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 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.RingQuot | {
"line": 433,
"column": 22
} | {
"line": 433,
"column": 24
} | {
"line": 433,
"column": 25
} | [
{
"pp": "case refine_3\nR : Type uR\ninst✝⁵ : Semiring R\nS : Type uS\ninst✝⁴ : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝³ : Semiring A\ninst✝² : Algebra S A\nr✝ : R → R → Prop\ninst✝¹ : Semiring T\nB : Type uR\ninst✝ : CommRing B\nr : B → B → Prop\nx : B\nh : x ∈ Ideal.ofRel r\na b : B\nhx✝ : a ∈ Submodu... | [
"case refine_3\nR : Type uR\ninst✝⁵ : Semiring R\nS : Type uS\ninst✝⁴ : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝³ : Semiring A\ninst✝² : Algebra S A\nr✝ : R → R → Prop\ninst✝¹ : Semiring T\nB : Type uR\ninst✝ : CommRing B\nr : B → B → Prop\nx : B\nh : x ∈ Ideal.ofRel r\na b : B\nhx✝ : a ∈ Submodule.span B {x... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.RingQuot | {
"line": 435,
"column": 8
} | {
"line": 436,
"column": 17
} | {
"line": 436,
"column": 17
} | [
{
"pp": "case refine_4\nR : Type uR\ninst✝⁵ : Semiring R\nS : Type uS\ninst✝⁴ : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝³ : Semiring A\ninst✝² : Algebra S A\nr✝ : R → R → Prop\ninst✝¹ : Semiring T\nB : Type uR\ninst✝ : CommRing B\nr : B → B → Prop\nx : B\nh : x ∈ Ideal.ofRel r\n⊢ ∀ (a x : B), x ∈ Submodu... | [] | intro a x _ hx
simp [hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.RingQuot | {
"line": 435,
"column": 8
} | {
"line": 436,
"column": 17
} | {
"line": 436,
"column": 17
} | [
{
"pp": "case refine_4\nR : Type uR\ninst✝⁵ : Semiring R\nS : Type uS\ninst✝⁴ : CommSemiring S\nT : Type uT\nA : Type uA\ninst✝³ : Semiring A\ninst✝² : Algebra S A\nr✝ : R → R → Prop\ninst✝¹ : Semiring T\nB : Type uR\ninst✝ : CommRing B\nr : B → B → Prop\nx : B\nh : x ∈ Ideal.ofRel r\n⊢ ∀ (a x : B), x ∈ Submodu... | [] | intro a x _ hx
simp [hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Star.RingQuot | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 20
} | {
"line": 32,
"column": 21
} | [
{
"pp": "case add_left\nR : Type u\ninst✝¹ : Semiring R\nr : R → R → Prop\ninst✝ : StarRing R\nhr : ∀ (a b : R), r a b → r (Star.star a) (Star.star b)\na b a✝¹ b✝ c✝ : R\na✝ : Rel r a✝¹ b✝\nh : Rel r (Star.star a✝¹) (Star.star b✝)\n⊢ Rel r (Star.star (a✝¹ + c✝)) (Star.star (b✝ + c✝))",
"ppTerm": "?add_left"... | [] | | add_left _ h => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 638,
"column": 44
} | {
"line": 639,
"column": 44
} | {
"line": 641,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nN : Type u_3\ninst✝¹ : AddCommMonoid N\ninst✝ : NonUnitalNonAssocSemiring k\ng : SkewMonoidAlgebra k G\nb : k\nh : G → k → N\nh0 : ∀ (i : G), h i 0 = 0\n⊢ (b • g).sum h = g.sum fun x1 x2 ↦ h x1 (b * x2)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
... | [] | by
simp [sum_def, Finsupp.sum_smul_index' h0] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 699,
"column": 34
} | {
"line": 699,
"column": 48
} | {
"line": 699,
"column": 48
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝⁵ : Mul G\nR : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : NonAssocSemiring k\ninst✝² : SMul G k\ng_hom : Type u_4\ninst✝¹ : FunLike g_hom G R\ninst✝ : MulHomClass g_hom G R\nf : k →+* R\ng : g_hom\na b : SkewMonoidAlgebra k G\nh_comm : ∀ {x y : G}, y ∈ a.support → f (y • b... | [
"k : Type u_1\nG : Type u_2\ninst✝⁵ : Mul G\nR : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : NonAssocSemiring k\ninst✝² : SMul G k\ng_hom : Type u_4\ninst✝¹ : FunLike g_hom G R\ninst✝ : MulHomClass g_hom G R\nf : k →+* R\ng : g_hom\na b : SkewMonoidAlgebra k G\nh_comm : ∀ {x y : G}, y ∈ a.support → f (y • b.coeff x) * ... | ← sum_single b | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 1279,
"column": 70
} | {
"line": 1280,
"column": 92
} | {
"line": 1282,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝³ : Monoid G\ninst✝² : CommSemiring k\ninst✝¹ : MulSemiringAction G k\ninst✝ : SMulCommClass G k k\na : G\nb : k\n⊢ single a b = (algebraMap k (SkewMonoidAlgebra k G)) b * (of k G) a",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"NonAssocSe... | [] | by
simp [coe_algebraMap, comp_apply, of_apply, single_mul_single, one_mul, smul_one, mul_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 1284,
"column": 87
} | {
"line": 1285,
"column": 92
} | {
"line": 1288,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝⁵ : Monoid G\ninst✝⁴ : CommSemiring k\nA : Type u_3\ninst✝³ : Semiring A\ninst✝² : Algebra k A\na : G\nb : k\ninst✝¹ : MulSemiringAction G A\ninst✝ : SMulCommClass G k A\n⊢ single a ((algebraMap k A) b) = (algebraMap k (SkewMonoidAlgebra A G)) b * (of A G) a",
"ppTe... | [] | by
simp [coe_algebraMap, comp_apply, of_apply, single_mul_single, one_mul, smul_one, mul_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Vertex.VertexOperator | {
"line": 96,
"column": 39
} | {
"line": 96,
"column": 59
} | {
"line": 96,
"column": 59
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nf : ℤ → Module.End R V\nhf : ∀ (x : V), BddBelow (Function.support fun y ↦ (f y) x)\nn : ℤ\nv : V\n⊢ ((HahnModule.of R).symm ((of_coeff f hf) v)).coeff (-n - 1) = (f (-n - 1)) v",
"ppTerm": "?m.72",
"a... | [
"R : Type u_1\nV : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nf : ℤ → Module.End R V\nhf : ∀ (x : V), BddBelow (Function.support fun y ↦ (f y) x)\nn : ℤ\nv : V\n⊢ (f (-n - 1)) v = (f (-n - 1)) v"
] | of_coeff_apply_coeff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.RingedSpace.PresheafedSpace | {
"line": 337,
"column": 2
} | {
"line": 337,
"column": 32
} | {
"line": 338,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : PresheafedSpace C\nU✝ : Opens ↑↑X\n⊢ X.presheaf.map (⋯.adjunction.counit.app U✝).op = (eqToHom ⋯).app (op U✝)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"Lattice.toSemilatti... | [
"case p\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : PresheafedSpace C\nU✝ : Opens ↑↑X\n⊢ op U✝ = op (⋯.functor.obj ((Opens.map ⊤.inclusion').obj U✝))"
] | erw [eqToHom_map, eqToHom_app] | Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_tacticErw____1 | Lean.Parser.Tactic.tacticErw___ |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 526,
"column": 2
} | {
"line": 537,
"column": 86
} | {
"line": 539,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n⊢ IsIso f ↔ IsIso f.base ∧ ∀ (x : ↥X), IsIso (Scheme.Hom.stalkMap f x)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Scheme",
"Categ... | [] | rw [isIso_iff_isOpenImmersion_and_epi_base,
IsOpenImmersion.iff_isIso_stalkMap, and_comm, ← and_assoc]
refine and_congr ⟨?_, ?_⟩ Iff.rfl
· rintro ⟨h₁, h₂⟩
convert_to!
IsIso
(TopCat.isoOfHomeo
(Equiv.toHomeomorphOfContinuousOpen
(.ofBijective _ ⟨h₂.injective, (TopCat.epi_i... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 526,
"column": 2
} | {
"line": 537,
"column": 86
} | {
"line": 539,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n⊢ IsIso f ↔ IsIso f.base ∧ ∀ (x : ↥X), IsIso (Scheme.Hom.stalkMap f x)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Scheme",
"Categ... | [] | rw [isIso_iff_isOpenImmersion_and_epi_base,
IsOpenImmersion.iff_isIso_stalkMap, and_comm, ← and_assoc]
refine and_congr ⟨?_, ?_⟩ Iff.rfl
· rintro ⟨h₁, h₂⟩
convert_to!
IsIso
(TopCat.isoOfHomeo
(Equiv.toHomeomorphOfContinuousOpen
(.ofBijective _ ⟨h₂.injective, (TopCat.epi_i... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 823,
"column": 33
} | {
"line": 823,
"column": 52
} | {
"line": 823,
"column": 52
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsOpenImmersion f\nhf : Scheme.Hom.opensRange f = ⊤\n⊢ Function.Surjective ⇑f",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Set.range_eq_univ",
"Eq.mpr",
"AlgebraicGeometry.PresheafedSpace.carrier",
"congrArg",
"Cat... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsOpenImmersion f\nhf : Scheme.Hom.opensRange f = ⊤\n⊢ Set.range ⇑f = Set.univ"
] | ← Set.range_eq_univ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 421,
"column": 4
} | {
"line": 421,
"column": 89
} | {
"line": 421,
"column": 89
} | [
{
"pp": "R 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\ns : R\nf : M\ng : ↥(Submonoid.powers s)\n⊢ basicOpen s ≤ basicOpen ↑g",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.b... | [] | have := PrimeSpectrum.le_basicOpen_pow s; aesop (add simp [Submonoid.mem_powers_iff]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 421,
"column": 4
} | {
"line": 421,
"column": 89
} | {
"line": 421,
"column": 89
} | [
{
"pp": "R 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\ns : R\nf : M\ng : ↥(Submonoid.powers s)\n⊢ basicOpen s ≤ basicOpen ↑g",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.b... | [] | have := PrimeSpectrum.le_basicOpen_pow s; aesop (add simp [Submonoid.mem_powers_iff]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 542,
"column": 6
} | {
"line": 542,
"column": 38
} | {
"line": 542,
"column": 38
} | [
{
"pp": "R : CommRingCat\nf g : ↑R\n⊢ (basicOpenIsoSpecAway (f * g)).inv ≫ (Spec R).homOfLE ⋯ =\n Spec.map (CommRingCat.ofHom (IsLocalization.Away.awayToAwayRight f g)) ≫ (basicOpenIsoSpecAway f).inv",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometr... | [
"R : CommRingCat\nf g : ↑R\n⊢ ((basicOpenIsoSpecAway (f * g)).inv ≫ (Spec R).homOfLE ⋯) ≫ Scheme.Opens.ι (PrimeSpectrum.basicOpen f) =\n (Spec.map (CommRingCat.ofHom (IsLocalization.Away.awayToAwayRight f g)) ≫ (basicOpenIsoSpecAway f).inv) ≫\n Scheme.Opens.ι (PrimeSpectrum.basicOpen f)"
] | ← cancel_mono (Scheme.Opens.ι _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 477,
"column": 2
} | {
"line": 477,
"column": 24
} | {
"line": 478,
"column": 2
} | [
{
"pp": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Opens ↑(PrimeSpectrum.Top R)\nhU : IsCompact ↑U\ns : (structureSheafInType R M).obj.obj (op U)\ng : ↥U → R\nhxg : ∀ (x : ↥U), ↑x ∈ basicOpen (g x)\nigU : ∀ (x : ↥U), basicOpen (g x) ≤ U\nf : ↥U → M\nH :\n ∀ (x : ↥U),\n... | [
"R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Opens ↑(PrimeSpectrum.Top R)\nhU : IsCompact ↑U\ns : (structureSheafInType R M).obj.obj (op U)\ng : ↥U → R\nhxg : ∀ (x : ↥U), ↑x ∈ basicOpen (g x)\nigU : ∀ (x : ↥U), basicOpen (g x) ≤ U\nf : ↥U → M\nH :\n ∀ (x : ↥U),\n const (f... | choose n hn using this | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 233,
"column": 8
} | {
"line": 233,
"column": 35
} | {
"line": 233,
"column": 36
} | [
{
"pp": "X : LocallyRingedSpace\nr✝ : ↑(Γ.obj (op X))\nx : ↑X.toTopCat\np : PrimeSpectrum ↑(Γ.obj (op X)) := ⋯\nS : CommRingCat := ⋯\nt : ↑S\nht : IsUnit ((CommRingCat.Hom.hom (PresheafedSpace.Hom.stalkMap X.toΓSpecSheafedSpace.hom x)) t)\nr : ↑(Γ.obj (op X))\ns : ↥p.asIdeal.primeCompl\nt' : ↑S := ⋯\nhe : t * t... | [
"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 ... | ← toStalk_stalkMap_toΓSpec, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 900,
"column": 4
} | {
"line": 900,
"column": 74
} | {
"line": 901,
"column": 4
} | [
{
"pp": "case refine_1\nC : Type u₁\ninst✝ : Category.{v, u₁} C\nX : Scheme\n𝒰 : X.OpenCover\nU : X.Opens\ni : 𝒰.I₀\n⊢ Set.range ⇑(𝒰.f i ⁻¹ᵁ U).ι = Set.range ⇑(pullback.snd U.ι (𝒰.f i))",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.pul... | [
"case refine_1\nC : Type u₁\ninst✝ : Category.{v, u₁} C\nX : Scheme\n𝒰 : X.OpenCover\nU : X.Opens\ni : 𝒰.I₀\n⊢ Set.range ⇑(𝒰.f i ⁻¹ᵁ U).ι = ↑(𝒰.f i ⁻¹ᵁ U)"
] | rw [IsOpenImmersion.range_pullbackSnd U.ι (𝒰.f i), Opens.opensRange_ι] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer | {
"line": 30,
"column": 14
} | {
"line": 30,
"column": 52
} | {
"line": 30,
"column": 52
} | [
{
"pp": "C : 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⊢ equalizer.ι f g ≫ prod.lift f g = (equalizer.ι f g ≫ f) ≫ prod.lift (𝟙 Y) (𝟙 Y)",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.C... | [] | ext <;> simp [equalizer.condition f g] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer | {
"line": 30,
"column": 14
} | {
"line": 30,
"column": 52
} | {
"line": 30,
"column": 52
} | [
{
"pp": "C : 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⊢ equalizer.ι f g ≫ prod.lift f g = (equalizer.ι f g ≫ f) ≫ prod.lift (𝟙 Y) (𝟙 Y)",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.C... | [] | ext <;> simp [equalizer.condition f g] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer | {
"line": 30,
"column": 14
} | {
"line": 30,
"column": 52
} | {
"line": 30,
"column": 52
} | [
{
"pp": "C : 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⊢ equalizer.ι f g ≫ prod.lift f g = (equalizer.ι f g ≫ f) ≫ prod.lift (𝟙 Y) (𝟙 Y)",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.C... | [] | ext <;> simp [equalizer.condition f g] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 300,
"column": 32
} | {
"line": 300,
"column": 54
} | {
"line": 300,
"column": 54
} | [
{
"pp": "case w\nX Y : LocallyRingedSpace\nf : X ⟶ Y\nx : ↑↑((𝟭 LocallyRingedSpace).obj X).toPresheafedSpace\n⊢ PrimeSpectrum.comap (CommRingCat.Hom.hom (f.c.app (op ⊤) ≫ X.presheaf.Γgerm x))\n (IsLocalRing.closedPoint ↑(X.presheaf.stalk x)) =\n PrimeSpectrum.comap\n ((CommRingCat.Hom.hom (Hom.sta... | [
"case w\nX Y : LocallyRingedSpace\nf : X ⟶ Y\nx : ↑↑((𝟭 LocallyRingedSpace).obj X).toPresheafedSpace\n⊢ PrimeSpectrum.comap (CommRingCat.Hom.hom (f.c.app (op ⊤) ≫ X.presheaf.Γgerm x))\n (IsLocalRing.closedPoint ↑(X.presheaf.stalk x)) =\n PrimeSpectrum.comap (CommRingCat.Hom.hom (Y.presheaf.Γgerm ((Concrete... | ← CommRingCat.hom_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 640,
"column": 29
} | {
"line": 640,
"column": 81
} | {
"line": 640,
"column": 81
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nV : X.Opens\nx : ↥V\nh : ↑x ∈ U\nthis : IsAffine ↑U\nr : ↑Γ(↑U, ⊤)\nh₁ : ↑x ∈ X.basicOpen r\nh₂ : X.basicOpen r ≤ V\n⊢ X.basicOpen ((CommRingCat.Hom.hom U.topIso.hom) r) ≤ V ∧ ↑x ∈ X.basicOpen ((CommRingCat.Hom.hom U.topIso.hom) r)",
"ppTerm": "?m.215",... | [] | by simp [Scheme.Opens.toScheme_presheaf_obj, h₁, h₂] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Types.Coequalizers | {
"line": 41,
"column": 9
} | {
"line": 41,
"column": 65
} | {
"line": 41,
"column": 65
} | [
{
"pp": "X Y Z : Type u\nf g : X ⟶ Y\ns : Cofork f g\n⊢ ⇑(hom s.π) ∘ ⇑(hom f) = ⇑(hom s.π) ∘ ⇑(hom g)",
"ppTerm": "?m.92",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.WalkingParallelPair",
"CategoryTheory.ConcreteCategory.hom",
"TypeCat.instFunLikeFun",
"Funct... | [] | by ext x; exact ConcreteCategory.congr_hom s.condition x | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 1054,
"column": 2
} | {
"line": 1054,
"column": 57
} | {
"line": 1055,
"column": 2
} | [
{
"pp": "R M : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nS : Type u\ninst✝² : CommRing S\nN : Type u\ninst✝¹ : AddCommGroup N\ninst✝ : Module S N\nσ : R →+* S\nf : M →ₛₗ[σ] N\nU : Opens ↑(PrimeSpectrum.Top R)\nV : Opens ↑(PrimeSpectrum.Top S)\nhUV : V.carrier ⊆ PrimeSpectrum.com... | [
"R M : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nS : Type u\ninst✝² : CommRing S\nN : Type u\ninst✝¹ : AddCommGroup N\ninst✝ : Module S N\nσ : R →+* S\nf : M →ₛₗ[σ] N\nU : Opens ↑(PrimeSpectrum.Top R)\nV : Opens ↑(PrimeSpectrum.Top S)\nhUV : V.carrier ⊆ PrimeSpectrum.comap σ ⁻¹' U.c... | obtain ⟨hs, H⟩ := h_frac ⟨PrimeSpectrum.comap σ q, hqW⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 951,
"column": 4
} | {
"line": 952,
"column": 77
} | {
"line": 953,
"column": 4
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\ns : Set ↑Γ(X, U)\n⊢ ⋃ i, (PrimeSpectrum.zeroLocus {↑i})ᶜ = Set.univ ↔ Ideal.span s = ⊤",
"ppTerm": "?m.393",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.compl_iInter",
"Semiring.toModule",
"Opposite",
"CommR... | [
"X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\ns : Set ↑Γ(X, U)\n⊢ PrimeSpectrum.zeroLocus (⋃ i, {↑i}) = ∅ ↔ PrimeSpectrum.zeroLocus s = ∅"
] | rw [← Set.compl_iInter, Set.compl_univ_iff, ← PrimeSpectrum.zeroLocus_iUnion, ←
PrimeSpectrum.zeroLocus_empty_iff_eq_top, PrimeSpectrum.zeroLocus_span] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits | {
"line": 136,
"column": 2
} | {
"line": 149,
"column": 29
} | {
"line": 151,
"column": 0
} | [
{
"pp": "X Y : LocallyRingedSpace\nf g : X ⟶ Y\nU : Opens ↑↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace\n⊢ IsLocalHom (CommRingCat.Hom.hom ((coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.c.app (op U)))",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Category... | [] | have := ι_comp_coequalizerComparison f.toShHom g.toShHom SheafedSpace.forgetToPresheafedSpace
dsimp at this
rw [← PreservesCoequalizer.iso_hom] at this
rw [← this, PresheafedSpace.comp_c_app,
← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note (https://github.com/leanprover-comm... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits | {
"line": 136,
"column": 2
} | {
"line": 149,
"column": 29
} | {
"line": 151,
"column": 0
} | [
{
"pp": "X Y : LocallyRingedSpace\nf g : X ⟶ Y\nU : Opens ↑↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace\n⊢ IsLocalHom (CommRingCat.Hom.hom ((coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.c.app (op U)))",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Category... | [] | have := ι_comp_coequalizerComparison f.toShHom g.toShHom SheafedSpace.forgetToPresheafedSpace
dsimp at this
rw [← PreservesCoequalizer.iso_hom] at this
rw [← this, PresheafedSpace.comp_c_app,
← PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_π]
-- Porting note (https://github.com/leanprover-comm... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1121,
"column": 68
} | {
"line": 1131,
"column": 40
} | {
"line": 1133,
"column": 0
} | [
{
"pp": "X : Scheme\ninst✝ : IsAffine X\ns : Set ↥X\n⊢ IsClosed s ↔ ∃ I, s = X.zeroLocus ↑I",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.LocallyRingedSpace.toΓSpecFun",
"AlgebraicGeometry.SheafedSpace.instTop... | [] | by
refine ⟨fun hs ↦ ?_, ?_⟩
· let Z : Set (Spec Γ(X, ⊤)) := X.toΓSpecFun '' s
have hZ : IsClosed Z := (X.isoSpec.hom.homeomorph).isClosedMap _ hs
obtain ⟨I, (hI : Z = _)⟩ := (PrimeSpectrum.isClosed_iff_zeroLocus_ideal _).mp hZ
use I
simp only [← Scheme.toSpecΓ_preimage_zeroLocus, ← hI, Z]
symm
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits | {
"line": 302,
"column": 33
} | {
"line": 302,
"column": 55
} | {
"line": 302,
"column": 55
} | [
{
"pp": "X Y : LocallyRingedSpace\nf g : X ⟶ Y\ns : Cofork f g\ne : Hom.toShHom f ≫ Hom.toShHom s.π = Hom.toShHom g ≫ Hom.toShHom s.π\ny : ↑↑Y.toPresheafedSpace\nh : ↑(s.pt.presheaf.stalk\n ((ConcreteCategory.hom (coequalizer.desc (Hom.toShHom s.π) e).hom.base)\n ((ConcreteCategory.hom (coequalizer.... | [
"X Y : LocallyRingedSpace\nf g : X ⟶ Y\ns : Cofork f g\ne : Hom.toShHom f ≫ Hom.toShHom s.π = Hom.toShHom g ≫ Hom.toShHom s.π\ny : ↑↑Y.toPresheafedSpace\nh : ↑(s.pt.presheaf.stalk\n ((ConcreteCategory.hom (coequalizer.desc (Hom.toShHom s.π) e).hom.base)\n ((ConcreteCategory.hom (coequalizer.π (Hom.toShH... | ← CommRingCat.hom_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 194,
"column": 41
} | {
"line": 194,
"column": 53
} | {
"line": 194,
"column": 54
} | [
{
"pp": "case w.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))\nthis :\n ⇑(TopCat.Hom.hom (pullback.snd (D.f i j) (D.f i k)).base) =\n ⇑(TopCat.Hom.hom (D.t k i).base) ∘\n ⇑(TopCat.Hom.hom (pullback.fst (D.f k i) (D.f k j)).base) ∘ ⇑(To... | [
"case w.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))\nthis :\n ⇑(TopCat.Hom.hom (pullback.snd (D.f i j) (D.f i k)).base) =\n ⇑(TopCat.Hom.hom (D.t k i).base) ∘\n ⇑(TopCat.Hom.hom (pullback.fst (D.f k i) (D.f k j)).base) ∘ ⇑(TopCat.Hom.hom... | ← comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 199,
"column": 41
} | {
"line": 199,
"column": 53
} | {
"line": 199,
"column": 54
} | [
{
"pp": "case w.refine_1\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))\nthis :\n ⇑(TopCat.Hom.hom (pullback.snd (D.f i j) (D.f i k)).base) =\n ⇑(TopCat.Hom.hom (D.t k i).base) ∘\n ⇑(TopCat.Hom.hom (pullback.fst (D.f k i) (D.f k j)).base)... | [
"case w.refine_1\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))\nthis :\n ⇑(TopCat.Hom.hom (pullback.snd (D.f i j) (D.f i k)).base) =\n ⇑(TopCat.Hom.hom (D.t k i).base) ∘\n ⇑(TopCat.Hom.hom (pullback.fst (D.f k i) (D.f k j)).base) ∘ ⇑(TopCat.... | ← comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 202,
"column": 41
} | {
"line": 202,
"column": 53
} | {
"line": 202,
"column": 54
} | [
{
"pp": "case w.refine_2\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))\nthis :\n ⇑(TopCat.Hom.hom (pullback.snd (D.f i j) (D.f i k)).base) =\n ⇑(TopCat.Hom.hom (D.t k i).base) ∘\n ⇑(TopCat.Hom.hom (pullback.fst (D.f k i) (D.f k j)).base)... | [
"case w.refine_2\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))\nthis :\n ⇑(TopCat.Hom.hom (pullback.snd (D.f i j) (D.f i k)).base) =\n ⇑(TopCat.Hom.hom (D.t k i).base) ∘\n ⇑(TopCat.Hom.hom (pullback.fst (D.f k i) (D.f k j)).base) ∘ ⇑(TopCat.... | ← comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 188,
"column": 33
} | {
"line": 188,
"column": 64
} | {
"line": 188,
"column": 64
} | [
{
"pp": "case h₀.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 j k : 𝒰.I₀\n⊢ t' 𝒰 f g i j k ≫\n t' 𝒰 f g j k i ≫\n t' 𝒰 f g k i j ≫\n pullback.fst (fV 𝒰 f g i j) (fV 𝒰 f g i k) ≫\n pullback.fst (pullback.... | [] | cocycle_fst_fst_fst 𝒰 f g i j k | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 485,
"column": 2
} | {
"line": 485,
"column": 25
} | {
"line": 486,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf g : X ⟶ Y\nH : ∀ (x : ↥X), ∃ U, x ∈ U ∧ U.ι ≫ f = U.ι ≫ g\n⊢ f = g",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Scheme",
"AlgebraicGeometry.PresheafedSpace.... | [
"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⊢ f = g"
] | choose U hxU hU using H | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.AlgebraicGeometry.Limits | {
"line": 304,
"column": 4
} | {
"line": 306,
"column": 47
} | {
"line": 308,
"column": 0
} | [
{
"pp": "σ : Type v\ng : σ → Scheme\ninst✝¹ : Small.{u, v} σ\nX : Scheme\nα : (i : σ) → g i ⟶ X\ninst✝ : ∀ (i : σ), IsOpenImmersion (α i)\nhα : _root_.Pairwise (Disjoint on fun x ↦ Set.range ⇑(α x))\nι : Type u\ne : σ ≃ ι\n⊢ IsOpenImmersion (Sigma.desc fun i ↦ α (e.symm i))",
"ppTerm": "?m.71",
"assigne... | [] | apply isOpenImmersion_sigmaDesc_aux
intro i j hij
exact hα (fun h ↦ hij (e.symm.injective h)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Limits | {
"line": 304,
"column": 4
} | {
"line": 306,
"column": 47
} | {
"line": 308,
"column": 0
} | [
{
"pp": "σ : Type v\ng : σ → Scheme\ninst✝¹ : Small.{u, v} σ\nX : Scheme\nα : (i : σ) → g i ⟶ X\ninst✝ : ∀ (i : σ), IsOpenImmersion (α i)\nhα : _root_.Pairwise (Disjoint on fun x ↦ Set.range ⇑(α x))\nι : Type u\ne : σ ≃ ι\n⊢ IsOpenImmersion (Sigma.desc fun i ↦ α (e.symm i))",
"ppTerm": "?m.71",
"assigne... | [] | apply isOpenImmersion_sigmaDesc_aux
intro i j hij
exact hα (fun h ↦ hij (e.symm.injective h)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 687,
"column": 42
} | {
"line": 687,
"column": 74
} | {
"line": 687,
"column": 74
} | [
{
"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\nx : failed to pretty print expression (use 'set_option pp.rawOnError t... | [
"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\nx : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw... | ← cancel_mono (Scheme.Opens.ι _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 529,
"column": 2
} | {
"line": 529,
"column": 25
} | {
"line": 530,
"column": 2
} | [
{
"pp": "R S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nx : ↥(Spec (CommRingCat.of R)) ⊕ ↥(Spec (CommRingCat.of S))\n⊢ (coprodSpec R S) ((coprodMk (Spec (CommRingCat.of R)) (Spec (CommRingCat.of S))) x) =\n (PrimeSpectrum.primeSpectrumProd R S).symm x",
"ppTerm": "?m.14",
"assigned": true,
... | [
"R S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nx : ↥(Spec (CommRingCat.of R)) ⊕ ↥(Spec (CommRingCat.of S))\n⊢ ((coprodSpec R S) ((coprodMk (Spec (CommRingCat.of R)) (Spec (CommRingCat.of S))) x)).asIdeal =\n ((PrimeSpectrum.primeSpectrumProd R S).symm x).asIdeal"
] | apply PrimeSpectrum.ext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.Limits | {
"line": 533,
"column": 4
} | {
"line": 533,
"column": 45
} | {
"line": 534,
"column": 4
} | [
{
"pp": "case inl\nR S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nx : ↥(Spec (CommRingCat.of R))\n⊢ ((Spec.map (CommRingCat.ofHom (RingHom.fst R S))) x).asIdeal =\n ((PrimeSpectrum.primeSpectrumProd R S).symm (Sum.inl x)).asIdeal",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
... | [
"case inl\nR S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nx : ↥(Spec (CommRingCat.of R))\n⊢ Ideal.comap (CommRingCat.Hom.hom (CommRingCat.ofHom (RingHom.fst R S))) x.asIdeal = x.asIdeal.prod ⊤"
] | change Ideal.comap _ _ = x.asIdeal.prod ⊤ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 180,
"column": 76
} | {
"line": 181,
"column": 52
} | {
"line": 182,
"column": 4
} | [
{
"pp": "P : MorphismProperty Scheme\ninst✝¹ : IsZariskiLocalAtTarget P\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : P.RespectsRight IsOpenImmersion\nι : Type u_1\nU : ι → Y.Opens\nH : Set.range ⇑f ⊆ ↑(⨆ i, U i)\nhf : ∀ (i : ι), P (f ∣_ U i)\ng : X ⟶ ↑(⨆ i, U i) := IsOpenImmersion.lift (⨆ i, U i).ι f ⋯\ni : ι\nheq : g ⁻¹ᵁ... | [] | by
simp [← CategoryTheory.cancel_mono (U i).ι, g] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 25
} | {
"line": 191,
"column": 2
} | [
{
"pp": "P : MorphismProperty Scheme\ninst✝ : IsZariskiLocalAtTarget P\nX Y : Scheme\nf : X ⟶ Y\nH : ∀ (x : ↥Y), ∃ U, x ∈ U ∧ P (f ∣_ U)\n⊢ P f",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometr... | [
"P : MorphismProperty Scheme\ninst✝ : IsZariskiLocalAtTarget P\nX Y : Scheme\nf : X ⟶ Y\nU : ↥Y → Y.Opens\nhxU : ∀ (x : ↥Y), x ∈ U x\nhU : ∀ (x : ↥Y), P (f ∣_ U x)\n⊢ P f"
] | choose U hxU hU using H | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 203,
"column": 4
} | {
"line": 204,
"column": 93
} | {
"line": 206,
"column": 0
} | [
{
"pp": "case hf\nP : MorphismProperty Scheme\ninst✝² : IsZariskiLocalAtTarget P\nX Y : Scheme\ninst✝¹ : P.RespectsRight IsOpenImmersion\ninst✝ : P.HasOfPostcompProperty IsOpenImmersion\nf : X ⟶ Y\nU : ↥X → Y.Opens\nh₁ : ∀ (x : ↥X), f x ∈ U x\nh₂ : ∀ (x : ↥X), P ((f ⁻¹ᵁ U x).ι ≫ f)\n⊢ ∀ (i : ↥X), P (f ∣_ U i)",... | [] | intro x
exact P.of_postcomp (f ∣_ U x) (U x).ι (inferInstance : IsOpenImmersion _) (by simp [h₂]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 203,
"column": 4
} | {
"line": 204,
"column": 93
} | {
"line": 206,
"column": 0
} | [
{
"pp": "case hf\nP : MorphismProperty Scheme\ninst✝² : IsZariskiLocalAtTarget P\nX Y : Scheme\ninst✝¹ : P.RespectsRight IsOpenImmersion\ninst✝ : P.HasOfPostcompProperty IsOpenImmersion\nf : X ⟶ Y\nU : ↥X → Y.Opens\nh₁ : ∀ (x : ↥X), f x ∈ U x\nh₂ : ∀ (x : ↥X), P ((f ⁻¹ᵁ U x).ι ≫ f)\n⊢ ∀ (i : ↥X), P (f ∣_ U i)",... | [] | intro x
exact P.of_postcomp (f ∣_ U x) (U x).ι (inferInstance : IsOpenImmersion _) (by simp [h₂]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 302,
"column": 4
} | {
"line": 302,
"column": 53
} | {
"line": 304,
"column": 0
} | [
{
"pp": "case refine_2\nP : MorphismProperty Scheme\ninst✝¹ : IsZariskiLocalAtSource P\ninst✝ : P.IsMultiplicative\nhP : ∀ {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) [IsOpenImmersion g], P (f ≫ g) → P f\nX Y : Scheme\nf : X ⟶ Y\n𝒰 : Scheme.zariskiPrecoverage.ZeroHypercover Y\nh : ∀ (i : 𝒰.I₀), P (pullback.snd f... | [] | exact P.comp_mem _ _ (h i) (of_isOpenImmersion _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Localization.Away.Lemmas | {
"line": 40,
"column": 10
} | {
"line": 40,
"column": 12
} | {
"line": 41,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ns : Set R\nhsone : Ideal.span s = ⊤\nRₜ : ↑s → Type u_2\ninst✝² : (t : ↑s) → CommRing (Rₜ t)\ninst✝¹ : (t : ↑s) → Algebra R (Rₜ t)\ninst✝ : ∀ (t : ↑s), Away (↑t) (Rₜ t)\np : (t : ↑s) → Set (Rₜ t)\nhtone : ∀ (r : ↑s), Ideal.span (p r) = ⊤\na : R\n⊢ a ∈ s → a ∈ ↑(Ideal.... | [
"R : Type u_1\ninst✝³ : CommRing R\ns : Set R\nhsone : Ideal.span s = ⊤\nRₜ : ↑s → Type u_2\ninst✝² : (t : ↑s) → CommRing (Rₜ t)\ninst✝¹ : (t : ↑s) → Algebra R (Rₜ t)\ninst✝ : ∀ (t : ↑s), Away (↑t) (Rₜ t)\np : (t : ↑s) → Set (Rₜ t)\nhtone : ∀ (r : ↑s), Ideal.span (p r) = ⊤\na : R\nha : a ∈ s\n⊢ a ∈ ↑(Ideal.span (Se... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 294,
"column": 4
} | {
"line": 294,
"column": 35
} | {
"line": 295,
"column": 4
} | [
{
"pp": "case hP₂\nα β : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nx✝¹ : Continuous[inst✝¹, inst✝] f\nx✝ : IsOpen[inst✝] s\nhf : ∀ (x : α), closure[inst✝] {f x} ⊆ f '' closure[inst✝¹] {x}\nx : α\nhx : x ∈ f ⁻¹' s\ny : β\nhy : y ∈ s\nhcl : y ∈ closure[inst✝] {f x}\n... | [
"case hP₂\nα β : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nx✝¹ : Continuous[inst✝¹, inst✝] f\nx✝ : IsOpen[inst✝] s\nhf : ∀ (x : α), closure[inst✝] {f x} ⊆ f '' closure[inst✝¹] {x}\nx : α\nhx : x ∈ f ⁻¹' s\ny : β\nhy : y ∈ s\nhcl : y ∈ closure[inst✝] {f x}\na : α\nha : ... | obtain ⟨a, ha, hay⟩ := hf x hcl | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 296,
"column": 38
} | {
"line": 296,
"column": 61
} | {
"line": 296,
"column": 61
} | [
{
"pp": "α β : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nx✝¹ : Continuous[inst✝¹, inst✝] f\nx✝ : IsOpen[inst✝] s\nhf : ∀ (x : α), closure[inst✝] {f x} ⊆ f '' closure[inst✝¹] {x}\nx : α\nhx : x ∈ f ⁻¹' s\ny : β\nhy : y ∈ s\nhcl : f x ⤳ y\na : α\nha : a ∈ closure[ins... | [] | simpa [closure_subtype] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 296,
"column": 38
} | {
"line": 296,
"column": 61
} | {
"line": 296,
"column": 61
} | [
{
"pp": "α β : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nx✝¹ : Continuous[inst✝¹, inst✝] f\nx✝ : IsOpen[inst✝] s\nhf : ∀ (x : α), closure[inst✝] {f x} ⊆ f '' closure[inst✝¹] {x}\nx : α\nhx : x ∈ f ⁻¹' s\ny : β\nhy : y ∈ s\nhcl : f x ⤳ y\na : α\nha : a ∈ closure[ins... | [] | simpa [closure_subtype] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 296,
"column": 38
} | {
"line": 296,
"column": 61
} | {
"line": 296,
"column": 61
} | [
{
"pp": "α β : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nx✝¹ : Continuous[inst✝¹, inst✝] f\nx✝ : IsOpen[inst✝] s\nhf : ∀ (x : α), closure[inst✝] {f x} ⊆ f '' closure[inst✝¹] {x}\nx : α\nhx : x ∈ f ⁻¹' s\ny : β\nhy : y ∈ s\nhcl : f x ⤳ y\na : α\nha : a ∈ closure[ins... | [] | simpa [closure_subtype] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RingHom.Surjective | {
"line": 51,
"column": 2
} | {
"line": 56,
"column": 88
} | {
"line": 58,
"column": 0
} | [
{
"pp": "R 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 ⊗[R] T\n⊢ ∃ a, (algebraMap S (S ⊗[R] T)) a = x",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [] | induction x with
| zero => exact ⟨0, map_zero _⟩
| tmul x y =>
obtain ⟨y, rfl⟩ := h y; use y • x; dsimp
rw [TensorProduct.smul_tmul, Algebra.algebraMap_eq_smul_one]
| add x y ex ey => obtain ⟨⟨x, rfl⟩, ⟨y, rfl⟩⟩ := ex, ey; exact ⟨x + y, map_add _ x y⟩ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 230,
"column": 14
} | {
"line": 230,
"column": 46
} | {
"line": 230,
"column": 46
} | [
{
"pp": "case of_sSup_eq_top.a.refine_2\nP : MorphismProperty Scheme\nhP₂ : ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), IsOpenCover U → (∀ (i : ι), P (f ∣_ U i)) → P f\nX Y : Scheme\nf : X ⟶ Y\nι : Type u\nU : ι → Y.Opens\nhU : iSup U = ⊤\nH : ∀ (i : ι), P.universally (f ∣_ U i)\nX' Y' : Scheme... | [
"case of_sSup_eq_top.a.refine_2\nP : MorphismProperty Scheme\nhP₂ : ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), IsOpenCover U → (∀ (i : ι), P (f ∣_ U i)) → P f\nX Y : Scheme\nf : X ⟶ Y\nι : Type u\nU : ι → Y.Opens\nhU : iSup U = ⊤\nH : ∀ (i : ι), P.universally (f ∣_ U i)\nX' Y' : Scheme\ni₁ : X' ⟶ ... | ← cancel_mono (Scheme.Opens.ι _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 335,
"column": 4
} | {
"line": 335,
"column": 16
} | {
"line": 336,
"column": 4
} | [
{
"pp": "case of_sSup_eq_top\nP : {α β : Type u} → [TopologicalSpace α] → [TopologicalSpace β] → (α → β) → Prop\ninst✝ : (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P).RespectsIso\nhP₁ :\n ∀ {X Y : Type u} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (f : X → Y),\n Cont... | [
"case of_sSup_eq_top\nP : {α β : Type u} → [TopologicalSpace α] → [TopologicalSpace β] → (α → β) → Prop\ninst✝ : (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P).RespectsIso\nhP₁ :\n ∀ {X Y : Type u} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (f : X → Y),\n Continuous f → ∀... | introv hU hf | Mathlib.Tactic.evalIntrov | Mathlib.Tactic.introv |
Mathlib.AlgebraicGeometry.Morphisms.SurjectiveOnStalks | {
"line": 125,
"column": 8
} | {
"line": 125,
"column": 62
} | {
"line": 125,
"column": 62
} | [
{
"pp": "X 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)\nR A B : CommRingCat\niX : Spec A ⟶ X\niY : Spec B ⟶ Y\niS : Spec R ⟶ S\na✝² : IsOpenImmersion iX\na✝¹ : IsOpenImmersion iY\na✝ : IsOpenImmersion iS\... | [
"X 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)\nR A B : CommRingCat\niX : Spec A ⟶ X\niY : Spec B ⟶ Y\niS : Spec R ⟶ S\na✝² : IsOpenImmersion iX\na✝¹ : IsOpenImmersion iY\na✝ : IsOpenImmersion iS\nφ : R ⟶ A\n... | HasRingHomProperty.Spec_iff (P := @SurjectiveOnStalks) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 208,
"column": 6
} | {
"line": 208,
"column": 25
} | {
"line": 208,
"column": 25
} | [
{
"pp": "X : Scheme\nU : X.Opens\nx✝ : ∃ R f, Set.range ⇑f = ↑U\nR : CommRingCat\nf : Spec R ⟶ X\nhf : Set.range ⇑f = ↑U\n⊢ Function.Surjective ⇑(IsOpenImmersion.lift U.ι f ⋯)",
"ppTerm": "?m.101",
"assigned": true,
"usedConstants": [
"Set.range_eq_univ",
"Eq.mpr",
"AlgebraicGeomet... | [
"X : Scheme\nU : X.Opens\nx✝ : ∃ R f, Set.range ⇑f = ↑U\nR : CommRingCat\nf : Spec R ⟶ X\nhf : Set.range ⇑f = ↑U\n⊢ Set.range ⇑(IsOpenImmersion.lift U.ι f ⋯) = Set.univ"
] | ← Set.range_eq_univ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 256,
"column": 12
} | {
"line": 256,
"column": 41
} | {
"line": 256,
"column": 41
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nx f : ↑Γ(X, U)\nH : (x |_ X.basicOpen f) ⋯ = (CommRingCat.Hom.hom (X.presheaf.map (homOfLE ⋯).op)) 0\nn : ℕ\ne : f ^ n * x = f ^ n * 0\n⊢ f ^ n * x = 0",
"ppTerm": "?m.107",
"assigned": true,
"usedConstants": [
"Opposite",
"HMul.hMul... | [] | by simpa [mul_comm x] using e | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Height | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 44
} | {
"line": 195,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : FiniteRingKrullDim R\nl : LTSeries (PrimeSpectrum R) := LTSeries.longestOf (PrimeSpectrum R)\nm : Ideal R\nhm : m.IsMaximal\nhle : (RelSeries.last l).asIdeal ≤ m\n⊢ ringKrullDim R ≤ ↑m.height",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": ... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : FiniteRingKrullDim R\nl : LTSeries (PrimeSpectrum R) := ⋯\nm : Ideal R\nhm : m.IsMaximal\nhle : (RelSeries.last l).asIdeal ≤ m\n⊢ ringKrullDim R ≤ ↑(RelSeries.last l).asIdeal.height",
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : FiniteRingKrullDim R\nl : LTSeries (PrimeS... | trans (l.last.asIdeal.height : WithBot ℕ∞) | Batteries.Tactic._aux_Batteries_Tactic_Trans___elabRules_Batteries_Tactic_tacticTrans____1 | Batteries.Tactic.tacticTrans___ |
Mathlib.RingTheory.Ideal.Height | {
"line": 390,
"column": 2
} | {
"line": 391,
"column": 60
} | {
"line": 392,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nS : Submonoid R\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : IsLocalization S A\nJ : Ideal A\ninst✝ : J.IsPrime\ne : { p // p.IsPrime } ≃o { p // p.IsPrime ∧ Disjoint ↑S ↑p } := orderIsoOfPrime S A\n⊢ Order.krullDim ↑(Set.Iic { asIdeal := J, isPr... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nS : Submonoid R\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : IsLocalization S A\nJ : Ideal A\ninst✝ : J.IsPrime\ne : { p // p.IsPrime } ≃o { p // p.IsPrime ∧ Disjoint ↑S ↑p } := orderIsoOfPrime S A\nH : ∀ p ≤ comap (algebraMap R A) J, Disjoint ↑S ↑p\n⊢ Order... | have H (p : Ideal R) (hp : p ≤ J.comap (algebraMap R A)) : Disjoint (S : Set R) p :=
Set.disjoint_of_subset_right hp (e ⟨_, ‹J.IsPrime›⟩).2.2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 365,
"column": 4
} | {
"line": 365,
"column": 26
} | {
"line": 368,
"column": 4
} | [
{
"pp": "case refine_2.intro\nX : Scheme\nU✝ : X.Opens\nhU✝ : IsCompact U✝.carrier\nS : X.Opens\nhS : IsCompact S.carrier\nU : ↑X.affineOpens\nhU :\n IsQuasiSeparated S.carrier →\n ∀ (f : ↑Γ(X, S)) (x : ↑Γ(X, X.basicOpen f)),\n ∃ n y,\n (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) y =\... | [
"case refine_2.intro\nX : Scheme\nU✝ : X.Opens\nhU✝ : IsCompact U✝.carrier\nS : X.Opens\nhS : IsCompact S.carrier\nU : ↑X.affineOpens\nhU :\n IsQuasiSeparated S.carrier →\n ∀ (f : ↑Γ(X, S)) (x : ↑Γ(X, X.basicOpen f)),\n ∃ n y,\n (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) y =\n (... | choose n hn using this | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 584,
"column": 6
} | {
"line": 584,
"column": 34
} | {
"line": 585,
"column": 6
} | [
{
"pp": "case inr\nP : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝¹ : HasRingHomProperty P Q\nhQ : RingHom.StableUnderCompositionWithLocalizationAwaySource fun {R S} [CommRing R] [CommRing S] ↦ Q\nX Y : Scheme\ninst✝ : IsAffine Y\nU : Y.Ope... | [
"case inr\nP : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝¹ : HasRingHomProperty P Q\nhQ : RingHom.StableUnderCompositionWithLocalizationAwaySource fun {R S} [CommRing R] [CommRing S] ↦ Q\nX Y : Scheme\ninst✝ : IsAffine Y\nU : Y.Opens\nf : X ⟶ ... | apply f.iSup_preimage_eq_top | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.Properties | {
"line": 325,
"column": 2
} | {
"line": 325,
"column": 64
} | {
"line": 326,
"column": 2
} | [
{
"pp": "X Y : Scheme\nh : IsIntegral X\nf : X ⟶ Y\ninst✝ : IsIso f\n⊢ IsIntegral Y",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme",
"AlgebraicGeometry.PresheafedSpace.carrier",
"CommRingCat",
"CategoryTheory.IsIso.inv_isIso",
"CommRi... | [
"X Y : Scheme\nh : IsIntegral X\nf : X ⟶ Y\ninst✝ : IsIso f\n⊢ Nonempty ↥Y"
] | suffices Nonempty Y from isIntegral_of_isOpenImmersion (inv f) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.AlgebraicGeometry.Properties | {
"line": 381,
"column": 4
} | {
"line": 381,
"column": 46
} | {
"line": 382,
"column": 4
} | [
{
"pp": "case inr\nX : Scheme\nx : ↥X\nthis : ∀ {X : Scheme} (x : ↥X), (∃ R, X = Spec R) → ringKrullDim ↑(X.presheaf.stalk x) = ↑(coheight x)\nh : ¬∃ R, X = Spec R\nR : CommRingCat\nf : Spec R ⟶ X\nhf : IsOpenImmersion f\nhsub : x ∈ Set.range ⇑f ∧ Set.range ⇑f ⊆ ↑⊤\n⊢ ringKrullDim ↑(X.presheaf.stalk x) = ↑(cohe... | [
"case inr\nX : Scheme\nthis : ∀ {X : Scheme} (x : ↥X), (∃ R, X = Spec R) → ringKrullDim ↑(X.presheaf.stalk x) = ↑(coheight x)\nh : ¬∃ R, X = Spec R\nR : CommRingCat\nf : Spec R ⟶ X\nhf : IsOpenImmersion f\ny : ↥(Spec R)\nhsub : f y ∈ Set.range ⇑f ∧ Set.range ⇑f ⊆ ↑⊤\n⊢ ringKrullDim ↑(X.presheaf.stalk (f y)) = ↑(coh... | obtain ⟨y, rfl⟩ := Set.mem_range.mp hsub.1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 76,
"column": 4
} | {
"line": 76,
"column": 26
} | {
"line": 76,
"column": 26
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nU : ↑X.affineOpens\nφ : Γ(X, ↑U) ⟶ CommRingCat.of (↑Γ(X, ↑U) ⧸ I.ideal U) := CommRingCat.ofHom (Ideal.Quotient.mk (I.ideal U))\n⊢ RingHom.ker\n (CommRingCat.Hom.hom\n (Hom.appTop (Spec.map φ ≫ (IsAffineOpen.isoSpec ⋯).inv) ≫\n (ΓSpecIso (CommRingCa... | [
"X : Scheme\nI : X.IdealSheafData\nU : ↑X.affineOpens\nφ : Γ(X, ↑U) ⟶ CommRingCat.of (↑Γ(X, ↑U) ⧸ I.ideal U) := CommRingCat.ofHom (Ideal.Quotient.mk (I.ideal U))\n⊢ RingHom.ker\n (CommRingCat.Hom.hom\n (Hom.appTop (Spec.map φ ≫ (IsAffineOpen.isoSpec ⋯).inv) ≫\n (ΓSpecIso (CommRingCat.of (↑Γ(X, ... | ← CommRingCat.hom_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 741,
"column": 2
} | {
"line": 741,
"column": 61
} | {
"line": 742,
"column": 2
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝ : HasRingHomProperty P Q\nX Y : Scheme\nf : X ⟶ Y\nQ' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhQ' : RespectsIso fun {R S} [CommRing R] [Comm... | [
"P : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝ : HasRingHomProperty P Q\nX Y : Scheme\nf : X ⟶ Y\nQ' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhQ' : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q'... | rw [hQ'.arrow_mk_iso_iff (Scheme.arrowStalkMapSpecIso φ _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 26
} | {
"line": 183,
"column": 26
} | [
{
"pp": "case e'_2.e'_6\nX : 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) ∈... | [
"case e'_2.e'_6\nX : 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... | ← CommRingCat.hom_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 260,
"column": 10
} | {
"line": 260,
"column": 42
} | {
"line": 260,
"column": 42
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nU V W U₀ : ↑X.affineOpens\nhU₀ : ↑U ⊓ ↑W ≤ ↑U₀\n⊢ (pullback.fst (pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯)) ≫\n I.glueDataT U V ≫ pullback.fst (I.glueDataObjι V) (X.homOfLE ⋯)) ≫\n I.glueDataObjι V =\n ... | [
"X : Scheme\nI : X.IdealSheafData\nU V W U₀ : ↑X.affineOpens\nhU₀ : ↑U ⊓ ↑W ≤ ↑U₀\n⊢ ((pullback.fst (pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι U) (X.homOfLE ⋯)) ≫\n I.glueDataT U V ≫ pullback.fst (I.glueDataObjι V) (X.homOfLE ⋯)) ≫\n I.glueDataObjι V) ≫\n (↑V)... | ← cancel_mono (Scheme.Opens.ι _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 299,
"column": 10
} | {
"line": 299,
"column": 42
} | {
"line": 299,
"column": 42
} | [
{
"pp": "case h₁\nX : Scheme\nI : X.IdealSheafData\ni j k : ↑X.affineOpens\n⊢ (pullback.lift (I.glueDataT'Aux (i, j).1 (i, j).2 (i, k).2 (j, k).2 ⋯)\n (I.glueDataT'Aux (i, j).1 (i, j).2 (i, k).2 (j, i).2 ⋯) ⋯ ≫\n pullback.snd (pullback.fst (I.glueDataObjι (j, k).1) (X.homOfLE ⋯))\n (pul... | [
"case h₁\nX : Scheme\nI : X.IdealSheafData\ni j k : ↑X.affineOpens\n⊢ ((pullback.lift (I.glueDataT'Aux (i, j).1 (i, j).2 (i, k).2 (j, k).2 ⋯)\n (I.glueDataT'Aux (i, j).1 (i, j).2 (i, k).2 (j, i).2 ⋯) ⋯ ≫\n pullback.snd (pullback.fst (I.glueDataObjι (j, k).1) (X.homOfLE ⋯))\n (pullback... | ← cancel_mono (Scheme.Opens.ι _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 296,
"column": 4
} | {
"line": 300,
"column": 37
} | {
"line": 301,
"column": 2
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\ni j k : ↑X.affineOpens\n⊢ pullback.lift (I.glueDataT'Aux (i, j).1 (i, j).2 (i, k).2 (j, k).2 ⋯)\n (I.glueDataT'Aux (i, j).1 (i, j).2 (i, k).2 (j, i).2 ⋯) ⋯ ≫\n pullback.snd (pullback.fst (I.glueDataObjι (j, k).1) (X.homOfLE ⋯))\n (pullback.fst (I.glu... | [] | apply pullback.hom_ext
· rw [← cancel_mono (glueDataObjι _ _)]
simp
· rw [← cancel_mono (Scheme.Opens.ι _)]
simp [pullback.condition_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 296,
"column": 4
} | {
"line": 300,
"column": 37
} | {
"line": 301,
"column": 2
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\ni j k : ↑X.affineOpens\n⊢ pullback.lift (I.glueDataT'Aux (i, j).1 (i, j).2 (i, k).2 (j, k).2 ⋯)\n (I.glueDataT'Aux (i, j).1 (i, j).2 (i, k).2 (j, i).2 ⋯) ⋯ ≫\n pullback.snd (pullback.fst (I.glueDataObjι (j, k).1) (X.homOfLE ⋯))\n (pullback.fst (I.glu... | [] | apply pullback.hom_ext
· rw [← cancel_mono (glueDataObjι _ _)]
simp
· rw [← cancel_mono (Scheme.Opens.ι _)]
simp [pullback.condition_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 305,
"column": 46
} | {
"line": 305,
"column": 78
} | {
"line": 305,
"column": 78
} | [
{
"pp": "case h₀.h₀\nX : Scheme\nI : X.IdealSheafData\ni j k : ↑X.affineOpens\n⊢ (((pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫\n pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫\n pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDat... | [
"case h₀.h₀\nX : Scheme\nI : X.IdealSheafData\ni j k : ↑X.affineOpens\n⊢ ((((pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫\n pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫\n pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux ... | ← cancel_mono (Scheme.Opens.ι _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 311,
"column": 12
} | {
"line": 311,
"column": 44
} | {
"line": 311,
"column": 44
} | [
{
"pp": "case h₁\nX : Scheme\nI : X.IdealSheafData\ni j k : ↑X.affineOpens\n⊢ ((pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫\n pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫\n pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux k... | [
"case h₁\nX : Scheme\nI : X.IdealSheafData\ni j k : ↑X.affineOpens\n⊢ (((pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫\n pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫\n pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux k i j k ... | ← cancel_mono (Scheme.Opens.ι _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 314,
"column": 46
} | {
"line": 314,
"column": 78
} | {
"line": 314,
"column": 78
} | [
{
"pp": "case h₁.h₀\nX : Scheme\nI : X.IdealSheafData\ni j k : ↑X.affineOpens\n⊢ (((pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫\n pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫\n pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDat... | [
"case h₁.h₀\nX : Scheme\nI : X.IdealSheafData\ni j k : ↑X.affineOpens\n⊢ ((((pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫\n pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫\n pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux ... | ← cancel_mono (Scheme.Opens.ι _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 880,
"column": 6
} | {
"line": 880,
"column": 38
} | {
"line": 880,
"column": 38
} | [
{
"pp": "X Y : Scheme\nS R : CommRingCat\nφ : S ⟶ R\ninst✝ : QuasiCompact (Spec.map φ)\n⊢ closure (Set.range (PrimeSpectrum.comap (CommRingCat.Hom.hom ((ΓSpecIso S).inv ≫ appTop (Spec.map φ))))) ⊆\n closure (Set.range ⇑(Spec.map φ))",
"ppTerm": "?m.585",
"assigned": true,
"usedConstants": [
... | [
"X Y : Scheme\nS R : CommRingCat\nφ : S ⟶ R\ninst✝ : QuasiCompact (Spec.map φ)\n⊢ closure (Set.range (PrimeSpectrum.comap (CommRingCat.Hom.hom (φ ≫ (ΓSpecIso R).inv)))) ⊆\n closure (Set.range ⇑(Spec.map φ))"
] | ← Scheme.ΓSpecIso_inv_naturality | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 320,
"column": 12
} | {
"line": 320,
"column": 44
} | {
"line": 320,
"column": 44
} | [
{
"pp": "case h₁\nX : Scheme\nI : X.IdealSheafData\ni j k : ↑X.affineOpens\n⊢ ((pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫\n pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫\n pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux k... | [
"case h₁\nX : Scheme\nI : X.IdealSheafData\ni j k : ↑X.affineOpens\n⊢ (((pullback.lift (I.glueDataT'Aux i j k k ⋯) (I.glueDataT'Aux i j k i ⋯) ⋯ ≫\n pullback.lift (I.glueDataT'Aux j k i i ⋯) (I.glueDataT'Aux j k i j ⋯) ⋯ ≫\n pullback.lift (I.glueDataT'Aux k i j j ⋯) (I.glueDataT'Aux k i j k ... | ← cancel_mono (Scheme.Opens.ι _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 513,
"column": 4
} | {
"line": 513,
"column": 77
} | {
"line": 514,
"column": 4
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nx : ↥I.subscheme\n⊢ x ∈ Set.range ⇑(I.glueData.ι (Cover.idx (X.openCoverOfIsOpenCover (fun i ↦ ↑i) ⋯) ↑x) ≫ I.subschemeIso.inv)",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.iSup_affineOpens_eq_top",
"AlgebraicGeom... | [
"X : Scheme\nI : X.IdealSheafData\nx : ↥I.subscheme\nU : (X.openCoverOfIsOpenCover (fun i ↦ ↑i) ⋯).I₀ := Cover.idx (X.openCoverOfIsOpenCover (fun i ↦ ↑i) ⋯) ↑x\n⊢ x ∈ Set.range ⇑(I.glueData.ι (Cover.idx (X.openCoverOfIsOpenCover (fun i ↦ ↑i) ⋯) ↑x) ≫ I.subschemeIso.inv)"
] | let U := (X.openCoverOfIsOpenCover _ (iSup_affineOpens_eq_top X)).idx x.1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.RingHom.Finite | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 21
} | {
"line": 70,
"column": 2
} | [
{
"pp": "R S : Type u_5\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\nf : R →+* S\nM : Submonoid R\nR' S' : Type u_5\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\ninst✝³ : Algebra R R'\ninst✝² : Algebra S S'\ninst✝¹ : IsLocalization M R'\ninst✝ : IsLocalization (Submonoid.map f M) S'\nhf : f.Finite\nthis✝ : Algebra... | [
"R S : Type u_5\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\nf : R →+* S\nM : Submonoid R\nR' S' : Type u_5\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\ninst✝³ : Algebra R R'\ninst✝² : Algebra S S'\ninst✝¹ : IsLocalization M R'\ninst✝ : IsLocalization (Submonoid.map f M) S'\nhf : f.Finite\nthis✝¹ : Algebra R S := f.t... | let := f'.toAlgebra | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicGeometry.Morphisms.AffineAnd | {
"line": 301,
"column": 4
} | {
"line": 301,
"column": 26
} | {
"line": 301,
"column": 26
} | [
{
"pp": "Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nP : MorphismProperty Scheme\nhP : HasAffineProperty P (affineAnd fun {R S} [CommRing R] [CommRing S] ↦ Q)\nhQi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q\nhQ :\n ∀ {R S T : Type u} [inst : CommRing ... | [
"Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nP : MorphismProperty Scheme\nhP : HasAffineProperty P (affineAnd fun {R S} [CommRing R] [CommRing S] ↦ Q)\nhQi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q\nhQ :\n ∀ {R S T : Type u} [inst : CommRing R] [inst_1 :... | ← CommRingCat.hom_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 116,
"column": 2
} | {
"line": 128,
"column": 39
} | {
"line": 129,
"column": 2
} | [
{
"pp": "case inst\nX Y Z : Scheme\nf : X ⟶ Y\n⊢ (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] f ↦ IsLocallyClosed (Set.range f)).RespectsIso",
"ppTerm": "?inst✝",
"assigned": true,
"usedConstants": [
"IsLocallyClosed.image",
"Homeomorph.isInducing",
"Iff.mpr",... | [
"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), IsLocallyClosed (Se... | · refine { precomp := ?_, postcomp := ?_ }
· intro X Y Z i hi f hf
change IsIso i at hi
change IsLocallyClosed _
simpa only [Scheme.Hom.comp_base, TopCat.coe_comp, Set.range_comp,
Set.range_eq_univ.mpr i.surjective, Set.image_univ]
· intro X Y Z i hi f hf
change IsIso i at hi
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Finiteness.FinitePresentationLocal | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 72
} | {
"line": 88,
"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 →... | [
"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 →ₐ[R] S\nhf :... | simp only [Finset.univ_eq_attach, I, Ideal.mem_span_singleton] at hp | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 94
} | {
"line": 229,
"column": 2
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Y\n⊢ IsImmersion (prod.lift (𝟙 X) (𝟙 X))",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"CategoryTheory.MorphismProperty.IsLocalAtTarget.toRespects",
"AlgebraicGeometry.Scheme",
"A... | [
"X Y Z : Scheme\nf : X ⟶ Y\n⊢ IsImmersion (prod.lift (𝟙 X) (𝟙 X) ≫ (prodIsoPullback X X).hom)"
] | rw [← MorphismProperty.cancel_right_of_respectsIso @IsImmersion _ (prodIsoPullback X X).hom] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Spectrum.Prime.ConstructibleSet | {
"line": 155,
"column": 62
} | {
"line": 155,
"column": 95
} | {
"line": 156,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\ns : ConstructibleSetData R\nhs : IsConstructible s.toSet\ne_2✝ : ↥s = ↑↑s\nC : ↥s\nI : Ideal R := Ideal.span (Set.range (↑C).g)\nf : R ⧸ I := (Ideal.Quotient.mk I) (↑C).f\n⊢ comap (Ideal.Quotient.mk I) ''\n ↑((TopologicalSpace.Opens.comap { toFun := comap (Ideal.Quo... | [
"R : Type u\ninst✝ : CommRing R\ns : ConstructibleSetData R\nhs : IsConstructible s.toSet\ne_2✝ : ↥s = ↑↑s\nC : ↥s\nI : Ideal R := Ideal.span (Set.range (↑C).g)\nf : R ⧸ I := (Ideal.Quotient.mk I) (↑C).f\n⊢ comap (Ideal.Quotient.mk I) ''\n ⇑{ toFun := comap (Ideal.Quotient.mk I), continuous_toFun := ⋯ } ⁻¹' ↑(... | TopologicalSpace.Opens.coe_comap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Separated | {
"line": 312,
"column": 2
} | {
"line": 312,
"column": 44
} | {
"line": 313,
"column": 2
} | [
{
"pp": "W X Y Z : Scheme\ninst✝² : IsReduced X\nf g : X ⟶ Y\ns : Y ⟶ Z\ninst✝¹ : IsSeparated s\nh : f ≫ s = g ≫ s\nι : W ⟶ X\ninst✝ : IsDominant ι\nhU : ι ≫ f = ι ≫ g\nX' : Over Z := Over.mk (f ≫ s)\nY' : Over Z := Over.mk s\nU' : Over Z := Over.mk (ι ≫ f ≫ s)\nf' : X' ⟶ Y' := Over.homMk f ⋯\ng' : X' ⟶ Y' := O... | [
"W X Y Z : Scheme\ninst✝² : IsReduced X\nf g : X ⟶ Y\ns : Y ⟶ Z\ninst✝¹ : IsSeparated s\nh : f ≫ s = g ≫ s\nι : W ⟶ X\ninst✝ : IsDominant ι\nhU : ι ≫ f = ι ≫ g\nX' : Over Z := Over.mk (f ≫ s)\nY' : Over Z := Over.mk s\nU' : Over Z := Over.mk (ι ≫ f ≫ s)\nf' : X' ⟶ Y' := Over.homMk f ⋯\ng' : X' ⟶ Y' := Over.homMk g ... | rw [← cancel_epi (equalizer.ι f' g').left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 384,
"column": 2
} | {
"line": 389,
"column": 45
} | {
"line": 391,
"column": 0
} | [
{
"pp": "X Y Z : Scheme\n⊢ MorphismProperty.IsStableUnderBaseChange @IsClosedImmersion",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.instIsAffinePullbackSchemeOfIsAffineHom_1",
"CategoryTheory.Limits.pullback",
"RingHom.IsStableUnderBaseChange.pullback... | [] | apply HasAffineProperty.isStableUnderBaseChange
have := HasAffineProperty.isLocal_affineProperty @IsClosedImmersion
apply AffineTargetMorphismProperty.IsStableUnderBaseChange.mk
intro X Y S _ _ f g ⟨ha, hsurj⟩
exact ⟨inferInstance, RingHom.surjective_isStableUnderBaseChange.pullback_fst_appTop _
RingHom.sur... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 384,
"column": 2
} | {
"line": 389,
"column": 45
} | {
"line": 391,
"column": 0
} | [
{
"pp": "X Y Z : Scheme\n⊢ MorphismProperty.IsStableUnderBaseChange @IsClosedImmersion",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.instIsAffinePullbackSchemeOfIsAffineHom_1",
"CategoryTheory.Limits.pullback",
"RingHom.IsStableUnderBaseChange.pullback... | [] | apply HasAffineProperty.isStableUnderBaseChange
have := HasAffineProperty.isLocal_affineProperty @IsClosedImmersion
apply AffineTargetMorphismProperty.IsStableUnderBaseChange.mk
intro X Y S _ _ f g ⟨ha, hsurj⟩
exact ⟨inferInstance, RingHom.surjective_isStableUnderBaseChange.pullback_fst_appTop _
RingHom.sur... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 464,
"column": 2
} | {
"line": 464,
"column": 30
} | {
"line": 466,
"column": 0
} | [
{
"pp": "R S : CommRingCat\ninst✝ : Subsingleton ↥(Spec S)\nφ : S ⟶ R\nψ : R ⟶ S\nhg : φ ≫ ψ = 𝟙 S\n⊢ Function.LeftInverse ⇑(ConcreteCategory.hom ψ) ⇑(ConcreteCategory.hom φ)",
"ppTerm": "?m.414",
"assigned": true,
"usedConstants": [
"CommRingCat.carrier",
"congrArg",
"CommSemirin... | [] | exact fun x ↦ congr($hg.1 x) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 66,
"column": 10
} | {
"line": 66,
"column": 36
} | {
"line": 66,
"column": 37
} | [
{
"pp": "case inr.mpr.inl\nR : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Module.Free R A\ninst✝¹ : Module.Finite R A\nf : A\nI : Ideal R\ninst✝ : I.IsPrime\nh✝ : Nontrivial R\nthis : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A... | [
"case inr.mpr.inl\nR : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : Module.Free R A\ninst✝¹ : Module.Finite R A\nf : A\nI : Ideal R\ninst✝ : I.IsPrime\nh✝ : Nontrivial R\nthis : Module.finrank I.ResidueField (I.ResidueField ⊗[R] A) = Module.finrank R A\nH : ∀ i < ... | ← Polynomial.leadingCoeff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 152,
"column": 65
} | {
"line": 155,
"column": 29
} | {
"line": 157,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R[X]\n⊢ comap C '' ↑(basicOpen f) = (zeroLocus (Set.range f.coeff))ᶜ",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Set.ext",
"Polynomial.mem_image_comap_C_basicOpen",
"Eq.mpr",
"Polynomial.C",
"SetLike.mem_coe.... | [] | by
ext p
rw [mem_image_comap_C_basicOpen]
simp [Set.range_subset_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 23
} | {
"line": 161,
"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)\n⊢ IsOpen (comap C '' ⋃₀ S)",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"CommSemiri... | [
"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)\n⊢ IsOpen (⋃₀ (Set.image (comap C) '' S))"
] | rw [Set.image_sUnion] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 215,
"column": 76
} | {
"line": 218,
"column": 29
} | {
"line": 220,
"column": 0
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\nσ : Type u_1\nf : MvPolynomial σ R\n⊢ comap C '' ↑(basicOpen f) = (zeroLocus (Set.range fun m ↦ coeff m f))ᶜ",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Nat.instMulZeroCl... | [] | by
ext p
rw [mem_image_comap_C_basicOpen]
simp [Set.range_subset_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.Polynomial | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 23
} | {
"line": 224,
"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)\n⊢ IsOpen (comap C '' ⋃₀ S)",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",... | [
"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)\n⊢ IsOpen (⋃₀ (Set.image (comap C) '' S))"
] | rw [Set.image_sUnion] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Geometrically.Basic | {
"line": 110,
"column": 20
} | {
"line": 110,
"column": 23
} | {
"line": 110,
"column": 24
} | [
{
"pp": "P : ObjectProperty Scheme\nX Y : Scheme\nf : X ⟶ Y\nH : ∀ (y : ↥Y), geometrically P (Scheme.Hom.fiberToSpecResidueField f y)\nK : Type u\ninst✝ : Field K\ny : Spec (of K) ⟶ Y\nZ : Scheme\nfst : Z ⟶ X\n⊢ ∀ (snd : Z ⟶ Spec (of K)), IsPullback fst snd f y → P Z",
"ppTerm": "?m.37",
"assigned": tru... | [
"P : ObjectProperty Scheme\nX Y : Scheme\nf : X ⟶ Y\nH : ∀ (y : ↥Y), geometrically P (Scheme.Hom.fiberToSpecResidueField f y)\nK : Type u\ninst✝ : Field K\ny : Spec (of K) ⟶ Y\nZ : Scheme\nfst : Z ⟶ X\nsnd : Z ⟶ Spec (of K)\n⊢ IsPullback fst snd f y → P Z"
] | snd | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.AlgebraicGeometry.Morphisms.Finite | {
"line": 114,
"column": 85
} | {
"line": 115,
"column": 65
} | {
"line": 117,
"column": 0
} | [
{
"pp": "⊢ @IsFinite = @IsIntegralHom ⊓ @LocallyOfFiniteType",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.IsIntegralHom",
"CategoryTheory.MorphismProperty",
"AlgebraicGeometry.Scheme",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom"... | [] | by
ext; exact IsFinite.iff_isIntegralHom_and_locallyOfFiniteType _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.Finite | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 24
} | {
"line": 194,
"column": 2
} | [
{
"pp": "case inr\nX : Scheme\ninst✝² : Subsingleton ↥X\ninst✝¹ : IsReduced X\nS : CommRingCat\nf : X ⟶ Spec S\ninst✝ : JacobsonSpace ↥(Spec S)\nthis :\n ∀ {X : Scheme} [Subsingleton ↥X] [IsReduced X] {f : X ⟶ Spec S},\n (∃ R, X = Spec R) → (IsFinite f ↔ LocallyOfFiniteType f)\nhX : ¬∃ R, X = Spec R\ninst :... | [] | refine this ⟨_, rfl⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.QuasiAffine | {
"line": 142,
"column": 8
} | {
"line": 142,
"column": 40
} | {
"line": 142,
"column": 40
} | [
{
"pp": "case refine_1.e_a\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsAffineHom f\ninst✝ : Y.IsQuasiAffine\nthis✝¹ : CompactSpace ↥X\nthis✝ : X.IsQuasiAffine\nthis :\n ∀ (r : ↑Γ(Y, ⊤)),\n IsPushout (Hom.appTop f) (Y.presheaf.map (homOfLE ⋯).op) (X.presheaf.map (homOfLE ⋯).op)\n (Hom.appLE f (Y.basicOpen r) (... | [
"case refine_1.e_a\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsAffineHom f\ninst✝ : Y.IsQuasiAffine\nthis✝¹ : CompactSpace ↥X\nthis✝ : X.IsQuasiAffine\nthis :\n ∀ (r : ↑Γ(Y, ⊤)),\n IsPushout (Hom.appTop f) (Y.presheaf.map (homOfLE ⋯).op) (X.presheaf.map (homOfLE ⋯).op)\n (Hom.appLE f (Y.basicOpen r) (X.basicOpen ... | ← cancel_mono (Scheme.Opens.ι _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 245,
"column": 10
} | {
"line": 245,
"column": 13
} | {
"line": 245,
"column": 14
} | [
{
"pp": "n : ℕ\nP : (R : Type u) → [inst : CommRing R] → InductionObj R n → Prop\nhP₁ : ∀ (R : Type u) [inst : CommRing R], P R { val := 0 }\nhP₂ :\n ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i : Fin n),\n (e.val i).Monic → (∀ (j : Fin n), j ≠ i → e.val j = 0) → P R e\nhP₃ :\n ∀ (R : Type ... | [
"n : ℕ\nP : (R : Type u) → [inst : CommRing R] → InductionObj R n → Prop\nhP₁ : ∀ (R : Type u) [inst : CommRing R], P R { val := 0 }\nhP₂ :\n ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i : Fin n),\n (e.val i).Monic → (∀ (j : Fin n), j ≠ i → e.val j = 0) → P R e\nhP₃ :\n ∀ (R : Type u) [inst : C... | hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 284,
"column": 2
} | {
"line": 286,
"column": 66
} | {
"line": 288,
"column": 0
} | [
{
"pp": "n : Type u\nS T : Scheme\nf : S ⟶ T\n⊢ map n f ≫ toSpecMvPoly n T = toSpecMvPoly n S",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"Nat.instMulZeroClass",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarri... | [] | apply (toSpecMvPolyIntEquiv _).injective
ext i
rw [toSpecMvPolyIntEquiv_comp, ← coord, map_appTop_coord, coord] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 284,
"column": 2
} | {
"line": 286,
"column": 66
} | {
"line": 288,
"column": 0
} | [
{
"pp": "n : Type u\nS T : Scheme\nf : S ⟶ T\n⊢ map n f ≫ toSpecMvPoly n T = toSpecMvPoly n S",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"Nat.instMulZeroClass",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarri... | [] | apply (toSpecMvPolyIntEquiv _).injective
ext i
rw [toSpecMvPolyIntEquiv_comp, ← coord, map_appTop_coord, coord] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered | {
"line": 74,
"column": 54
} | {
"line": 82,
"column": 22
} | {
"line": 84,
"column": 0
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v, u} J\ninst✝ : IsFiltered J\nF : J ⥤ Type w₀\nj : J\nx y : F.obj j\n⊢ F.ιColimitType j x = F.ιColimitType j y ↔\n ∃ k f, (ConcreteCategory.hom (F.map f)) x = (ConcreteCategory.hom (F.map f)) y",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
... | [] | by
rw [ιColimitType_eq_iff_of_isFiltered]
constructor
· rintro ⟨k, f, f', h⟩
refine ⟨coeq f f', f ≫ coeqHom f f', ?_⟩
nth_rw 2 [coeq_condition]
simp [h]
· rintro ⟨k, f, h⟩
exact ⟨k, f, f, h⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.AlgClosed.Basic | {
"line": 108,
"column": 4
} | {
"line": 109,
"column": 29
} | {
"line": 110,
"column": 2
} | [
{
"pp": "case refine_1\nX Y : Scheme\nK : Type u\ninst✝⁵ : Field K\ninst✝⁴ : IsAlgClosed K\nf g : X ⟶ Y\ni : Y ⟶ Spec (CommRingCat.of K)\ninst✝³ : IsSeparated i\ninst✝² : LocallyOfFiniteType i\ninst✝¹ : IsReduced X\ninst✝ : LocallyOfFiniteType (f ≫ i)\nS : Set ↥X\nhS : IsLocallyClosed S\nhS' : Dense S\nH : ∀ x ... | [] | rwa [dense_iff_closure_eq, JacobsonSpace.closure_inter_closedPoints_eq_closure hS,
← dense_iff_closure_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
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