nk : (Opens.iSupOpenCover fun i_1 ↦ Hom.opensRange (F.map i_... | rw [homOfLE_tAux_assoc F ↓i ↓j k.2.1 k.2.2, Iso.eq_inv_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Limits | {
"line": 465,
"column": 73
} | {
"line": 465,
"column": 81
} | [
{
"pp": "case refine_1.refine_1.h.h.inl\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\nX' Y' : Scheme\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : (BinaryCofan.mk coprod.inl coprod.inr).pt ⟶ (BinaryCofan.mk coprod.inl coprod.inr).pt\nh₁ : αX ≫ co... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.AlgebraicGeometry.Limits | {
"line": 465,
"column": 73
} | {
"line": 465,
"column": 81
} | [
{
"pp": "case refine_1.refine_1.h.h.inr\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\nX' Y' : Scheme\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : (BinaryCofan.mk coprod.inl coprod.inr).pt ⟶ (BinaryCofan.mk coprod.inl coprod.inr).pt\nh₁ : αX ≫ co... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.AlgebraicGeometry.Limits | {
"line": 465,
"column": 73
} | {
"line": 465,
"column": 81
} | [
{
"pp": "case refine_1.refine_2.h.h.inl\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\nX' Y' : Scheme\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : (BinaryCofan.mk coprod.inl coprod.inr).pt ⟶ (BinaryCofan.mk coprod.inl coprod.inr).pt\nh₁ : αX ≫ co... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.AlgebraicGeometry.Limits | {
"line": 465,
"column": 73
} | {
"line": 465,
"column": 81
} | [
{
"pp": "case refine_1.refine_2.h.h.inr\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\nX' Y' : Scheme\nαX : X' ⟶ X\nαY : Y' ⟶ Y\nf : (BinaryCofan.mk coprod.inl coprod.inr).pt ⟶ (BinaryCofan.mk coprod.inl coprod.inr).pt\nh₁ : αX ≫ co... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.AlgebraicGeometry.Limits | {
"line": 690,
"column": 85
} | {
"line": 690,
"column": 93
} | [
{
"pp": "X : Scheme\nU V : X.Opens\nhU : IsAffineOpen U\nhV : IsAffineOpen V\nH : Disjoint U V\n⊢ ∀ (i : Unit ⊕ Unit), IsAffineOpen ((fun i ↦ Sum.elim (fun x ↦ U) (fun x ↦ V) i) i)",
"usedConstants": [
"congrArg",
"and_self",
"AlgebraicGeometry.IsAffineOpen",
"Sum",
"Sum.inl",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.AlgebraicGeometry.Limits | {
"line": 690,
"column": 85
} | {
"line": 690,
"column": 93
} | [
{
"pp": "X : Scheme\nU V : X.Opens\nhU : IsAffineOpen U\nhV : IsAffineOpen V\nH : Disjoint U V\n⊢ ∀ (i : Unit ⊕ Unit), IsAffineOpen ((fun i ↦ Sum.elim (fun x ↦ U) (fun x ↦ V) i) i)",
"usedConstants": [
"congrArg",
"and_self",
"AlgebraicGeometry.IsAffineOpen",
"Sum",
"Sum.inl",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Limits | {
"line": 690,
"column": 85
} | {
"line": 690,
"column": 93
} | [
{
"pp": "X : Scheme\nU V : X.Opens\nhU : IsAffineOpen U\nhV : IsAffineOpen V\nH : Disjoint U V\n⊢ ∀ (i : Unit ⊕ Unit), IsAffineOpen ((fun i ↦ Sum.elim (fun x ↦ U) (fun x ↦ V) i) i)",
"usedConstants": [
"congrArg",
"and_self",
"AlgebraicGeometry.IsAffineOpen",
"Sum",
"Sum.inl",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 453,
"column": 8
} | {
"line": 453,
"column": 18
} | [
{
"pp": "case inst\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : GlueData C\ninst✝ : HasLimits C\ni j : D.J\nU : Opens ↑↑(D.U i)\n⊢ Mono\n ((D.t i j ≫ D.f j i).c.app\n (op\n ((Opens.map\n (colimit.ι D.diagram.multispan\n (unop (op (WalkingMultispan.right ((MultispanSh... | comp_c_app | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 542,
"column": 4
} | {
"line": 542,
"column": 93
} | [
{
"pp": "case basicOpen\nP : MorphismProperty Scheme\nQ : AffineTargetMorphismProperty\ninst✝ : HasAffineProperty P Q\nX Y : Scheme\nf : X ⟶ Y\nι : Sort u_1\nU✝ : ι → ↑Y.affineOpens\nhU : ⨆ i, ↑(U✝ i) = ⊤\nhU' : ∀ (i : ι), Q (f ∣_ ↑(U✝ i))\nthis : Q.IsLocal := isLocal_affineProperty P\nU : ↑Y.affineOpens\nr : ↑... | have := AffineTargetMorphismProperty.IsLocal.to_basicOpen (f ∣_ U.1) (U.1.topIso.inv r) h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 45,
"column": 65
} | {
"line": 49,
"column": 88
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ IsZariskiLocalAtTarget (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ Function.Injective)",
"usedConstants": [
"Set.restrictPreimage",
"Lattice.toSemilatticeSup",
"Continuous",
"AlgebraicGeometry.topologically_isZ... | by
refine topologically_isZariskiLocalAtTarget _ (fun _ s _ _ h ↦ h.restrictPreimage s)
fun f ι U H _ hf x₁ x₂ e ↦ ?_
obtain ⟨i, hxi⟩ : ∃ i, f x₁ ∈ U i := by simpa using congr(f x₁ ∈ $H)
exact congr(($(@hf i ⟨x₁, hxi⟩ ⟨x₂, show f x₂ ∈ U i from e ▸ hxi⟩ (Subtype.ext e))).1) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 282,
"column": 4
} | {
"line": 283,
"column": 20
} | [
{
"pp": "case hP₂\n⊢ ∀ {α β γ : Type u_1} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] [inst_2 : TopologicalSpace γ]\n (f : α → β) (g : β → γ), SpecializingMap f → SpecializingMap g → SpecializingMap (g ∘ f)",
"usedConstants": [
"SpecializingMap",
"TopologicalSpace",
"Speci... | introv hf hg
exact hf.comp hg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 282,
"column": 4
} | {
"line": 283,
"column": 20
} | [
{
"pp": "case hP₂\n⊢ ∀ {α β γ : Type u_1} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] [inst_2 : TopologicalSpace γ]\n (f : α → β) (g : β → γ), SpecializingMap f → SpecializingMap g → SpecializingMap (g ∘ f)",
"usedConstants": [
"SpecializingMap",
"TopologicalSpace",
"Speci... | introv hf hg
exact hf.comp hg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 100
} | [
{
"pp": "case refine_1\nP : MorphismProperty Scheme\ninst✝² : P.HasOfPostcompProperty IsOpenImmersion\ninst✝¹ : P.RespectsRight IsOpenImmersion\ninst✝ : IsZariskiLocalAtSource P\ng : {X Y : Scheme} → (f : X ⟶ Y) → (U : X.Opens) → pullback (U.ι ≫ f) (U.ι ≫ f) ⟶ pullback f f :=\n fun {X Y} f U ↦ pullback.map (U.... | apply P.of_postcomp (W' := @IsOpenImmersion) (pullback.diagonal (U.ι ≫ f)) (g f U) inferInstance | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 262,
"column": 32
} | {
"line": 264,
"column": 22
} | [
{
"pp": "P : {α β : Type u} → [TopologicalSpace α] → [TopologicalSpace β] → (α → β) → Prop\nhP :\n ∀ {α β γ : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] [inst_2 : TopologicalSpace γ] (f : α → β)\n (g : β → γ), P f → P g → P (g ∘ f)\nX Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\nhf : topologic... | by
simp only [topologically, Scheme.Hom.comp_base, TopCat.coe_comp]
exact hP _ _ hf hg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RingHom.Surjective | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 26
} | [
{
"pp": "case tmul\nR S T : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nh : Function.Surjective ⇑(algebraMap R T)\nx : S\ny : T\n⊢ ∃ a, (algebraMap S (S ⊗[R] T)) a = x ⊗ₜ[R] y",
"usedConstants": []
}
] | obtain ⟨y, rfl⟩ := h y | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.RingHom.Surjective | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 50
} | [
{
"pp": "R S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\ns : Set R\ne : Ideal.span s = ⊤\nH : ∀ (r : ↑s), (fun {X Y} [CommRing X] [CommRing Y] f ↦ Function.Surjective ⇑f) (Localization.awayMap f ↑r)\n⊢ Function.Surjective ⇑f",
"usedConstants": [
"Set.range_eq_univ",
"Eq.mpr... | rw [← Set.range_eq_univ, Set.eq_univ_iff_forall] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 202,
"column": 2
} | {
"line": 203,
"column": 18
} | [
{
"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... | obtain ⟨r, r', hBrr', hBfx⟩ := exists_basicOpen_le_affine_inter U₁.2 U₂.2 (f x)
⟨hfx₁, e₂ hx₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 357,
"column": 2
} | {
"line": 358,
"column": 98
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝ : HasRingHomProperty P Q\nR S : CommRingCat\nφ : R ⟶ S\nH : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q\n⊢ P (Spec.map φ) ↔ Q (CommRingCat.Hom.hom φ)",
"usedConsta... | rw [iff_of_isAffine (P := P), ← H.cancel_right_isIso _ (Scheme.ΓSpecIso _).hom,
← CommRingCat.hom_comp, Scheme.ΓSpecIso_naturality, CommRingCat.hom_comp, H.cancel_left_isIso] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Height | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 20
} | [
{
"pp": "case h.e'_5\nR : Type u_1\ninst✝² : CommRing R\nI J : Ideal R\ne : I ≤ J\ninst✝¹ : J.IsPrime\ninst✝ : J.FiniteHeight\ne' : J.height ≤ I.height\np : Ideal R\nh₁ : p ∈ I.minimalPrimes\nh₂ : p ≤ J\nh₃ : p < J\n⊢ False",
"usedConstants": [
"CommSemiring.toSemiring",
"Ideal.IsMinimalPrime.is... | have := h₁.isPrime | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Morphisms.SurjectiveOnStalks | {
"line": 185,
"column": 10
} | {
"line": 185,
"column": 17
} | [
{
"pp": "case refine_2.refine_3\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\ninst✝ : SurjectiveOnStalks g\nL : ↥(pullback f g) → ↥X × ↥Y := fun x ↦ ((pullback.fst f g) x, (pullback.snd f g) x)\nH :\n ∀ (R A B : CommRingCat) (f' : Spec A ⟶ Spec R) (g' : Spec B ⟶ Spec R) (iX : Spec A ⟶ X) (iY : Spec B ⟶ Y)\n (iS : ... | ← hx₁', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 179,
"column": 2
} | {
"line": 180,
"column": 55
} | [
{
"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 : ... | refine AffineTargetMorphismProperty.diagonal_of_openCover_source _
(Scheme.openCoverOfIsOpenCover _ _ hU) fun i j ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 181,
"column": 2
} | {
"line": 184,
"column": 51
} | [
{
"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 : ... | rw [← isCompact_univ_iff, (pullback.fst ((X.openCoverOfIsOpenCover U hU).f i)
((X.openCoverOfIsOpenCover U hU).f j) ≫
(X.openCoverOfIsOpenCover U hU).f i).isOpenEmbedding.isCompact_iff, Set.image_univ,
IsOpenImmersion.range_pullback_to_base_of_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 268,
"column": 2
} | {
"line": 273,
"column": 28
} | [
{
"pp": "X : TopCat\nF : TopCat.Presheaf CommRingCat X\nU₁ U₂ U₃ U₄ U₅ U₆ U₇ : Opens ↑X\nn₁ n₂ : ℕ\ny₁ : ↑(F.obj (op U₁))\ny₂ : ↑(F.obj (op U₂))\nf : ↑(F.obj (op (U₁ ⊔ U₂)))\nx : ↑(F.obj (op U₃))\nh₄₁ : U₄ ≤ U₁\nh₄₂ : U₄ ≤ U₂\nh₅₁ : U₅ ≤ U₁\nh₅₃ : U₅ ≤ U₃\nh₆₂ : U₆ ≤ U₂\nh₆₃ : U₆ ≤ U₃\nh₇₄ : U₇ ≤ U₄\nh₇₅ : U₇ ≤... | apply_fun (fun x : F.obj (op U₅) ↦ x |_ U₇) at e₁
apply_fun (fun x : F.obj (op U₆) ↦ x |_ U₇) at e₂
dsimp only [TopCat.Presheaf.restrictOpenCommRingCat_apply] at e₁ e₂ ⊢
simp only [map_mul, map_pow, ← op_comp, ← F.map_comp, homOfLE_comp, ← CommRingCat.comp_apply]
at e₁ e₂ ⊢
rw [e₁, e₂, mul_left_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 268,
"column": 2
} | {
"line": 273,
"column": 28
} | [
{
"pp": "X : TopCat\nF : TopCat.Presheaf CommRingCat X\nU₁ U₂ U₃ U₄ U₅ U₆ U₇ : Opens ↑X\nn₁ n₂ : ℕ\ny₁ : ↑(F.obj (op U₁))\ny₂ : ↑(F.obj (op U₂))\nf : ↑(F.obj (op (U₁ ⊔ U₂)))\nx : ↑(F.obj (op U₃))\nh₄₁ : U₄ ≤ U₁\nh₄₂ : U₄ ≤ U₂\nh₅₁ : U₅ ≤ U₁\nh₅₃ : U₅ ≤ U₃\nh₆₂ : U₆ ≤ U₂\nh₆₃ : U₆ ≤ U₃\nh₇₄ : U₇ ≤ U₄\nh₇₅ : U₇ ≤... | apply_fun (fun x : F.obj (op U₅) ↦ x |_ U₇) at e₁
apply_fun (fun x : F.obj (op U₆) ↦ x |_ U₇) at e₂
dsimp only [TopCat.Presheaf.restrictOpenCommRingCat_apply] at e₁ e₂ ⊢
simp only [map_mul, map_pow, ← op_comp, ← F.map_comp, homOfLE_comp, ← CommRingCat.comp_apply]
at e₁ e₂ ⊢
rw [e₁, e₂, mul_left_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 413,
"column": 2
} | {
"line": 413,
"column": 82
} | [
{
"pp": "X : Scheme\nU : Opens ↥X\nhU : IsCompact U.carrier\nhU' : IsQuasiSeparated U.carrier\nf g s : ↑Γ(X, U)\nhfg : (f |_ X.basicOpen s) ⋯ = (g |_ X.basicOpen s) ⋯\n⊢ ∃ n, s ^ n * f = s ^ n * g",
"usedConstants": [
"Opposite",
"CommRingCat.carrier",
"AlgebraicGeometry.PresheafedSpace.ca... | obtain ⟨n, hc⟩ := (isLocalization_basicOpen_of_qcqs hU hU' s).exists_of_eq s hfg | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 46
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nU V : ↑X.affineOpens\nx : ↑Γ(X, ↑V)\nhx : x ∈ I.ideal V\np : ↥(I.glueDataObj U)\nhp : p ∈ Opposite.unop (Opposite.op (I.glueDataObjι U ⁻¹ᵁ (↑U).ι ⁻¹ᵁ ↑V))\nf : ↑Γ(X, ↑U)\ng : ↑Γ(X, ↑V)\nhfg : X.basicOpen f = X.basicOpen g\nhf : ↑((I.glueDataObjι U) p) ∈ X.basicOpen f\n... | simp only [Scheme.homOfLE_ι, ← top_le_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 424,
"column": 87
} | {
"line": 424,
"column": 95
} | [
{
"pp": "case zero\nX : Scheme\nI : X.IdealSheafData\nhn : 0 ≠ 0\n⊢ (I ^ 0).support = I.support",
"usedConstants": [
"AlgebraicGeometry.Scheme.IdealSheafData.support",
"False",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.car... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 424,
"column": 87
} | {
"line": 424,
"column": 95
} | [
{
"pp": "case succ\nX : Scheme\nI : X.IdealSheafData\nn✝ : ℕ\nhn : n✝ + 1 ≠ 0\n⊢ (I ^ (n✝ + 1)).support = I.support",
"usedConstants": [
"AlgebraicGeometry.Scheme.IdealSheafData.support",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpac... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 712,
"column": 4
} | {
"line": 712,
"column": 57
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : ↑Y.affineOpens\n⊢ ⊥.subschemeι ≫ Hom.toImage (𝟙 X) ≫ IdealSheafData.inclusion ⋯ = 𝟙 ⊥.subscheme",
"usedConstants": [
"AlgebraicGeometry.Scheme.IdealSheafData.inclusion",
"CategoryTheory.Category.assoc",
"AlgebraicGeometry.Scheme.IdealSheafData.instIs... | by simp [← cancel_mono (IdealSheafData.subschemeι _)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 94,
"column": 49
} | {
"line": 94,
"column": 87
} | [
{
"pp": "X : Scheme\ns : Set ↑Γ(X, ⊤)\nhs : Ideal.span s = ⊤\nhs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)\nU V : ↑X.affineOpens\ns' : Finset ↑Γ(X, ⊤)\nhs' : ↑s' ⊆ s\ne : Ideal.span ↑s' = ⊤\n⊢ IsCompact (↑↑U ∩ ↑↑V ∩ ↑⊤)",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologica... | ← iSup_basicOpen_of_span_eq_top _ _ e, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 101,
"column": 4
} | {
"line": 102,
"column": 37
} | [
{
"pp": "case h.refine_1\nX : Scheme\ns : Set ↑Γ(X, ⊤)\nhs : Ideal.span s = ⊤\nhs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)\nU V : ↑X.affineOpens\ns' : Finset ↑Γ(X, ⊤)\nhs' : ↑s' ⊆ s\ne : Ideal.span ↑s' = ⊤\ni : ↑↑s'\n⊢ IsCompact (↑↑U ∩ ↑(X.basicOpen ↑i))",
"usedConstants": [
"Eq.mpr",
"Algebraic... | rw [← Opens.coe_inf, ← X.basicOpen_res _ (homOfLE le_top).op]
exact (U.2.basicOpen _).isCompact | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 101,
"column": 4
} | {
"line": 102,
"column": 37
} | [
{
"pp": "case h.refine_1\nX : Scheme\ns : Set ↑Γ(X, ⊤)\nhs : Ideal.span s = ⊤\nhs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)\nU V : ↑X.affineOpens\ns' : Finset ↑Γ(X, ⊤)\nhs' : ↑s' ⊆ s\ne : Ideal.span ↑s' = ⊤\ni : ↑↑s'\n⊢ IsCompact (↑↑U ∩ ↑(X.basicOpen ↑i))",
"usedConstants": [
"Eq.mpr",
"Algebraic... | rw [← Opens.coe_inf, ← X.basicOpen_res _ (homOfLE le_top).op]
exact (U.2.basicOpen _).isCompact | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 13
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : ↥Y → Y.Opens\nhxU : ∀ (x : ↥Y), x ∈ U x\nhU : ∀ (x : ↥Y), IsAffineOpen (U x)\nhfU : ∀ (x : ↥Y), IsAffineOpen (f ⁻¹ᵁ U x)\n⊢ ∀ (i : ↥Y), IsAffine ↑(f ⁻¹ᵁ ↑⟨U i, ⋯⟩)",
"usedConstants": []
}
] | exact hfU | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 13
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : ↥Y → Y.Opens\nhxU : ∀ (x : ↥Y), x ∈ U x\nhU : ∀ (x : ↥Y), IsAffineOpen (U x)\nhfU : ∀ (x : ↥Y), IsAffineOpen (f ⁻¹ᵁ U x)\n⊢ ∀ (i : ↥Y), IsAffine ↑(f ⁻¹ᵁ ↑⟨U i, ⋯⟩)",
"usedConstants": []
}
] | exact hfU | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 13
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : ↥Y → Y.Opens\nhxU : ∀ (x : ↥Y), x ∈ U x\nhU : ∀ (x : ↥Y), IsAffineOpen (U x)\nhfU : ∀ (x : ↥Y), IsAffineOpen (f ⁻¹ᵁ U x)\n⊢ ∀ (i : ↥Y), IsAffine ↑(f ⁻¹ᵁ ↑⟨U i, ⋯⟩)",
"usedConstants": []
}
] | exact hfU | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.AffineAnd | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 30
} | [
{
"pp": "Q W : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhQW : ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] {f : R →+* S}, Q f → W f\nX Y : Scheme\nf : X ⟶ Y\nh : targetAffineLocally (affineAnd fun {R S} [CommRing R] [CommRing S] ↦ Q) f\nU : ↑Y.affineOpens\n⊢... | exact ⟨(h U).1, hQW (h U).2⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Artinian.Ring | {
"line": 61,
"column": 11
} | {
"line": 61,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsArtinianRing R\nI : Ideal R\n⊢ I.jacobson = I.radical",
"usedConstants": [
"Ideal.jacobson",
"id",
"Ideal",
"CommRing.toCommSemiring",
"CommRing.toRing",
"Ring.toSemiring",
"Eq",
"Ideal.radical"
]
}
] | Ideal.jacobson, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Spectrum.Prime.Jacobson | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 31
} | [
{
"pp": "case h\nR : Type u_1\ninst✝ : CommRing R\nH : JacobsonSpace (PrimeSpectrum R)\nI : Ideal R\nhI : I.IsRadical\nx : PrimeSpectrum R\nhx : I ≤ x.asIdeal\nhx' : x ∈ closedPoints (PrimeSpectrum R)\n⊢ x ∈ zeroLocus ↑I.jacobson",
"usedConstants": [
"Semiring.toModule",
"PrimeSpectrum.zeroLocus... | change jacobson I ≤ x.asIdeal | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.Spectrum.Prime.Jacobson | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 65
} | [
{
"pp": "case h.mp\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsJacobsonRing R\np : PrimeSpectrum R\ntfae_1_to_2 : IsOpen {p} → IsClopen {p}\ntfae_2_to_3 : IsClopen {p} → IsClosed {p} ∧ StableUnderGeneralization {p}\nh₁ : IsMax p\nh₂ : StableUnderGeneralization {p}\ni✝ : PrimeSpect... | rw [stableUnderGeneralization_singleton, ← isMin_iff] at h₂ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.Separated | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 25
} | [
{
"pp": "case h.h.h.a\nx✝² x✝¹ : Scheme\nx✝ : x✝² ⟶ x✝¹\n⊢ IsSeparated x✝ ↔ MorphismProperty.diagonal (@IsClosedImmersion) x✝",
"usedConstants": [
"AlgebraicGeometry.isSeparated_iff"
]
}
] | exact isSeparated_iff _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 39
} | [
{
"pp": "⊢ @UniversallyClosed = (topologically @IsClosedMap).universally",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"Iff.rfl",
"id",
"CategoryTheory.MorphismProperty.universall... | ext X Y f; rw [universallyClosed_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 39
} | [
{
"pp": "⊢ @UniversallyClosed = (topologically @IsClosedMap).universally",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"Iff.rfl",
"id",
"CategoryTheory.MorphismProperty.universall... | ext X Y f; rw [universallyClosed_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Finite | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 84
} | [
{
"pp": "case inr\nS : CommRingCat\ninst✝² : JacobsonSpace ↥(Spec S)\nR : CommRingCat\ninst✝¹ : Subsingleton ↥(Spec R)\ninst✝ : IsReduced (Spec R)\nh✝ : Nonempty ↥(Spec R)\nthis✝⁴ : IrreducibleSpace ↥(Spec R)\nthis✝³ : IsDomain ↑R\nφ : S ⟶ R\nthis✝² : IsField ↑R\nthis✝¹ : Field ↑R := this✝².toField\nthis✝ : Alg... | exact ⟨fun _ ↦ inferInstance, fun _ ↦ finite_of_finite_type_of_isJacobsonRing _ _⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 128,
"column": 4
} | {
"line": 128,
"column": 40
} | [
{
"pp": "case hP\nX Y Z : Scheme\nf : X ⟶ Y\n⊢ ∀ {α β : Type u_1} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α → β) {ι : Type u_1}\n (U : ι → TopologicalSpace.Opens β),\n TopologicalSpace.IsOpenCover U →\n Continuous f →\n (IsLocallyClosed (Set.range f) ↔ ∀ (i : ι), IsLocal... | simp_rw [Set.range_restrictPreimage] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.AlgebraicGeometry.AlgClosed.Basic | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 75
} | [
{
"pp": "case a\nX Y : Scheme\nK : Type u\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\nf : X ⟶ Spec (CommRingCat.of K)\ninst✝ : LocallyOfFiniteType f\nx : ↥X\nhx : IsClosed {x}\np : { p // p ≫ f = 𝟙 (Spec (CommRingCat.of K)) }\npf✝¹ : IsLocalRing ↑(CommRingCat.of K)\nh : IsClosed {↑p (IsLocalRing.closedPoint K)}... | simp only [Spec.map_id, Spec.map_comp, SpecMap_residueFieldIsoBase_inv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.HopkinsLevitzki | {
"line": 190,
"column": 31
} | {
"line": 190,
"column": 47
} | [
{
"pp": "R : Type u_3\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\n⊢ nilradical R = IsLocalRing.maximalIdeal R ↔ IsLocalRing.maximalIdeal R ≤ nilradical R",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"Pa... | le_antisymm_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
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