module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 108,
"column": 4
} | {
"line": 110,
"column": 91
} | {
"line": 111,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u\nS : Type v\nT : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra.FiniteType R T\ninst✝¹ : Algebra.IsIntegral R S\nf : S →ₐ[R] T\ng : S\nhg : Function.Surjective ⇑(awayMapₐ f g)\np : Ideal R\n... | [
"case refine_1\nR : Type u\nS : Type v\nT : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra.FiniteType R T\ninst✝¹ : Algebra.IsIntegral R S\nf : S →ₐ[R] T\ng : S\nhg : Function.Surjective ⇑(awayMapₐ f g)\np : Ideal R\ninst✝ : p.Is... | have : IsLocalization.Away (f.toRingHom (algebraMap R S r))
(Localization.Away (algebraMap R T r)) := by
simp only [AlgHom.toRingHom_eq_coe, RingHom.coe_coe, AlgHom.commutes]; infer_instance | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Finiteness.Descent | {
"line": 63,
"column": 6
} | {
"line": 70,
"column": 68
} | {
"line": 71,
"column": 2
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Module.FaithfullyFlat R S\nI : Ideal R\nhI : (Ideal.map (algebraMap R S) I).FG\nf : S ⊗[R] ↥I →ₗ[S] S := ↑(AlgebraTensorModule.rid R S S) ∘ₗ (AlgebraTensorModule.lTensor S S) (Submodule.su... | [] | induction x with
| zero => simp
| add _ _ _ _ => simp_all [Ideal.add_mem]
| tmul s x =>
have : f (s ⊗ₜ[R] x) = s • f (1 ⊗ₜ x) := by simp [f]
rw [this]
apply Ideal.mul_mem_left
simpa [f, Algebra.smul_def] using Ideal.mem_map_of_mem _ x.2 | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 82
} | {
"line": 226,
"column": 4
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : FiniteType R S\np : Ideal R\ninst✝³ : p.IsPrime\nq✝ : Ideal S\ninst✝² : q✝.IsPrime\ninst✝¹ : q✝.LiesOver p\ninst✝ : QuasiFiniteAt R q✝\ns : S\nhsq : s ∉ q✝\nhRs : IsIntegral R s\nhs : ∀ (q' : Ideal S), q'.I... | [
"R : Type u\nS : Type v\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : FiniteType R S\np : Ideal R\ninst✝³ : p.IsPrime\nq✝ : Ideal S\ninst✝² : q✝.IsPrime\ninst✝¹ : q✝.LiesOver p\ninst✝ : QuasiFiniteAt R q✝\ns : S\nhsq : s ∉ q✝\nhRs : IsIntegral R s\nhs : ∀ (q' : Ideal S), q'.IsPrime → q' ... | set m' := rootMultiplicity 0 ((minpoly R s).map (algebraMap R p.ResidueField)) | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 57
} | {
"line": 282,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nV : { x // QuasiFiniteAt f x } → (normalization f).Opens\nhxV✝ : ∀ (x : { x // QuasiFiniteAt f x }), (toNormalization f) ↑x ∈ V x\nhV : ∀ (x : { x // QuasiFiniteAt f x }), IsIso (toNormalization f ∣... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nV : { x // QuasiFiniteAt f x } → (normalization f).Opens\nhxV✝ : ∀ (x : { x // QuasiFiniteAt f x }), (toNormalization f) ↑x ∈ V x\nhV : ∀ (x : { x // QuasiFiniteAt f x }), IsIso (toNormalization f ∣_ V x)\n𝒰 :... | refine .of_comp (g := (Y.presheaf.germ U _ hxU).hom) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Tactic.CategoryTheory.Bicategory.Normalize | {
"line": 47,
"column": 44
} | {
"line": 48,
"column": 12
} | {
"line": 50,
"column": 0
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b : B\nf g h i j : a ⟶ b\nα : f ≅ g\nη : g ⟶ h\nηs : h ⟶ i\nθ : i ⟶ j\nι : h ⟶ j\ne_ι : ηs ≫ θ = ι\n⊢ (α.hom ≫ η ≫ ηs) ≫ θ = α.hom ≫ η ≫ ι",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"CategoryTheory.Category.assoc",
"CategoryTheo... | [] | by
simp [e_ι] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 532,
"column": 2
} | {
"line": 599,
"column": 78
} | {
"line": 601,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type (max u v)\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\np : Ideal R\ninst✝ : p.IsPrime\n⊢ ∃ R' x x_1,\n ∃ (_ : Etale R R'),\n ∃ P,\n ∃ (x_3 : P.IsPrime) (x_4 : P.LiesOver p),\n ∃ n e,\n ∃ (_ : Complet... | [] | induction h : (p.primesOver S).ncard using Nat.strong_induction_on generalizing R S with
| h n IH =>
have : IsArtinianRing (p.ResidueField ⊗[R] S) := IsArtinianRing.of_finite p.ResidueField _
have hpSfin : (p.primesOver S).Finite :=
(PrimeSpectrum.primesOverOrderIsoFiber R S p).finite_iff.mpr inferInsta... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 532,
"column": 2
} | {
"line": 599,
"column": 78
} | {
"line": 601,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type (max u v)\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\np : Ideal R\ninst✝ : p.IsPrime\n⊢ ∃ R' x x_1,\n ∃ (_ : Etale R R'),\n ∃ P,\n ∃ (x_3 : P.IsPrime) (x_4 : P.LiesOver p),\n ∃ n e,\n ∃ (_ : Complet... | [] | induction h : (p.primesOver S).ncard using Nat.strong_induction_on generalizing R S with
| h n IH =>
have : IsArtinianRing (p.ResidueField ⊗[R] S) := IsArtinianRing.of_finite p.ResidueField _
have hpSfin : (p.primesOver S).Finite :=
(PrimeSpectrum.primesOverOrderIsoFiber R S p).finite_iff.mpr inferInsta... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 532,
"column": 2
} | {
"line": 599,
"column": 78
} | {
"line": 601,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type (max u v)\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\np : Ideal R\ninst✝ : p.IsPrime\n⊢ ∃ R' x x_1,\n ∃ (_ : Etale R R'),\n ∃ P,\n ∃ (x_3 : P.IsPrime) (x_4 : P.LiesOver p),\n ∃ n e,\n ∃ (_ : Complet... | [] | induction h : (p.primesOver S).ncard using Nat.strong_induction_on generalizing R S with
| h n IH =>
have : IsArtinianRing (p.ResidueField ⊗[R] S) := IsArtinianRing.of_finite p.ResidueField _
have hpSfin : (p.primesOver S).Finite :=
(PrimeSpectrum.primesOverOrderIsoFiber R S p).finite_iff.mpr inferInsta... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.FlatRank | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 68
} | {
"line": 102,
"column": 4
} | [
{
"pp": "X S T : Scheme\nf : X ⟶ S\ng : T ⟶ S\ninst✝² : IsAffine T\nt : ↥T\ninst✝¹ : Flat f\ninst✝ : IsFinite f\ni : Spec (S.affineOpenCover.X (S.affineOpenCover.idx (g t))) ⟶ S := ⋯\nR : CommRingCat\nu : Spec R ⟶ pullback i g\nhu : IsOpenImmersion u\nz : ↥(Spec R)\nhyl : (pullback.fst (S.affineOpenCover.f (S.a... | [
"case refine_1\nX S T : Scheme\nf : X ⟶ S\ng : T ⟶ S\ninst✝² : IsAffine T\nt : ↥T\ninst✝¹ : Flat f\ninst✝ : IsFinite f\ni : Spec (S.affineOpenCover.X (S.affineOpenCover.idx (g t))) ⟶ S := S.affineOpenCover.f (S.affineOpenCover.idx (g t))\nR : CommRingCat\nu : Spec R ⟶ pullback i g\nhu : IsOpenImmersion u\nz : ↥(Spe... | refine (IsAffine.finrank_of_isPullback _ _ ?_ ?_ ?_ _ _ ?_).symm | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Flat.Rank | {
"line": 122,
"column": 2
} | {
"line": 127,
"column": 7
} | {
"line": 129,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Flat R S\ninst✝ : Module.Finite R S\n⊢ rankAtStalk S = 1 ↔ Function.Bijective ⇑(algebraMap R S)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LinearOrdered... | [] | rw [Function.Bijective, ← rankAtStalk_pos_iff_algebraMap_injective,
← rankAtStalk_le_one_iff_surjective]
refine ⟨fun h ↦ by simp [h], fun h ↦ ?_⟩
ext p
rw [Pi.one_apply]
grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Flat.Rank | {
"line": 122,
"column": 2
} | {
"line": 127,
"column": 7
} | {
"line": 129,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Flat R S\ninst✝ : Module.Finite R S\n⊢ rankAtStalk S = 1 ↔ Function.Bijective ⇑(algebraMap R S)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LinearOrdered... | [] | rw [Function.Bijective, ← rankAtStalk_pos_iff_algebraMap_injective,
← rankAtStalk_le_one_iff_surjective]
refine ⟨fun h ↦ by simp [h], fun h ↦ ?_⟩
ext p
rw [Pi.one_apply]
grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.SmoothFiber | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 43
} | {
"line": 47,
"column": 2
} | [
{
"pp": "X✝ Y : Scheme\nf✝ : X✝ ⟶ Y\nX : Scheme\nR : CommRingCat\nf : X ⟶ Spec R\ninst✝¹ : LocallyOfFinitePresentation f\ninst✝ : Flat f\nh : ∀ (y : ↥(Spec R)), Smooth (Scheme.Hom.fiberToSpecResidueField f y)\n⊢ Smooth f",
"ppTerm": "?m.111",
"assigned": true,
"usedConstants": [
"AlgebraicGeom... | [
"case inr\nX✝ Y : Scheme\nf✝ : X✝ ⟶ Y\nX : Scheme\nR : CommRingCat\nf : X ⟶ Spec R\ninst✝¹ : LocallyOfFinitePresentation f\ninst✝ : Flat f\nh✝ : ∀ (y : ↥(Spec R)), Smooth (Scheme.Hom.fiberToSpecResidueField f y)\nthis :\n ∀ {X : Scheme} (f : X ⟶ Spec R) [LocallyOfFinitePresentation f] [Flat f],\n (∀ (y : ↥(Spec... | wlog h : ∃ S, X = Spec S generalizing f X | Mathlib.Tactic._aux_Mathlib_Tactic_WLOG___elabRules_Mathlib_Tactic_wlog_1 | Mathlib.Tactic.wlog |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 72,
"column": 2
} | {
"line": 73,
"column": 69
} | {
"line": 75,
"column": 0
} | [
{
"pp": "ι : Type u\nR : ι → CommRingCat\nI : Ideal ((i : ι) → ↑(R i))\nf : (∐ fun i ↦ Spec (R i)) ⟶ Spec (CommRingCat.of (((i : ι) → ↑(R i)) ⧸ I))\nhf : f ≫ Spec.map (CommRingCat.ofHom (Ideal.Quotient.mk I)) = sigmaSpec R\nx : (i : ι) → ↑(R i)\nhx : x ∈ I\ni : ι\n⊢ x i = 0 i",
"ppTerm": "?m.51",
"assig... | [] | simpa [← Category.assoc, Ideal.Quotient.eq_zero_iff_mem.mpr hx] using
congr((Spec.preimage (Sigma.ι (Spec <| R ·) i ≫ $hf)).hom x).symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology | {
"line": 380,
"column": 6
} | {
"line": 380,
"column": 30
} | {
"line": 381,
"column": 6
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\n⊢ f ∉ z.asHomogeneousIdeal ↔ ∃ u ∈ Set.range fun i ↦ basicOpen 𝒜 ((GradedRing.proj 𝒜 i) f), z ∈ u",
"ppTerm": "?m.72",
"a... | [
"case mp\nA : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\nhz : f ∉ z.asHomogeneousIdeal\n⊢ ∃ u ∈ Set.range fun i ↦ basicOpen 𝒜 ((GradedRing.proj 𝒜 i) f), z ∈ u",
"case mpr\nA : Type u_1\nσ... | constructor <;> intro hz | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 13
} | {
"line": 58,
"column": 4
} | [
{
"pp": "R : CommRingCat\nM✝ : ModuleCat ↑R\nM N : (Spec R).Modules\nf : modulesSpecToSheaf.obj M ⟶ modulesSpecToSheaf.obj N\nU : (Opens ↥(Spec R))ᵒᵖ\n⊢ ∀ (m : ↑((Spec R).ringCatSheaf.obj.obj U)) (x : ↑(M.val.1 U)),\n (ModuleCat.Hom.hom (f.hom.app U)).toFun (m • x) =\n (RingHom.id ↑((Spec R).ringCatShea... | [
"R : CommRingCat\nM✝ : ModuleCat ↑R\nM N : (Spec R).Modules\nf : modulesSpecToSheaf.obj M ⟶ modulesSpecToSheaf.obj N\nU : (Opens ↥(Spec R))ᵒᵖ\nt : ↑((Spec R).ringCatSheaf.obj.obj U)\nm : ↑(M.val.1 U)\n⊢ (ModuleCat.Hom.hom (f.hom.app U)).toFun (t • m) =\n (RingHom.id ↑((Spec R).ringCatSheaf.obj.obj U)) t • (Modul... | intro t m | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 182,
"column": 14
} | {
"line": 182,
"column": 41
} | {
"line": 182,
"column": 42
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nx : ↑(Proj.restrict ⋯).toTopCat\nz : HomogeneousLocalization.NumDenSameDeg 𝒜 (Submonoid.powers f)\n⊢ IsLocalization.mk' (Localization (Submonoid.powers f)) 1 ... | [
"A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nx : ↑(Proj.restrict ⋯).toTopCat\nz : HomogeneousLocalization.NumDenSameDeg 𝒜 (Submonoid.powers f)\n⊢ (algebraMap A (Localization (Submonoid.powers f))) ↑z.num ∈\n Id... | Ideal.unit_mul_mem_iff_mem, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 579,
"column": 13
} | {
"line": 579,
"column": 61
} | {
"line": 579,
"column": 61
} | [
{
"pp": "ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : SetLike σ A\ninst✝⁴ : AddSubgroupClass σ A\n𝒜 : ι → σ\ninst✝³ : AddCommMonoid ι\ninst✝² : DecidableEq ι\ninst✝¹ : GradedRing 𝒜\n𝔭 : Ideal A\ninst✝ : 𝔭.IsPrime\nf : NumDenSameDeg 𝒜 𝔭.primeCompl\nb : A\ns : ↥𝔭.primeCompl\neq0 ... | [
"ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : SetLike σ A\ninst✝⁴ : AddSubgroupClass σ A\n𝒜 : ι → σ\ninst✝³ : AddCommMonoid ι\ninst✝² : DecidableEq ι\ninst✝¹ : GradedRing 𝒜\n𝔭 : Ideal A\ninst✝ : 𝔭.IsPrime\nf : NumDenSameDeg 𝒜 𝔭.primeCompl\nb : A\ns : ↥𝔭.primeCompl\neq0 : ∃ c, ↑c * ... | IsLocalization.eq_iff_exists (M := 𝔭.primeCompl) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 580,
"column": 2
} | {
"line": 580,
"column": 51
} | {
"line": 581,
"column": 2
} | [
{
"pp": "ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : SetLike σ A\ninst✝⁴ : AddSubgroupClass σ A\n𝒜 : ι → σ\ninst✝³ : AddCommMonoid ι\ninst✝² : DecidableEq ι\ninst✝¹ : GradedRing 𝒜\n𝔭 : Ideal A\ninst✝ : 𝔭.IsPrime\nf : NumDenSameDeg 𝒜 𝔭.primeCompl\nb : A\ns : ↥𝔭.primeCompl\neq0 ... | [
"ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : SetLike σ A\ninst✝⁴ : AddSubgroupClass σ A\n𝒜 : ι → σ\ninst✝³ : AddCommMonoid ι\ninst✝² : DecidableEq ι\ninst✝¹ : GradedRing 𝒜\n𝔭 : Ideal A\ninst✝ : 𝔭.IsPrime\nf : NumDenSameDeg 𝒜 𝔭.primeCompl\nb : A\ns : ↥𝔭.primeCompl\neq1 : IsLocaliza... | obtain ⟨c, hc : _ = c.1 * (f.den.1 * s.1)⟩ := eq0 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 604,
"column": 8
} | {
"line": 604,
"column": 14
} | {
"line": 604,
"column": 15
} | [
{
"pp": "case refine_2\nX : Scheme\nM : X.Modules\ninst✝ : SheafOfModules.IsQuasicoherent M\nI : Type u\nW : I → TopologicalSpace.Opens ↥X\ncov : iSup W = ⊤\npres : (i : I) → (SheafOfModules.over M (W i)).Presentation\nκ : I → Set (TopologicalSpace.Opens ↥X)\nhsub : ∀ (i : I), κ i ⊆ X.affineOpens\nheq : ∀ (i : ... | [
"case refine_2\nX : Scheme\nM : X.Modules\ninst✝ : SheafOfModules.IsQuasicoherent M\nI : Type u\nW : I → TopologicalSpace.Opens ↥X\ncov : iSup W = ⊤\npres : (i : I) → (SheafOfModules.over M (W i)).Presentation\nκ : I → Set (TopologicalSpace.Opens ↥X)\nhsub : ∀ (i : I), κ i ⊆ X.affineOpens\nheq : ∀ (i : I), W i = sS... | heq i, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 680,
"column": 39
} | {
"line": 681,
"column": 85
} | {
"line": 682,
"column": 2
} | [
{
"pp": "ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝¹³ : CommRing A\ninst✝¹² : SetLike σ A\ninst✝¹¹ : AddSubgroupClass σ A\ninst✝¹⁰ : AddCommMonoid ι\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → σ\ninst✝⁸ : GradedRing 𝒜\nB : Type u_4\nτ : Type u_5\ninst✝⁷ : CommRing B\ninst✝⁶ : SetLike τ B\ninst✝⁵ : AddSubgroupClass ... | [] | by
simp only [← mk_add, Quotient.map'_mk'', num_add, map_add, map_mul, den_add]; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 456,
"column": 73
} | {
"line": 489,
"column": 8
} | {
"line": 491,
"column": 0
} | [
{
"pp": "σ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nX : Scheme\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nr : A\nn : ℕ\nhn : 0 < n\nhr : r ∈ 𝒜 n\n⊢ fromOfGlobalSections 𝒜 f hf ⁻... | [] | by
apply le_antisymm
· intro x hx
obtain ⟨i, x, rfl⟩ := (openCoverOfMapIrrelevantEqTop 𝒜 f hf).exists_eq x
rw [← SetLike.mem_coe] at hx -- TODO : mem version of TopologicalSpace.Opens.map_coe
simp only [TopologicalSpace.Opens.map_coe, Set.mem_preimage, SetLike.mem_coe,
← Scheme.Hom.comp_apply, fr... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 820,
"column": 39
} | {
"line": 825,
"column": 39
} | {
"line": 827,
"column": 0
} | [
{
"pp": "ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubgroupClass σ A\ninst✝² : AddCommMonoid ι\ninst✝¹ : DecidableEq ι\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\ne : ι\nf g : A\nhg : g ∈ 𝒜 e\nx : A\nhx : x = f * g\n⊢ Away 𝒜 f →+* Away 𝒜 x",
"ppTerm": "?m.31... | [] | by
let e := RingEquiv.ofLeftInverse (f := algebraMap (Away 𝒜 x) (Localization.Away x))
(h := (val_injective _).hasLeftInverse.choose_spec)
refine RingHom.comp (e.symm.toRingHom.comp (Subring.inclusion ?_))
(awayMapAux 𝒜 (f := f) ⟨_, hx⟩).rangeRestrict
exact range_awayMapAux_subset 𝒜 hg hx | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LocalRing.LocalSubring | {
"line": 113,
"column": 2
} | {
"line": 115,
"column": 91
} | {
"line": 116,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nK : Type u_3\ninst✝¹ : Field K\nA : Subring K\nP : Ideal ↥A\ninst✝ : P.IsPrime\nx : ↥A\ns : ↥P.primeCompl\ny : ↥A\nt : ↥P.primeCompl\ne :\n (IsLocalization.liftAlgHom ⋯) (IsLocalization.mk' (Localization.AtPrime P) x s) =\n (IsLo... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nK : Type u_3\ninst✝¹ : Field K\nA : Subring K\nP : Ideal ↥A\ninst✝ : P.IsPrime\nx : ↥A\ns : ↥P.primeCompl\ny : ↥A\nt : ↥P.primeCompl\ne :\n (IsLocalization.liftAlgHom ⋯) (IsLocalization.mk' (Localization.AtPrime P) x s) =\n (IsLocalization.l... | have : x.1 = y.1 * t.1.1⁻¹ * s.1.1 := by
simpa [IsLocalization.lift_mk', Algebra.ofId_apply, H,
Algebra.algebraMap_ofSubsemiring_apply, IsUnit.coe_liftRight] using congr($e * s.1.1) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 622,
"column": 2
} | {
"line": 622,
"column": 68
} | {
"line": 623,
"column": 2
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\np : ↥(unop (op (pbo f)))\nx : HomogeneousLocalization.NumDenSameDeg 𝒜 (Submonoid.powers f)\n⊢ ((HomogeneousLocalization.mapId 𝒜 ⋯) (HomogeneousLocalization.m... | [
"A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\np : ↥(unop (op (pbo f)))\nx : HomogeneousLocalization.NumDenSameDeg 𝒜 (Submonoid.powers f)\n⊢ Localization.mk ↑x.num ⟨↑x.den, ⋯⟩ =\n (IsLocalization.map (Localization ... | dsimp [HomogeneousLocalization.mapId, HomogeneousLocalization.map] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 821,
"column": 2
} | {
"line": 822,
"column": 55
} | {
"line": 823,
"column": 2
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nthis : IsIso (toSpec 𝒜 f).base\n⊢ IsIso (toSpec 𝒜 f)",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants":... | [
"A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nthis : IsIso (toSpec 𝒜 f).base\nx✝ : ∀ (x : ↑(Proj.restrict ⋯).toTopCat), IsIso (LocallyRingedSpace.Hom.stalkMap (toSpec 𝒜 f) x)\n⊢ ... | haveI _ (x) : IsIso ((toSpec 𝒜 f).stalkMap x) := by
rw [stalkMap_toSpec 𝒜 f x f_deg hm]; infer_instance | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.AlgebraicGeometry.ValuativeCriterion | {
"line": 208,
"column": 37
} | {
"line": 208,
"column": 60
} | {
"line": 208,
"column": 60
} | [
{
"pp": "X✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nY' X X' Y : Scheme\nY'_to_Y : Y' ⟶ Y\nf : X ⟶ Y\nX'_to_X : X' ⟶ X\nf' : X' ⟶ Y'\nhP : IsPullback X'_to_X f' f Y'_to_Y\nhf : Existence f\ncommSq : ValuativeCommSq f'\ncommSq' : ValuativeCommSq f :=\n { R := commSq.R, commRing := commSq.commRing, domain := ⋯, valuationRing ... | [] | simp_all only [commSq'] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.AlgebraicGeometry.ValuativeCriterion | {
"line": 208,
"column": 37
} | {
"line": 208,
"column": 60
} | {
"line": 208,
"column": 60
} | [
{
"pp": "X✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nY' X X' Y : Scheme\nY'_to_Y : Y' ⟶ Y\nf : X ⟶ Y\nX'_to_X : X' ⟶ X\nf' : X' ⟶ Y'\nhP : IsPullback X'_to_X f' f Y'_to_Y\nhf : Existence f\ncommSq : ValuativeCommSq f'\ncommSq' : ValuativeCommSq f :=\n { R := commSq.R, commRing := commSq.commRing, domain := ⋯, valuationRing ... | [] | simp_all only [commSq'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.ValuativeCriterion | {
"line": 208,
"column": 37
} | {
"line": 208,
"column": 60
} | {
"line": 208,
"column": 60
} | [
{
"pp": "X✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nY' X X' Y : Scheme\nY'_to_Y : Y' ⟶ Y\nf : X ⟶ Y\nX'_to_X : X' ⟶ X\nf' : X' ⟶ Y'\nhP : IsPullback X'_to_X f' f Y'_to_Y\nhf : Existence f\ncommSq : ValuativeCommSq f'\ncommSq' : ValuativeCommSq f :=\n { R := commSq.R, commRing := commSq.commRing, domain := ⋯, valuationRing ... | [] | simp_all only [commSq'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Sites.Small | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 37
} | {
"line": 237,
"column": 2
} | [
{
"pp": "P Q : MorphismProperty Scheme\nS : Scheme\ninst✝⁵ : P.IsStableUnderBaseChange\ninst✝⁴ : P.IsMultiplicative\ninst✝³ : P.RespectsIso\ninst✝² : Q.IsStableUnderComposition\ninst✝¹ : Q.IsStableUnderBaseChange\ninst✝ : Q.HasOfPostcompProperty Q\nX : Q.Over ⊤ S\nR : Sieve X\n⊢ R ∈ (smallPretopology P Q).toGro... | [
"P Q : MorphismProperty Scheme\nS : Scheme\ninst✝⁵ : P.IsStableUnderBaseChange\ninst✝⁴ : P.IsMultiplicative\ninst✝³ : P.RespectsIso\ninst✝² : Q.IsStableUnderComposition\ninst✝¹ : Q.IsStableUnderBaseChange\ninst✝ : Q.HasOfPostcompProperty Q\nX : Q.Over ⊤ S\nR : Sieve X\n⊢ (∃ R_1 ∈ (smallPretopology P Q).coverings X,... | rw [Pretopology.mem_toGrothendieck] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 223,
"column": 45
} | {
"line": 226,
"column": 87
} | {
"line": 226,
"column": 87
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝² : Category.{v₀, u₀} C₀\ninst✝¹ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\ninst✝ : IsDenseSubsite J₀ J F\nX : C\ndata : F.OneHypercoverDenseData J₀ J X\ni₁ i₂ : data.I₀\nW : C\np₁ : W ⟶ F.obj (data.X i₁)\np₂ : W ⟶ F.obj (da... | [] | by
letI str := Presieve.getFunctorPushforwardStructure hg.bindStruct.hg
exact Sieve.pullback str.lift
(Sieve.functorPushforward F (data.sieve₁₀ str.cover.1.choose str.cover.2.choose)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 235,
"column": 4
} | {
"line": 235,
"column": 74
} | {
"line": 236,
"column": 2
} | [
{
"pp": "case right.add\nσ : Type u_1\nA : Type u_2\ninst✝¹⁰ : CommRing A\ninst✝⁹ : SetLike σ A\ninst✝⁸ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝⁷ : GradedRing 𝒜\nO : Type u_3\ninst✝⁶ : CommRing O\ninst✝⁵ : IsDomain O\ninst✝⁴ : ValuationRing O\nK : Type u_4\ninst✝³ : Field K\ninst✝² : Algebra O K\ninst✝¹ : IsF... | [] | exact ((valuation O K).map_add _ _).trans <| sup_le_iff.mpr ⟨hhx, hhy⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Sites.Point.Basic | {
"line": 211,
"column": 2
} | {
"line": 214,
"column": 22
} | {
"line": 216,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nJ : GrothendieckTopology C\nΦ : J.Point\nA : Type u'\ninst✝⁵ : Category.{v', u'} A\ninst✝⁴ : HasColimitsOfSize.{w, w, v', u'} A\nFC : A → A → Type u_1\nCC : A → Type w'\ninst✝³ : (X Y : A) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝² : ConcreteCategory A FC\nP : Cᵒᵖ ... | [] | obtain ⟨⟨X, x⟩, z, rfl⟩ := Types.jointly_surjective_of_isColimit
(isColimitOfPreserves (forget A)
(colimit.isColimit ((CategoryOfElements.π Φ.fiber).op ⋙ P))) p
exact ⟨X, x, z, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.Point.Basic | {
"line": 211,
"column": 2
} | {
"line": 214,
"column": 22
} | {
"line": 216,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nJ : GrothendieckTopology C\nΦ : J.Point\nA : Type u'\ninst✝⁵ : Category.{v', u'} A\ninst✝⁴ : HasColimitsOfSize.{w, w, v', u'} A\nFC : A → A → Type u_1\nCC : A → Type w'\ninst✝³ : (X Y : A) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝² : ConcreteCategory A FC\nP : Cᵒᵖ ... | [] | obtain ⟨⟨X, x⟩, z, rfl⟩ := Types.jointly_surjective_of_isColimit
(isColimitOfPreserves (forget A)
(colimit.isColimit ((CategoryOfElements.π Φ.fiber).op ⋙ P))) p
exact ⟨X, x, z, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Point.Basic | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 27
} | {
"line": 319,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nΦ : J.Point\ninst✝¹ : LocallySmall.{w, v, u} C\nU T : C\nf : U ⟶ T\ninst✝ : Mono f\n⊢ Function.Injective ⇑(ConcreteCategory.hom (Φ.fiber.map f))",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"C : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nΦ : J.Point\ninst✝¹ : LocallySmall.{w, v, u} C\nU T : C\nf : U ⟶ T\ninst✝ : Mono f\n⊢ Mono (Φ.fiber.map f)"
] | rw [← mono_iff_injective] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 290,
"column": 2
} | {
"line": 310,
"column": 43
} | {
"line": 312,
"column": 0
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝² : Category.{v₀, u₀} C₀\ninst✝¹ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\ninst✝ : IsDenseSubsite J₀ J F\nX : C\ndata : F.OneHypercoverDenseData J₀ J X\nX₀ : C₀\nf : F.obj X₀ ⟶ X\n⊢ data.sieve f ∈ J₀ X₀",
"ppTerm": "?m.... | [] | have := IsDenseSubsite.isCoverDense J₀ J F
have := IsDenseSubsite.isLocallyFull J₀ J F
rw [← functorPushforward_mem_iff J₀ J F]
let R : ⦃W : C⦄ → ⦃p : W ⟶ F.obj X₀⦄ →
(Sieve.pullback f data.toOneHypercover.sieve₀).arrows p → Sieve W := fun W p hp ↦
Sieve.bind (Sieve.coverByImage F W).arrows (fun U π hπ ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 290,
"column": 2
} | {
"line": 310,
"column": 43
} | {
"line": 312,
"column": 0
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝² : Category.{v₀, u₀} C₀\ninst✝¹ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\ninst✝ : IsDenseSubsite J₀ J F\nX : C\ndata : F.OneHypercoverDenseData J₀ J X\nX₀ : C₀\nf : F.obj X₀ ⟶ X\n⊢ data.sieve f ∈ J₀ X₀",
"ppTerm": "?m.... | [] | have := IsDenseSubsite.isCoverDense J₀ J F
have := IsDenseSubsite.isLocallyFull J₀ J F
rw [← functorPushforward_mem_iff J₀ J F]
let R : ⦃W : C⦄ → ⦃p : W ⟶ F.obj X₀⦄ →
(Sieve.pullback f data.toOneHypercover.sieve₀).arrows p → Sieve W := fun W p hp ↦
Sieve.bind (Sieve.coverByImage F W).arrows (fun U π hπ ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Hypercover.ZeroFamily | {
"line": 105,
"column": 60
} | {
"line": 114,
"column": 43
} | {
"line": 116,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nP : PreZeroHypercoverFamily C\nh₁ : ∀ {X : C}, P.property.IsClosedUnderIsomorphisms\nh₂ :\n ∀ {X Y : C} (f : X ⟶ Y) (E : PreZeroHypercover Y) [inst : ∀ (i : E.I₀), HasPullback f (E.f i)],\n P.property E → P.property (PreZeroHypercover.pullback₁ f E)\nι : Type ... | [] | by
let E : PreZeroHypercover S := ⟨ι, X, f⟩
have (i : E.I₀) : HasPullback g (E.f i) := (h i).hasPullback
let F : PreZeroHypercover Y := ⟨_, _, p₁⟩
let e : F ≅ E.pullback₁ g :=
PreZeroHypercover.isoMk (Equiv.refl _) (fun i ↦ (h i).isoPullback)
change F.presieve₀ ∈ _
rw [F.presieve₀_mem_prec... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 386,
"column": 20
} | {
"line": 386,
"column": 80
} | {
"line": 387,
"column": 8
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝³ : Category.{v₀, u₀} C₀\ninst✝² : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\nG : Cᵒᵖ ⥤ A\nhG₀ : Presh... | [
"C₀ : Type u₀\nC : Type u\ninst✝³ : Category.{v₀, u₀} C₀\ninst✝² : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\nG : Cᵒᵖ ⥤ A\nhG₀ : Presheaf.IsSheaf ... | simp only [op_comp, Functor.map_comp, assoc, lift_map_assoc] | Lean.Elab.Tactic.Conv.evalSimp | Lean.Parser.Tactic.Conv.simp |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 386,
"column": 20
} | {
"line": 386,
"column": 80
} | {
"line": 387,
"column": 8
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝³ : Category.{v₀, u₀} C₀\ninst✝² : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\nG : Cᵒᵖ ⥤ A\nhG₀ : Presh... | [
"C₀ : Type u₀\nC : Type u\ninst✝³ : Category.{v₀, u₀} C₀\ninst✝² : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\nG : Cᵒᵖ ⥤ A\nhG₀ : Presheaf.IsSheaf ... | simp only [op_comp, Functor.map_comp, assoc, lift_map_assoc] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 386,
"column": 20
} | {
"line": 386,
"column": 80
} | {
"line": 387,
"column": 8
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝³ : Category.{v₀, u₀} C₀\ninst✝² : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\nG : Cᵒᵖ ⥤ A\nhG₀ : Presh... | [
"C₀ : Type u₀\nC : Type u\ninst✝³ : Category.{v₀, u₀} C₀\ninst✝² : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\nG : Cᵒᵖ ⥤ A\nhG₀ : Presheaf.IsSheaf ... | simp only [op_comp, Functor.map_comp, assoc, lift_map_assoc] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 304,
"column": 18
} | {
"line": 304,
"column": 25
} | {
"line": 305,
"column": 6
} | [
{
"pp": "case succ\nσ : Type u_1\nA : Type u_2\ninst✝¹⁰ : CommRing A\ninst✝⁹ : SetLike σ A\ninst✝⁸ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝⁷ : GradedRing 𝒜\nO : Type u_3\ninst✝⁶ : CommRing O\ninst✝⁵ : IsDomain O\ninst✝⁴ : ValuationRing O\nK : Type u_4\ninst✝³ : Field K\ninst✝² : Algebra O K\ninst✝¹ : IsFracti... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 358,
"column": 8
} | {
"line": 358,
"column": 32
} | {
"line": 358,
"column": 33
} | [
{
"pp": "case refine_2\nσ : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : SetLike σ A\ninst✝² : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) A\nO : Type u_2\ncommRing✝ : CommRing O\ndomain✝ : IsDomain O\nvaluationRing✝ : ValuationRing O\nK : Type u_2\nfi... | [
"case refine_2\nσ : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : SetLike σ A\ninst✝² : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) A\nO : Type u_2\ncommRing✝ : CommRing O\ndomain✝ : IsDomain O\nvaluationRing✝ : ValuationRing O\nK : Type u_2\nfield✝ : Field... | awayMap_fromZeroRingHom, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.SmallObject.Iteration.ExtendToSucc | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 22
} | {
"line": 109,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nJ : Type u\ninst✝¹ : LinearOrder J\ninst✝ : SuccOrder J\nj : J\nhj : ¬IsMax j\nF : ↑(Set.Iic j) ⥤ C\nX : C\nτ : F.obj ⟨j, ⋯⟩ ⟶ X\ni₁ i₂ i₃ : J\nh₁₂ : i₁ ≤ i₂\nh₂₃ : i₂ ≤ i₃\nh : i₃ ≤ Order.succ j\n⊢ map hj F τ i₁ i₃ ⋯ h = map hj F τ i₁ i₂ h₁₂ ⋯ ≫ map hj F τ... | [
"case pos\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\nJ : Type u\ninst✝¹ : LinearOrder J\ninst✝ : SuccOrder J\nj : J\nhj : ¬IsMax j\nF : ↑(Set.Iic j) ⥤ C\nX : C\nτ : F.obj ⟨j, ⋯⟩ ⟶ X\ni₁ i₂ i₃ : J\nh₁₂ : i₁ ≤ i₂\nh₂₃ : i₂ ≤ i₃\nh : i₃ ≤ Order.succ j\nh₁ : i₃ ≤ j\n⊢ map hj F τ i₁ i₃ ⋯ h = map hj F τ i₁ i₂ h₁₂ ⋯ ≫... | by_cases h₁ : i₃ ≤ j | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.CategoryTheory.MorphismProperty.TransfiniteComposition | {
"line": 360,
"column": 2
} | {
"line": 360,
"column": 24
} | {
"line": 361,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\n⊢ Monotone transfiniteCompositions.{w, v, u}",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"CategoryTheory.MorphismProperty",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.MorphismProperty.in... | [
"C : Type u\ninst✝ : Category.{v, u} C\nW₁ W₂ : MorphismProperty C\nh : W₁ ≤ W₂\nX Y : C\nf : X ⟶ Y\nhf : transfiniteCompositions.{w, v, u} W₁ f\n⊢ transfiniteCompositions.{w, v, u} W₂ f"
] | intro W₁ W₂ h X Y f hf | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.SmallObject.Iteration.Basic | {
"line": 332,
"column": 15
} | {
"line": 332,
"column": 28
} | {
"line": 332,
"column": 28
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nJ : Type w\nΦ : SuccStruct C\ninst✝⁵ : LinearOrder J\ninst✝⁴ : SuccOrder J\ninst✝³ : OrderBot J\ninst✝² : HasIterationOfShape J C\ninst✝¹ : WellFoundedLT J\nj : J\nK : Type w\ninst✝ : LinearOrder K\nx : K\nF G : ↑(Set.Iic x) ⥤ C\nk₁ k₂ : K\nh₁₂ : k₁ ≤ k₂\nh₂ : k₂... | [] | by rw [h.src] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.SmallObject.Construction | {
"line": 171,
"column": 2
} | {
"line": 181,
"column": 29
} | {
"line": 183,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nI : Type w\nA B : I → C\nf : (i : I) → A i ⟶ B i\nS X : C\nπX : X ⟶ S\ninst✝³ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C\ninst✝² : HasPushout (functorObjTop f πX) (functorObjLeft f πX)\ninst✝¹ : LocallySmall.{t, v, u} C\ninst✝ : Small.{t, w} I\n⊢ Sm... | [] | let φ (x : FunctorObjIndex f πX) :
Σ (i : Shrink.{t} I),
Shrink.{t} ((A ((equivShrink _).symm i) ⟶ X) ×
(B ((equivShrink _).symm i) ⟶ S)) :=
⟨equivShrink _ x.i, equivShrink _
⟨eqToHom (by simp) ≫ x.t, eqToHom (by simp) ≫ x.b⟩⟩
have hφ : Function.Injective φ := by
rintro ⟨i₁, t₁... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.SmallObject.Construction | {
"line": 171,
"column": 2
} | {
"line": 181,
"column": 29
} | {
"line": 183,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nI : Type w\nA B : I → C\nf : (i : I) → A i ⟶ B i\nS X : C\nπX : X ⟶ S\ninst✝³ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C\ninst✝² : HasPushout (functorObjTop f πX) (functorObjLeft f πX)\ninst✝¹ : LocallySmall.{t, v, u} C\ninst✝ : Small.{t, w} I\n⊢ Sm... | [] | let φ (x : FunctorObjIndex f πX) :
Σ (i : Shrink.{t} I),
Shrink.{t} ((A ((equivShrink _).symm i) ⟶ X) ×
(B ((equivShrink _).symm i) ⟶ S)) :=
⟨equivShrink _ x.i, equivShrink _
⟨eqToHom (by simp) ≫ x.t, eqToHom (by simp) ≫ x.b⟩⟩
have hφ : Function.Injective φ := by
rintro ⟨i₁, t₁... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 44
} | {
"line": 118,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nG : C\ninst✝¹ : Abelian C\nhG : IsSeparator G\nX Y : C\np : X ⟶ Y\ninst✝ : Mono p\nhp : ¬IsIso p\n⊢ ∃ X' i p', ∃ (_ : (generatingMonomorphisms G).pushouts i) (_ : ¬IsIso i) (_ : Mono p'), i ≫ p' = p",
"ppTerm": "?m.44",
"assigned": true,
"usedConstant... | [
"C : Type u\ninst✝² : Category.{v, u} C\nG : C\ninst✝¹ : Abelian C\nhG : IsSeparator G\nX Y : C\np : X ⟶ Y\ninst✝ : Mono p\nhp :\n ¬∀ (G_1 : C),\n ObjectProperty.singleton G G_1 →\n Function.Surjective ⇑(ConcreteCategory.hom ((coyoneda.obj (Opposite.op G_1)).map p))\n⊢ ∃ X' i p', ∃ (_ : (generatingMono... | rw [hG.isDetector.isIso_iff_of_mono] at hp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 442,
"column": 6
} | {
"line": 442,
"column": 32
} | {
"line": 442,
"column": 33
} | [
{
"pp": "p n : ℕ\nhp : Fact (Nat.Prime p)\nhn : n ≠ 0\n⊢ ¬p ^ (padicValNat p n + 1) ∣ n",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"congrArg",
"Nat.instMonoid",
"id",
"padicValNat",
"instOfNatNat",
"LE.le",
"in... | [
"p n : ℕ\nhp : Fact (Nat.Prime p)\nhn : n ≠ 0\n⊢ ¬padicValNat p n + 1 ≤ padicValNat p n"
] | padicValNat_dvd_iff_le hn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 521,
"column": 2
} | {
"line": 521,
"column": 65
} | {
"line": 523,
"column": 0
} | [
{
"pp": "p n : ℕ\nhn : n ≠ 0\nk : ℕ\nhk : k < padicValNat p n + 1\n⊢ p ^ k ∣ n ∧ n ≠ 0",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"_private.Mathlib.NumberTheory.Padics.PadicVal.Basic.0.range_pow_padicValNat_subset_divisors._proof_1_3",
"Dvd.dvd",
"Nat.instMonoid",
... | [] | exact ⟨(pow_dvd_pow p <| by lia).trans pow_padicValNat_dvd, hn⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 118,
"column": 74
} | {
"line": 118,
"column": 89
} | {
"line": 118,
"column": 89
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\n⊢ ↑x = ↑0 ↔ x = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"PadicInt",
"congrArg",
"CommSemiring.toSemiring",
"Subtype.coe_inj",
"i... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\n⊢ x = 0 ↔ x = 0"
] | Subtype.coe_inj | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 264,
"column": 27
} | {
"line": 264,
"column": 39
} | {
"line": 264,
"column": 39
} | [
{
"pp": "case h\np : ℕ\nhp : Fact (Nat.Prime p)\nε : ℝ\nhε : 0 < ε\nk : ℕ\nhk : ε⁻¹ < ↑k\n⊢ ε⁻¹ < ↑p ^ ↑k",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"GroupWithZero.toDivisionMonoid... | [
"case h\np : ℕ\nhp : Fact (Nat.Prime p)\nε : ℝ\nhε : 0 < ε\nk : ℕ\nhk : ε⁻¹ < ↑k\n⊢ ε⁻¹ < ↑p ^ k"
] | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 225,
"column": 2
} | {
"line": 226,
"column": 18
} | {
"line": 228,
"column": 0
} | [
{
"pp": "case mpr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\n⊢ f ≈ 0 → f.norm = 0",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"padicNorm.instIsAbsoluteValueRat",
"dite_cond_eq_true",
"Rat.instOfNat",
"congrArg",
"Rat",
"Rat.linearOrder",
... | [] | · intro h
simp [norm, h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 592,
"column": 31
} | {
"line": 592,
"column": 46
} | {
"line": 592,
"column": 46
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℤ_[p]\n⊢ ∀ {x y : ℤ_[p]}, ↑x = ↑y → ∃ c, ↑c * x = ↑c * y",
"ppTerm": "?m.298",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"PadicInt",
"Monoid.toMulOneClass",
... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℤ_[p]\n⊢ ∀ {x y : ℤ_[p]}, x = y → ∃ c, ↑c * x = ↑c * y"
] | Subtype.coe_inj | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 211,
"column": 4
} | {
"line": 211,
"column": 75
} | {
"line": 212,
"column": 2
} | [
{
"pp": "case inr\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝¹ : LocallySmall.{w, v, u} C\nhP :\n ∀ ⦃X : C⦄ (S : Sieve X),\n (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),\n ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fi... | [] | simpa using shrinkYonedaEquiv_symm_app_shrinkYonedaObjObjEquiv_symm s g | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 855,
"column": 6
} | {
"line": 855,
"column": 31
} | {
"line": 855,
"column": 31
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ ‖↑p‖ = (↑p)⁻¹",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"DivisionRing.toRatCast",
"congrArg",
"Real.instInv",
"Rat",
"AddGroupWithOne.toAddMonoidWithOne",
"i... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ ‖↑p‖ = (↑↑p)⁻¹"
] | ← @Rat.cast_natCast ℝ _ p | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 870,
"column": 21
} | {
"line": 870,
"column": 33
} | {
"line": 870,
"column": 33
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ ‖↑p ^ n‖ = ‖↑p ^ ↑n‖",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Norm.norm",
"Eq.mpr",
"Real",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"DivInvMonoid.toZPow",
"Fi... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\n⊢ ‖↑p ^ n‖ = ‖↑p ^ n‖"
] | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 912,
"column": 8
} | {
"line": 912,
"column": 42
} | {
"line": 912,
"column": 43
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\nhnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0\n⊢ ↑p ^\n (↑(padicValNat p { num := n, den := d, den_nz := hn, reduced := hd }.den) -\n ↑(padicValInt p { num := n, de... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\nhnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0\n⊢ ↑p ^ (↑0 - ↑(padicValInt p { num := n, den := d, den_nz := hn, reduced := hd }.num)) ≤ 1"
] | padicValNat.eq_zero_of_not_dvd hq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 912,
"column": 78
} | {
"line": 912,
"column": 90
} | {
"line": 912,
"column": 90
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\nhnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0\n⊢ (↑p ^ ↑(padicValInt p { num := n, den := d, den_nz := hn, reduced := hd }.num))⁻¹ ≤ 1",
"ppTerm": "?m.90",
"assigned": ... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nn : ℤ\nd : ℕ\nhn : d ≠ 0\nhd : n.natAbs.Coprime d\nhq : ¬p ∣ d\nhnz : ¬n = 0\nhnz' : { num := n, den := d, den_nz := hn, reduced := hd } ≠ 0\n⊢ (↑p ^ padicValInt p { num := n, den := d, den_nz := hn, reduced := hd }.num)⁻¹ ≤ 1"
] | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.DoldKan.Decomposition | {
"line": 67,
"column": 15
} | {
"line": 67,
"column": 37
} | {
"line": 67,
"column": 37
} | [
{
"pp": "case pos.e_s\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn q : ℕ\nhq : (Q q).f (n + 1) = ∑ i with ↑i < q, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev\nhqn : n < q\nx : ℕ\nhx : x < n + 1\n⊢ ⟨x, hx⟩ ∈ {i | ↑i < q} ↔ ⟨x, hx⟩ ∈ {i | ↑i < q + 1}",
"ppT... | [
"case pos.e_s\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn q : ℕ\nhq : (Q q).f (n + 1) = ∑ i with ↑i < q, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev\nhqn : n < q\nx : ℕ\nhx : x < n + 1\n⊢ x < q ↔ x < q + 1"
] | Finset.mem_filter_univ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicTopology.SimplicialObject.Split | {
"line": 338,
"column": 98
} | {
"line": 349,
"column": 13
} | {
"line": 351,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nS₁ S₂ : Split C\nΦ₁ Φ₂ : S₁.Hom S₂\nh : ∀ (n : ℕ), Φ₁.f n = Φ₂.f n\n⊢ Φ₁ = Φ₂",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"CategoryTheory.SimplicialObject.Split.s",
"Eq.mpr",
"CategoryTheory.SimplicialObject.Splitt... | [] | by
rcases Φ₁ with ⟨F₁, f₁, c₁⟩
rcases Φ₂ with ⟨F₂, f₂, c₂⟩
have h' : f₁ = f₂ := by
ext
apply h
subst h'
simp only [mk.injEq, and_true]
apply S₁.s.hom_ext
intro n
dsimp
rw [c₁, c₂] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.DoldKan.Degeneracies | {
"line": 111,
"column": 34
} | {
"line": 113,
"column": 13
} | {
"line": 113,
"column": 13
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nq n : ℕ\nhq : ∀ (i : Fin (n + 1 + 1)), n + 1 + 1 ≤ ↑i + q → X.σ i ≫ (P q).f (n + 1 + 1) = 0\ni : Fin (n + 1 + 1)\nh : ¬n + 1 + 1 ≤ ↑i + q\nhi : n + 1 = ↑i + q\nv : HigherFacesVanish q ((P q).f n.succ ≫ X.σ i)\n... | [] | by
simp only [← hk, Fin.rev_eq j hk.symm, Fin.succ_mk, Fin.val_mk]
lia | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject | {
"line": 128,
"column": 2
} | {
"line": 131,
"column": 13
} | {
"line": 133,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nX : SimplicialObject C\ns : X.Splitting\ninst✝ : Preadditive C\nn : ℕ\n⊢ PInfty.f n ≫ s.πSummand (IndexSet.id (op ⦋n⦌)) = s.πSummand (IndexSet.id (op ⦋n⦌))",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMo... | [] | conv_rhs => rw [← id_comp (s.πSummand _)]
symm
rw [← sub_eq_zero, ← sub_comp, ← comp_PInfty_eq_zero_iff, sub_comp, id_comp, PInfty_f_idem,
sub_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject | {
"line": 128,
"column": 2
} | {
"line": 131,
"column": 13
} | {
"line": 133,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nX : SimplicialObject C\ns : X.Splitting\ninst✝ : Preadditive C\nn : ℕ\n⊢ PInfty.f n ≫ s.πSummand (IndexSet.id (op ⦋n⦌)) = s.πSummand (IndexSet.id (op ⦋n⦌))",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMo... | [] | conv_rhs => rw [← id_comp (s.πSummand _)]
symm
rw [← sub_eq_zero, ← sub_comp, ← comp_PInfty_eq_zero_iff, sub_comp, id_comp, PInfty_f_idem,
sub_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.DoldKan.GammaCompN | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 24
} | {
"line": 158,
"column": 0
} | [
{
"pp": "case e'_2.e'_7\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nX : ChainComplex C ℕ\nn : ℕ\nA : Splitting.IndexSet (op ⦋n⦌)\n⊢ 𝟙 (X.X (unop A.fst).len) ≫ ((Γ₀.splitting X).cofan (op ⦋n⦌)).inj A =\n ((Γ₀.splitting X).cofan (op ⦋n⦌)).inj A ≫ 𝟙 (Γ... | [] | erw [comp_id, id_comp] | Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_tacticErw____1 | Lean.Parser.Tactic.tacticErw___ |
Mathlib.AlgebraicTopology.DoldKan.EquivalenceAdditive | {
"line": 63,
"column": 36
} | {
"line": 63,
"column": 52
} | {
"line": 63,
"column": 53
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nα : N.obj ((𝟭 (Karoubi (SimplicialObject C))).obj P) ≅ N.obj ((N₂ ⋙ Γ₂).obj P) := N.mapIso (Γ₂N₂.app P)\nβ : (Γ₂ ⋙ N₂).obj (N.obj P) ≅ (𝟭 (Karoubi (ChainComplex C ℕ))... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nα : N.obj ((𝟭 (Karoubi (SimplicialObject C))).obj P) ≅ N.obj ((N₂ ⋙ Γ₂).obj P) := N.mapIso (Γ₂N₂.app P)\nβ : (Γ₂ ⋙ N₂).obj (N.obj P) ≅ (𝟭 (Karoubi (ChainComplex C ℕ))).obj (N.obj... | ← comp_id β.hom, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.DoldKan.NCompGamma | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 21
} | {
"line": 90,
"column": 2
} | [
{
"pp": "case mk.mk\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nX : SimplicialObject C\nn n' : ℕ\ni : ⦋n⦌ ⟶ ⦋n'⦌\ninst✝ : Mono i\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ PInfty.f n = PInfty.f n' ≫ X.map i.op",
"ppTerm": "?mk.mk",
"assigned": true,
"usedConstants": [
"Oppo... | [
"case pos\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nX : SimplicialObject C\nn n' : ℕ\ni : ⦋n⦌ ⟶ ⦋n'⦌\ninst✝ : Mono i\nh : n = n'\n⊢ Γ₀.Obj.Termwise.mapMono K[X] i ≫ PInfty.f n = PInfty.f n' ≫ X.map i.op",
"case neg\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\... | by_cases h : n = n' | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.AlgebraicTopology.DoldKan.NCompGamma | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 36
} | {
"line": 101,
"column": 4
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nX : SimplicialObject C\nn : ℕ\ni : ⦋n⦌ ⟶ ⦋n + 1⦌\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ PInfty.f (n + 1) ≫ ∑ i, (-1) ^ ↑i • X.δ i = PInfty.f (n + 1) ≫ X.map i.op",
"ppTerm": "?pos✝",
"assigned": true,
"... | [
"case pos\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nX : SimplicialObject C\nn : ℕ\ni : ⦋n⦌ ⟶ ⦋n + 1⦌\ninst✝ : Mono i\nh : ¬n = n + 1\nhi : Isδ₀ i\n⊢ ∑ j, PInfty.f (n + 1) ≫ ((-1) ^ ↑j • X.δ j) = PInfty.f (n + 1) ≫ X.map i.op"
] | simp only [Preadditive.comp_sum] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.DoldKan.NCompGamma | {
"line": 189,
"column": 70
} | {
"line": 196,
"column": 40
} | {
"line": 198,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nX : SimplicialObject C\n⊢ Γ₂N₁.natTrans.app X = (Γ₂N₂ToKaroubiIso.app X).inv ≫ Γ₂N₂.natTrans.app ((toKaroubi (SimplicialObject C)).obj X)",
"ppTerm": "?m.50",
"assigned": true,
"usedConstant... | [] | by
rw [Γ₂N₂.natTrans_app_f_app]
dsimp only [Karoubi.decompId_i_toKaroubi, Karoubi.decompId_p_toKaroubi, Functor.comp_map,
NatTrans.comp_app]
rw [N₂.map_id, Γ₂.map_id, Iso.app_inv]
dsimp only [toKaroubi]
erw [id_comp]
rw [comp_id, Iso.inv_hom_id_app_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Quiver.Cast | {
"line": 108,
"column": 54
} | {
"line": 110,
"column": 30
} | {
"line": 112,
"column": 0
} | [
{
"pp": "U : Type u_1\ninst✝ : Quiver U\nu v u' v' : U\nhu : u = u'\nhv : v = v'\np : Path u v\np' : Path u' v'\n⊢ cast hu hv p = p' ↔ p ≍ p'",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Quiver.Path.cast_eq_cast",
"cast",
"id",
... | [] | by
rw [Path.cast_eq_cast]
exact _root_.cast_eq_iff_heq | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Quiver.SingleObj | {
"line": 115,
"column": 19
} | {
"line": 115,
"column": 65
} | {
"line": 115,
"column": 65
} | [
{
"pp": "case cons\nα : Type u_1\na : α\nl : List α\nih : pathToList (listToPath l) = l\n⊢ pathToList (listToPath (a :: l)) = a :: l",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"Quiver.SingleObj.pathToList",
"id",
"List.cons",
"List",
"Quiver.SingleObj.sta... | [
"case cons\nα : Type u_1\na : α\nl : List α\nih : pathToList (listToPath l) = l\n⊢ a :: pathToList (listToPath l) = a :: l"
] | change a :: pathToList (listToPath l) = a :: l | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.AlgebraicTopology.ModelCategory.CofibrantObjectHomotopy | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 83
} | {
"line": 297,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : ModelCategory C\nHcof : Type u_1 := (weakEquivalences (HoCat C)).Localization\nLcofπ : HoCat C ⥤ Hcof := (weakEquivalences (HoCat C)).Q\nLcof : CofibrantObject C ⥤ Hcof := toHoCat ⋙ Lcofπ\nH : Type u_1 := (weakEquivalences C).Localization\nL : C ⥤ H... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : ModelCategory C\nHcof : Type u_1 := (weakEquivalences (HoCat C)).Localization\nLcofπ : HoCat C ⥤ Hcof := (weakEquivalences (HoCat C)).Q\nLcof : CofibrantObject C ⥤ Hcof := toHoCat ⋙ Lcofπ\nH : Type u_1 := (weakEquivalences C).Localization\nL : C ⥤ H := (weakEqu... | let eF : ι ⋙ L ≅ Lcof ⋙ F := CatCommSq.iso (localizerMorphism C).functor Lcof L F | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic | {
"line": 71,
"column": 71
} | {
"line": 71,
"column": 78
} | {
"line": 72,
"column": 6
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx₀ x₁ : X\np : Path x₀ x₁\nx : ↑I\nhx : 2⁻¹ < ↑x\n⊢ p.extend (2 - 2 * ↑x) = p.extend (1 - (2 * ↑x - 1))",
"ppTerm": "?m.152",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero"... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.AlgebraicTopology.ModelCategory.BifibrantObjectHomotopy | {
"line": 469,
"column": 8
} | {
"line": 469,
"column": 37
} | {
"line": 469,
"column": 37
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : ModelCategory C\nX Y : CofibrantObject C\nf : X ⟶ Y\nhf : WeakEquivalence f\n⊢ IsIso (HoCat.bifibrantResolution.map (toHoCat.map f))",
"ppTerm": "?m.126",
"assigned": true,
"usedConstants": [
"HomotopicalAlgebra.BifibrantObject",
"... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : ModelCategory C\nX Y : CofibrantObject C\nf : X ⟶ Y\nhf : WeakEquivalence f\n⊢ IsIso (BifibrantObject.toHoCat.map (bifibrantResolutionMap f))"
] | HoCat.bifibrantResolution_map | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.ModelCategory.BifibrantObjectHomotopy | {
"line": 469,
"column": 4
} | {
"line": 469,
"column": 38
} | {
"line": 470,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : ModelCategory C\nX Y : CofibrantObject C\nf : X ⟶ Y\nhf : WeakEquivalence f\n⊢ IsIso (HoCat.bifibrantResolution.map (toHoCat.map f))",
"ppTerm": "?m.126",
"assigned": true,
"usedConstants": [
"HomotopicalAlgebra.BifibrantObject",
"... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : ModelCategory C\nX Y : CofibrantObject C\nf : X ⟶ Y\nhf : WeakEquivalence f\n⊢ IsIso (BifibrantObject.toHoCat.map (bifibrantResolutionMap f))"
] | rw [HoCat.bifibrantResolution_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.SimplicialSet.SubcomplexColimits | {
"line": 73,
"column": 8
} | {
"line": 73,
"column": 41
} | {
"line": 73,
"column": 41
} | [
{
"pp": "X : SSet\nA : X.Subcomplex\nι : Type u_1\nU : ι → X.Subcomplex\nV : ι → ι → X.Subcomplex\nh : A.MulticoequalizerDiagram U V\ninst✝ : LinearOrder ι\ni j : ι\n⊢ V i j = V j i",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CompleteLattice.MulticoequalizerDiagr... | [] | rw [h.eq_inf, h.eq_inf, inf_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Horn | {
"line": 78,
"column": 4
} | {
"line": 81,
"column": 7
} | {
"line": 82,
"column": 2
} | [
{
"pp": "n : ℕ\ni : Fin (n + 3)\nj : Δ[n + 2] _⦋0⦌\nS : Finset (Fin (n + 3)) := {i, j 0}\nhS : S = Finset.univ\n⊢ False",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Fintype.card_fin",
"Opposite",
"SimplexCategory.instFintypeToTypeOrderHomFinHAddNatLenOfNat",
"Fi... | [] | have := Finset.card_le_card hS.symm.le
simp only [Finset.card_univ, Fintype.card_fin, S] at this
have := this.trans Finset.card_le_two
lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.Horn | {
"line": 78,
"column": 4
} | {
"line": 81,
"column": 7
} | {
"line": 82,
"column": 2
} | [
{
"pp": "n : ℕ\ni : Fin (n + 3)\nj : Δ[n + 2] _⦋0⦌\nS : Finset (Fin (n + 3)) := {i, j 0}\nhS : S = Finset.univ\n⊢ False",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Fintype.card_fin",
"Opposite",
"SimplexCategory.instFintypeToTypeOrderHomFinHAddNatLenOfNat",
"Fi... | [] | have := Finset.card_le_card hS.symm.le
simp only [Finset.card_univ, Fintype.card_fin, S] at this
have := this.trans Finset.card_le_two
lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Horn | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 61
} | {
"line": 255,
"column": 2
} | [
{
"pp": "case refine_2\nn : ℕ\ni : Fin (n + 4)\nh₀ : 0 < i\nhₙ : i < Fin.last (n + 3)\nk : ℕ\nh : k < n + 2\n⊢ ⟨k + 1, ⋯⟩ ≤ ⟨k + 2, ⋯⟩",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Nat.instIsOrderedAddMonoid",
"SSet.horn.primitiveTriangle._simp_8",
"instIsLeftCance... | [
"case refine_3\nn : ℕ\ni : Fin (n + 4)\nh₀ : 0 < i\nhₙ : i < Fin.last (n + 3)\nk : ℕ\nh : k < n + 2\n⊢ stdSimplex.triangle ⟨k, ⋯⟩ ⟨k + 1, ⋯⟩ ⟨k + 2, ⋯⟩ ⋯ ⋯ ∈ Λ[n + 3, i].obj (op ⦋2⦌)"
] | · simp only [Fin.mk_le_mk, add_le_add_iff_left, one_le_two] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.SimplicialSet.HornColimits | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 50
} | {
"line": 192,
"column": 4
} | [
{
"pp": "case succ\nX : SSet\nn : ℕ\ni : Fin (n + 1 + 2)\ng : Λ[n + 1 + 1, i].toSSet ⟶ X\n⊢ horn.IsCompatible fun j hj ↦ ι i j hj ≫ g",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"SSet.Subcomplex.toSSet",
"Eq.mpr",
"Opposite",
"Fin.ne_zero_of_lt",
"Category... | [
"case succ\nX : SSet\nn : ℕ\ni : Fin (n + 1 + 2)\ng : Λ[n + 1 + 1, i].toSSet ⟶ X\n⊢ ∀ (j k : Fin (n + 3)) (hj : j ≠ i) (hk : k ≠ i) (hjk : j < k),\n (stdSimplex.δ (k.pred ⋯) ≫ ι i j hj) ≫ g = (stdSimplex.δ (j.castPred ⋯) ≫ ι i k hk) ≫ g"
] | simp only [isCompatible_iff, ← Category.assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Enriched.Basic | {
"line": 491,
"column": 8
} | {
"line": 495,
"column": 59
} | {
"line": 495,
"column": 60
} | [
{
"pp": "V : Type v\ninst✝⁴ : Category.{w, v} V\ninst✝³ : MonoidalCategory V\nC : Type u₁\ninst✝² : EnrichedCategory V C\nD : Type u₂\ninst✝¹ : EnrichedCategory V D\ninst✝ : BraidedCategory V\nF G : EnrichedFunctor V C D\nX✝ Y✝ : Vᵒᵖ\nf : X✝ ⟶ Y✝\nσ : GradedNatTrans ((Center.ofBraided V).obj (unop X✝)) F G\nX Y... | [] | have p := σ.naturality X Y
dsimp at p ⊢
rw [← id_tensor_comp_tensor_id (f.unop ≫ σ.app Y) _, id_tensor_comp, Category.assoc,
Category.assoc, ← braiding_naturality_assoc, id_tensor_comp_tensor_id_assoc, p,
tensorHom_comp_tensorHom_assoc, Category.id_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Enriched.Basic | {
"line": 491,
"column": 8
} | {
"line": 495,
"column": 59
} | {
"line": 495,
"column": 60
} | [
{
"pp": "V : Type v\ninst✝⁴ : Category.{w, v} V\ninst✝³ : MonoidalCategory V\nC : Type u₁\ninst✝² : EnrichedCategory V C\nD : Type u₂\ninst✝¹ : EnrichedCategory V D\ninst✝ : BraidedCategory V\nF G : EnrichedFunctor V C D\nX✝ Y✝ : Vᵒᵖ\nf : X✝ ⟶ Y✝\nσ : GradedNatTrans ((Center.ofBraided V).obj (unop X✝)) F G\nX Y... | [] | have p := σ.naturality X Y
dsimp at p ⊢
rw [← id_tensor_comp_tensor_id (f.unop ≫ σ.app Y) _, id_tensor_comp, Category.assoc,
Category.assoc, ← braiding_naturality_assoc, id_tensor_comp_tensor_id_assoc, p,
tensorHom_comp_tensorHom_assoc, Category.id_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Functor.FunctorHom | {
"line": 98,
"column": 12
} | {
"line": 98,
"column": 31
} | {
"line": 98,
"column": 31
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nF G : C ⥤ D\nA A' : (C ⥤ Type w)ᵒᵖ\nf : A ⟶ A'\nx : F.HomObj G (Opposite.unop A)\nX Y : C\nφ : X ⟶ Y\na : (Opposite.unop A').obj X\n⊢ F.map φ ≫ x.app Y ((ConcreteCategory.hom (f.unop.app Y)) ((ConcreteCategory.hom ((Oppos... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nF G : C ⥤ D\nA A' : (C ⥤ Type w)ᵒᵖ\nf : A ⟶ A'\nx : F.HomObj G (Opposite.unop A)\nX Y : C\nφ : X ⟶ Y\na : (Opposite.unop A').obj X\n⊢ F.map φ ≫ x.app Y ((ConcreteCategory.hom (f.unop.app Y)) ((ConcreteCategory.hom ((Opposite.unop A')... | ← HomObj.naturality | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated | {
"line": 140,
"column": 45
} | {
"line": 142,
"column": 53
} | {
"line": 144,
"column": 0
} | [
{
"pp": "X : Truncated 2\ninst✝ : X.Quasicategory₂\nx y : X.obj (Opposite.op { obj := { len := 0 }, property := Quasicategory₂._proof_1 })\nf g : Edge x y\nh : HomotopicL f g\n⊢ HomotopicR f g",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"SSet.Truncated.Quasicategory₂.fill32",
... | [] | by
rcases h with ⟨h⟩
exact Quasicategory₂.fill32 (idComp f) (compId f) h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated | {
"line": 285,
"column": 62
} | {
"line": 288,
"column": 73
} | {
"line": 290,
"column": 0
} | [
{
"pp": "A : Truncated 2\ninst✝ : A.Quasicategory₂\nx y z : A.obj (Opposite.op { obj := { len := 0 }, property := Quasicategory₂._proof_1 })\nf : Edge x y\ng : Edge y z\nh : Edge x z\nfac : homMk f ≫ homMk g = homMk h\n⊢ f.CompStruct g h",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
... | [] | by
dsimp [homMk, CategoryStruct.comp] at fac
rw [Quotient.eq_iff_equiv] at fac
exact (Quasicategory₂.fill32 (compStruct f g) (compId g) fac.some).some | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Coskeletal | {
"line": 120,
"column": 10
} | {
"line": 120,
"column": 15
} | {
"line": 120,
"column": 16
} | [
{
"pp": "case zero\nX : SSet\nsx : X.StrictSegal\nn : ℕ\ns : Cone (proj (op ⦋n⦌) (inclusion 2).op ⋙ (inclusion 2).op ⋙ X)\nx : s.pt\ni : ℕ\nhij : i ≤ i\nhj : i ≤ n\nthis : mkOfLe ⟨i, ⋯⟩ ⟨i, ⋯⟩ ⋯ = ⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩\nα : strArrowMk₂ (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) lift._proof_1 ⟶\n strArrowMk₂ (⦋1⦌.... | [
"case zero\nX : SSet\nsx : X.StrictSegal\nn : ℕ\ns : Cone (proj (op ⦋n⦌) (inclusion 2).op ⋙ (inclusion 2).op ⋙ X)\nx : s.pt\ni : ℕ\nhij : i ≤ i\nhj : i ≤ n\nthis : mkOfLe ⟨i, ⋯⟩ ⟨i, ⋯⟩ ⋯ = ⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩\nα : strArrowMk₂ (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) lift._proof_1 ⟶\n strArrowMk₂ (⦋1⦌.const ⦋0⦌ 0 ... | lift, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.EpiMono | {
"line": 60,
"column": 94
} | {
"line": 64,
"column": 46
} | {
"line": 66,
"column": 0
} | [
{
"pp": "x y : SimplexCategoryGenRel\ne : x ⟶ y\nhe : P_σ e\n⊢ IsSplitEpi e",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"SimplexCategoryGenRel.degeneracies.casesOn",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.MorphismProperty.multipli... | [] | by
induction he with
| of x hx => cases hx; infer_instance
| id => infer_instance
| comp_of _ _ _ h => cases h; infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.EpiMono | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 23
} | {
"line": 132,
"column": 0
} | [
{
"pp": "case «0».«1»\n⊢ δ ((fun i ↦ i) ⟨1, ⋯⟩) ≫ σ ((fun i ↦ i) ⟨0, ⋯⟩) = 𝟙 (mk 0)",
"ppTerm": "?«0».«1»",
"assigned": true,
"usedConstants": [
"Nat.le_refl",
"SimplexCategoryGenRel.δ_comp_σ_succ",
"Fin.mk",
"instOfNatNat",
"instHAdd",
"HAdd.hAdd",
"Simple... | [] | · exact δ_comp_σ_succ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.EpiMono | {
"line": 181,
"column": 4
} | {
"line": 200,
"column": 43
} | {
"line": 202,
"column": 0
} | [
{
"pp": "case comp_σ\nx y : SimplexCategoryGenRel\nn n' : ℕ\nj : Fin (n' + 1)\nz : SimplexCategoryGenRel\ne : mk n ⟶ z\nm : z ⟶ mk (n' + 1)\nhe : P_σ e\nhm : P_δ m\n⊢ ∃ z_1 e_1 m_1, ∃ (_ : P_σ e_1) (_ : P_δ m_1), (e ≫ m) ≫ σ j = e_1 ≫ m_1",
"ppTerm": "?comp_σ",
"assigned": true,
"usedConstants": [
... | [] | cases hm with
| of g hg =>
rcases hg with ⟨i⟩
obtain ⟨_, _, _, ⟨he₁, hm₁, h₁⟩⟩ := factor_δ_σ j i
exact ⟨_, _, _, P_σ.comp_mem _ _ he he₁, hm₁,
by simp [← h₁]⟩
| @id n =>
exact ⟨mk n', e ≫ σ j, 𝟙 _, P_σ.comp_mem _ _ he (P_σ.σ _), P_δ.id_mem _, by simp⟩
| comp_of f g hf hg =>
... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.NormalForms | {
"line": 83,
"column": 66
} | {
"line": 91,
"column": 84
} | {
"line": 93,
"column": 0
} | [
{
"pp": "m : ℕ\nL : List ℕ\n⊢ IsAdmissible m L ↔ List.IsChain (fun x1 x2 ↦ x1 < x2) L ∧ ∀ (k : ℕ) (h : k < L.length), L[k] ≤ m + k",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"_private.Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.NormalForms.0.SimplexCategoryGenR... | [] | by
induction L using List.twoStepInduction generalizing m with
| nil => grind
| singleton _ => simp
| cons_cons _ _ _ _ IH =>
simp_rw [isAdmissible_cons_cons_iff, IH, List.length_cons, and_assoc,
List.isChain_cons_cons, and_assoc, and_congr_right_iff, and_comm]
exact fun _ _ => ⟨fun h => by grind,... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplexCategory.ToMkOne | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 23
} | {
"line": 134,
"column": 4
} | [
{
"pp": "case pos\nn : ℕ\nf : ⦋n⦌ ⟶ ⦋1⦌\nS : Finset (Fin (n + 1)) := {i | (ConcreteCategory.hom f) i = 1}\nhS : S.Nonempty\ni : Fin (⦋n⦌.len + 1)\n⊢ (ConcreteCategory.hom (toMk₁ (S.min' hS).castSucc)) i = (ConcreteCategory.hom f) i",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Fin... | [
"case pos\nn : ℕ\nf : ⦋n⦌ ⟶ ⦋1⦌\nS : Finset (Fin (n + 1)) := {i | (ConcreteCategory.hom f) i = 1}\nhS : S.Nonempty\ni : Fin (⦋n⦌.len + 1)\n⊢ (if i.castSucc < (S.min' hS).castSucc then 0 else 1) = (ConcreteCategory.hom f) i"
] | dsimp [toMk₁_apply] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.AlgebraicTopology.SimplexCategory.ToMkOne | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 23
} | {
"line": 154,
"column": 4
} | [
{
"pp": "case neg\nn : ℕ\nf : ⦋n⦌ ⟶ ⦋1⦌\nS : Finset (Fin (n + 1)) := {i | (ConcreteCategory.hom f) i = 1}\nhS : ¬S.Nonempty\ni : Fin (⦋n⦌.len + 1)\n⊢ (ConcreteCategory.hom (toMk₁ (Fin.last (n + 1)))) i = (ConcreteCategory.hom f) i",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Simp... | [
"case neg\nn : ℕ\nf : ⦋n⦌ ⟶ ⦋1⦌\nS : Finset (Fin (n + 1)) := {i | (ConcreteCategory.hom f) i = 1}\nhS : ¬S.Nonempty\ni : Fin (⦋n⦌.len + 1)\n⊢ (if i.castSucc < Fin.last (n + 1) then 0 else 1) = (ConcreteCategory.hom f) i"
] | dsimp [toMk₁_apply] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Order.UpperLower.Relative | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 28
} | {
"line": 148,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nP : α → Prop\ninst✝ : LE α\ns : Set { x // P x }\nh : IsUpperSet s\na : α\nx : a ∈ Subtype.val '' s\n⊢ P a ∧ ∀ ⦃b : α⦄, a ≤ b → P b → b ∈ Subtype.val '' s",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Subtype",
... | [
"case refine_1\nα : Type u_1\nP : α → Prop\ninst✝ : LE α\ns : Set { x // P x }\nh : IsUpperSet s\na : { x // P x }\nma : a ∈ s\n⊢ P ↑a ∧ ∀ ⦃b : α⦄, ↑a ≤ b → P b → b ∈ Subtype.val '' s"
] | obtain ⟨a, ma, rfl⟩ := x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.UpperLower.Relative | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 28
} | {
"line": 156,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nP : α → Prop\ninst✝ : LE α\ns : Set { x // P x }\nh : IsLowerSet s\na : α\nx : a ∈ Subtype.val '' s\n⊢ P a ∧ ∀ ⦃b : α⦄, b ≤ a → P b → b ∈ Subtype.val '' s",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Subtype",
... | [
"case refine_1\nα : Type u_1\nP : α → Prop\ninst✝ : LE α\ns : Set { x // P x }\nh : IsLowerSet s\na : { x // P x }\nma : a ∈ s\n⊢ P ↑a ∧ ∀ ⦃b : α⦄, b ≤ ↑a → P b → b ∈ Subtype.val '' s"
] | obtain ⟨a, ma, rfl⟩ := x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.SimplicialObject.II | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 24
} | {
"line": 183,
"column": 4
} | [
{
"pp": "case inl\nn : ℕ\ni x : Fin (n + 1)\n⊢ map' { toFun := i.predAbove, monotone' := ⋯ } x.castSucc = i.succ.castSucc.succAbove x.castSucc",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
"Fin.succ",
"SimplexCategory.II.map'",
"PartialOrder.toPreo... | [
"case pos\nn : ℕ\ni x : Fin (n + 1)\nhi : i < x\n⊢ map' { toFun := i.predAbove, monotone' := ⋯ } x.castSucc = i.succ.castSucc.succAbove x.castSucc",
"case neg\nn : ℕ\ni x : Fin (n + 1)\nhi : x ≤ i\n⊢ map' { toFun := i.predAbove, monotone' := ⋯ } x.castSucc = i.succ.castSucc.succAbove x.castSucc"
] | by_cases! hi : i < x | Mathlib.Tactic.ByCases._aux_Mathlib_Tactic_ByCases___macroRules_Mathlib_Tactic_ByCases_byCases!_1 | Mathlib.Tactic.ByCases.byCases! |
Mathlib.AlgebraicTopology.SimplicialObject.II | {
"line": 187,
"column": 11
} | {
"line": 187,
"column": 63
} | {
"line": 187,
"column": 63
} | [
{
"pp": "n : ℕ\ni x : Fin (n + 1)\nhi : i < x\n⊢ i.castSucc < x.succ",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.AlgebraicTopology.SimplicialObject.II.0.SimplexCategory.II.map'_predAbove._simp_1_3",
"Fin.succ",
"PartialOrder.toPreorde... | [] | by simpa only [Fin.castSucc_lt_succ_iff] using hi.le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Monoidal | {
"line": 306,
"column": 4
} | {
"line": 314,
"column": 20
} | {
"line": 315,
"column": 2
} | [
{
"pp": "x✝ y✝ z✝ : AugmentedSimplexCategory\nx y z : SimplexCategory\n⊢ inl (WithInitial.of x) (WithInitial.of y) ≫\n (WithInitial.of x ⊗ WithInitial.of y).inl (WithInitial.of z) ≫\n (α_ (WithInitial.of x) (WithInitial.of y) (WithInitial.of z)).hom =\n inl (WithInitial.of x) (WithInitial.of y ⊗ ... | [] | change inl' _ _ ≫ inl' _ _ ≫ WithInitial.down _ = inl' _ _
ext i : 3
dsimp [MonoidalCategoryStruct.associator, associator]
have e₁ := inl'_eval x y i
have e₂ := inl'_eval x (tensorObjOf y z) i
have e₃ := inl'_eval (tensorObjOf x y) z <| Fin.cast (by simp +arith) <| i.castAdd (y.len + 1)
simp onl... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplexCategory.Augmented.Monoidal | {
"line": 306,
"column": 4
} | {
"line": 314,
"column": 20
} | {
"line": 315,
"column": 2
} | [
{
"pp": "x✝ y✝ z✝ : AugmentedSimplexCategory\nx y z : SimplexCategory\n⊢ inl (WithInitial.of x) (WithInitial.of y) ≫\n (WithInitial.of x ⊗ WithInitial.of y).inl (WithInitial.of z) ≫\n (α_ (WithInitial.of x) (WithInitial.of y) (WithInitial.of z)).hom =\n inl (WithInitial.of x) (WithInitial.of y ⊗ ... | [] | change inl' _ _ ≫ inl' _ _ ≫ WithInitial.down _ = inl' _ _
ext i : 3
dsimp [MonoidalCategoryStruct.associator, associator]
have e₁ := inl'_eval x y i
have e₂ := inl'_eval x (tensorObjOf y z) i
have e₃ := inl'_eval (tensorObjOf x y) z <| Fin.cast (by simp +arith) <| i.castAdd (y.len + 1)
simp onl... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialObject.II | {
"line": 212,
"column": 4
} | {
"line": 215,
"column": 10
} | {
"line": 215,
"column": 10
} | [
{
"pp": "n m : ℕ\nf : Fin (n + 1) →o Fin (m + 1)\nx y : Fin (m + 2)\nhxy : x ≤ y\nz : Fin (n + 2)\nhz : z ∈ finset f y\n⊢ z ∈ finset f x",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimplexCategory.II.finset",
"SimplexCategory.II.castSucc_mem_finset_iff._sim... | [] | obtain ⟨z, rfl⟩ | rfl := z.eq_castSucc_or_eq_last
· simp only [castSucc_mem_finset_iff] at hz ⊢
exact hxy.trans hz
· simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialObject.II | {
"line": 212,
"column": 4
} | {
"line": 215,
"column": 10
} | {
"line": 215,
"column": 10
} | [
{
"pp": "n m : ℕ\nf : Fin (n + 1) →o Fin (m + 1)\nx y : Fin (m + 2)\nhxy : x ≤ y\nz : Fin (n + 2)\nhz : z ∈ finset f y\n⊢ z ∈ finset f x",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimplexCategory.II.finset",
"SimplexCategory.II.castSucc_mem_finset_iff._sim... | [] | obtain ⟨z, rfl⟩ | rfl := z.eq_castSucc_or_eq_last
· simp only [castSucc_mem_finset_iff] at hz ⊢
exact hxy.trans hz
· simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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