module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Limits.Constructions.Equalizers | {
"line": 165,
"column": 4
} | {
"line": 172,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nG : C ⥤ D\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : HasPushouts C\nF : WalkingParallelPair ⥤ C\n⊢ ∀ (s : Cocone F) (m : (coequalizerCocone F).pt ⟶ s.pt),\n (∀ (j : WalkingParallelPair), (coequalizerCocone F).ι.app j ≫ ... | intro c m J
have J1 : pushoutInl F ≫ m = c.ι.app WalkingParallelPair.one := by
simpa using J WalkingParallelPair.one
apply pushout.hom_ext
· rw [colimit.ι_desc]
exact J1
· rw [colimit.ι_desc, ← pushoutInl_eq_pushout_inr]
exact J1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Constructions.Equalizers | {
"line": 165,
"column": 4
} | {
"line": 172,
"column": 14
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\nG : C ⥤ D\ninst✝¹ : HasBinaryCoproducts C\ninst✝ : HasPushouts C\nF : WalkingParallelPair ⥤ C\n⊢ ∀ (s : Cocone F) (m : (coequalizerCocone F).pt ⟶ s.pt),\n (∀ (j : WalkingParallelPair), (coequalizerCocone F).ι.app j ≫ ... | intro c m J
have J1 : pushoutInl F ≫ m = c.ι.app WalkingParallelPair.one := by
simpa using J WalkingParallelPair.one
apply pushout.hom_ext
· rw [colimit.ι_desc]
exact J1
· rw [colimit.ι_desc, ← pushoutInl_eq_pushout_inr]
exact J1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Constructions.Equalizers | {
"line": 201,
"column": 10
} | {
"line": 201,
"column": 87
} | [
{
"pp": "case refine_3\nC : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nG : C ⥤ D\ninst✝³ : HasBinaryCoproducts C\ninst✝² : HasPushouts C\ninst✝¹ : PreservesColimitsOfShape (Discrete WalkingPair) G\ninst✝ : PreservesColimitsOfShape WalkingSpan G\nK : WalkingParallelPair ⥤ C\nc... | apply (mapIsColimitOfPreservesOfIsColimit G _ _ (coprodIsCoprod _ _)).hom_ext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Monoidal.Cartesian.Basic | {
"line": 806,
"column": 17
} | {
"line": 806,
"column": 46
} | [
{
"pp": "C✝ : Type u\ninst✝⁴ : Category.{v, u} C✝\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : CartesianMonoidalCategory C\nP : ObjectProperty C\ninst✝¹ : P.IsClosedUnderLimitsOfShape (Discrete PEmpty.{1})\ninst✝ : P.IsClosedUnderLimitsOfShape (Discrete WalkingPair)\nX Y : P.FullSubcategory\n⊢ ObjectProper... | by ext; exact fst_def X.1 Y.1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.SplitEqualizer | {
"line": 98,
"column": 49
} | {
"line": 98,
"column": 74
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nG : C ⥤ D\nX Y : C\nf g : X ⟶ Y\nW : C\nι : W ⟶ X\nq : IsSplitEqualizer f g ι\nF : C ⥤ D\n⊢ F.map (g ≫ q.rightRetraction) = 𝟙 (F.obj X)",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.IsSplitEqualizer.bott... | q.bottom_rightRetraction, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.Basic | {
"line": 177,
"column": 2
} | {
"line": 178,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasKernels C\ninst✝² : HasCokernels C\ninst✝¹ : HasZeroObject C\nX Y : C\nf : X ⟶ Y\ninst✝ : Epi f\n⊢ IsIso (imageMonoFactorisation f).m",
"usedConstants": [
"CategoryTheory.Limits.MonoFactorisation.I",
"CategoryTh... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.Basic | {
"line": 177,
"column": 2
} | {
"line": 178,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Preadditive C\ninst✝³ : HasKernels C\ninst✝² : HasCokernels C\ninst✝¹ : HasZeroObject C\nX Y : C\nf : X ⟶ Y\ninst✝ : Epi f\n⊢ IsIso (imageMonoFactorisation f).m",
"usedConstants": [
"CategoryTheory.Limits.MonoFactorisation.I",
"CategoryTh... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.Basic | {
"line": 723,
"column": 90
} | {
"line": 726,
"column": 87
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasPullbacks C\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : Epi f\ns : PullbackCone f g\nhs : IsLimit s\n⊢ Epi s.snd",
"usedConstants": [
"CategoryTheory.Limits.Cone.π",
"CategoryTheory.Functor",
"CategoryTheory.Abe... | by
haveI : Epi (NatTrans.app (limit.cone (cospan f g)).π WalkingCospan.right) :=
Abelian.epi_pullback_of_epi_f f g
apply epi_of_epi_fac (IsLimit.conePointUniqueUpToIso_hom_comp (limit.isLimit _) hs _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.Grp.Colimits | {
"line": 124,
"column": 2
} | {
"line": 125,
"column": 59
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ AddCommGrpCat\ninst✝ : DecidableEq J\nj : J\nx : ↑(F.obj j)\n⊢ (QuotientAddGroup.lift (Relations F)\n (DFinsupp.sumAddHom fun j ↦\n ((QuotientAddGroup.mk' (Relations (F ⋙ uliftFunctor))).comp\n (DFinsupp.singleAddHom (fun j ... | conv_lhs => erw [AddMonoidHom.comp_apply (QuotientAddGroup.mk' (Relations F))
(DFinsupp.singleAddHom _ j), QuotientAddGroup.lift_mk'] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.Algebra.Category.Grp.Colimits | {
"line": 134,
"column": 80
} | {
"line": 140,
"column": 6
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v, u} J\nF : J ⥤ AddCommGrpCat\nc : Cocone F\ninst✝ : DecidableEq J\n⊢ Quot (F ⋙ uliftFunctor) →+ Quot F",
"usedConstants": [
"AddEquivClass.instAddMonoidHomClass",
"Eq.mpr",
"ULift.addZeroClass",
"SetLike.mem_coe._simp_1",
"AddMonoidHom.... | by
refine QuotientAddGroup.lift (Relations (F ⋙ uliftFunctor))
(DFinsupp.sumAddHom (fun j ↦ (Quot.ι _ j).comp AddEquiv.ulift.toAddMonoidHom)) ?_
rw [AddSubgroup.closure_le]
intro _ hx
obtain ⟨j, j', u, a, rfl⟩ := hx
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Module.ModuleTopology | {
"line": 229,
"column": 4
} | {
"line": 234,
"column": 10
} | [
{
"pp": "case h.h.mpr\nR : Type u_1\nS : Type u_2\nτR : TopologicalSpace R\nτS : TopologicalSpace S\ninst✝⁸ : Semiring R\ninst✝⁷ : Semiring S\nσ : R →+* S\nσ' : S →+* R\ninst✝⁶ : RingHomInvPair σ σ'\ninst✝⁵ : RingHomInvPair σ' σ\nA : Type u_3\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R A\nτA : TopologicalSpace... | · rintro ⟨h1, h2⟩
use τ.induced e
rw [induced_compose]
refine ⟨⟨continuousSMul_inducedₛₗ g hσ, continuousAdd_induced h⟩, ?_⟩
nth_rw 2 [← induced_id (t := τ)]
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 323,
"column": 35
} | {
"line": 323,
"column": 57
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.LeftHomologyData\nh₂ : S₂.LeftHomologyData\nφK : h₁.K ⟶ h₂.K := h₂.liftK (h₁.i ≫ φ.τ₂) ⋯\n⊢ S₁.f ≫ φ.τ₂ = φ.τ₁ ≫ h₂.f' ≫ h₂.i",
"usedConstants": [
"Eq.mpr",
"Categ... | LeftHomologyData.f'_i, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 1110,
"column": 4
} | {
"line": 1112,
"column": 31
} | [
{
"pp": "case pos\nR : Type u_1\nM : Type u_2\nM₂ : Type u_3\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : TopologicalSpace M₂\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R M₂\nh : IsInvertible 0\n⊢ inverse 0 = 0",
"usedConstants": [
"Continu... | rcases isInvertible_zero_iff.1 h with ⟨hM, hM₂⟩
ext x
exact Subsingleton.elim _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 1110,
"column": 4
} | {
"line": 1112,
"column": 31
} | [
{
"pp": "case pos\nR : Type u_1\nM : Type u_2\nM₂ : Type u_3\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : TopologicalSpace M₂\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R M₂\nh : IsInvertible 0\n⊢ inverse 0 = 0",
"usedConstants": [
"Continu... | rcases isInvertible_zero_iff.1 h with ⟨hM, hM₂⟩
ext x
exact Subsingleton.elim _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 842,
"column": 20
} | {
"line": 842,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₁.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂\nwi : i ≫ S₂.g = 0\nW'✝ : C\nx : W'✝ ⟶ S₂.X₂\nhx : x ≫ S₂.g = 0\n⊢ (fun ... | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 842,
"column": 20
} | {
"line": 842,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₁.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂\nwi : i ≫ S₂.g = 0\nW'✝ : C\nx : W'✝ ⟶ S₂.X₂\nhx : x ≫ S₂.g = 0\n⊢ (fun ... | simp [i] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 842,
"column": 20
} | {
"line": 842,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₁.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂\nwi : i ≫ S₂.g = 0\nW'✝ : C\nx : W'✝ ⟶ S₂.X₂\nhx : x ≫ S₂.g = 0\n⊢ (fun ... | simp [i] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 879,
"column": 20
} | {
"line": 879,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\nW'✝ : C\nx : W'✝ ⟶ S₁.X₂\nhx : x ≫ S₁.g = 0\n⊢ (... | simp [i] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 879,
"column": 20
} | {
"line": 879,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\nW'✝ : C\nx : W'✝ ⟶ S₁.X₂\nhx : x ≫ S₁.g = 0\n⊢ (... | simp [i] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 879,
"column": 20
} | {
"line": 879,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₂.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂\nwi : i ≫ S₁.g = 0\nW'✝ : C\nx : W'✝ ⟶ S₁.X₂\nhx : x ≫ S₁.g = 0\n⊢ (... | simp [i] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 406,
"column": 16
} | {
"line": 411,
"column": 9
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.RightHomologyData\nh₂ : S₂.RightHomologyData\nψ₁ ψ₂ : RightHomologyMapData φ h₁ h₂\n⊢ ψ₁ = ψ₂",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.HasZeroMorphism... | by
have hQ : ψ₁.φQ = ψ₂.φQ := by rw [← cancel_epi h₁.p, commp, commp]
have hH : ψ₁.φH = ψ₂.φH := by rw [← cancel_mono h₂.ι, commι, commι, hQ]
cases ψ₁
cases ψ₂
congr | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 1007,
"column": 2
} | {
"line": 1007,
"column": 33
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasLeftHomology\n⊢ S.opcyclesOpIso.hom ≫ S.toCycles.op = S.op.fromOpcycles",
"usedConstants": [
"CategoryTheory.ShortComplex.opcycles",
"Opposite",
"CategoryTheory.CategoryStru... | dsimp [opcyclesOpIso, toCycles] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 1010,
"column": 38
} | {
"line": 1010,
"column": 60
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasLeftHomology\n⊢ (S.leftHomologyData.f' ≫ S.leftHomologyData.i).op = S.op.g",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.ShortComplex.LeftHomologyData.f'",
"CategoryTheory.... | LeftHomologyData.f'_i, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 614,
"column": 39
} | {
"line": 614,
"column": 60
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh₁ h₁' : S.LeftHomologyData\nh₂ h₂' : S.RightHomologyData\n⊢ leftRightHomologyComparison' h₁ h₂ = leftRightHomologyComparison' h₁ h₂ ≫ rightHomologyMap' (𝟙 S) h₂ h₂",
"usedConstants": [
"Eq.mpr",
"C... | rightHomologyMap'_id, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 752,
"column": 48
} | {
"line": 754,
"column": 93
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\ninst✝¹ : S₁.HasHomology\ninst✝ : S₂.HasHomology\nφ : S₁ ⟶ S₂\n⊢ S₁.leftHomologyIso.inv ≫ leftHomologyMap φ = homologyMap φ ≫ S₂.leftHomologyIso.inv",
"usedConstants": [
"CategoryTheory.ShortComplex.le... | by
simpa only [LeftHomologyData.homologyIso_leftHomologyData, Iso.symm_inv] using
LeftHomologyData.leftHomologyIso_hom_naturality φ S₁.leftHomologyData S₂.leftHomologyData | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 1119,
"column": 8
} | {
"line": 1119,
"column": 35
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\ninst✝¹ : S₁.HasHomology\ninst✝ : S₂.HasHomology\nh₁ : IsIso (opcyclesMap φ)\nh₂ : Mono φ.τ₃\nh : (S₂.homologyι ≫ inv (opcyclesMap φ)) ≫ S₁.fromOpcycles = 0\nz : S₂.homology ⟶ (Kernel... | ← cancel_mono S₂.homologyι, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.Abelian | {
"line": 96,
"column": 2
} | {
"line": 99,
"column": 12
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nS : ShortComplex C\nD : Type u'\ninst✝¹ : Category.{v', u'} D\ninst✝ : HasZeroMorphisms D\nγ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f\nf' : S.X₁ ⟶ kernel S.g := kernel.lift S.g S.f ⋯\nhf' : f' = kernel.lift γ f' ⋯ ≫ kernel.... | have fac : f' = Abelian.factorThruImage S.f ≫ e.hom ≫ kernel.ι γ := by
rw [hf', he]
simp only [γ, f', kernel.lift_ι, abelianImageToKernel, ← cancel_mono (kernel.ι S.g),
assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 580,
"column": 65
} | {
"line": 580,
"column": 87
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nH₁ : S₁.LeftHomologyData\nH₂ : S₂.LeftHomologyData\nh₀ : S₁.X₁ ⟶ S₂.X₁\nh₀_f : h₀ ≫ S₂.f = 0\nh₁ : S₁.X₂ ⟶ S₂.X₁\nh₂ : S₁.X₃ ⟶ S₂.X₂\nh₃ : S₁.X₃ ⟶ S₂.X₃\ng_h₃ : S₁.g ≫ h₃ = 0\n⊢ S₁.f ≫... | LeftHomologyData.f'_i, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.Topology.Basic | {
"line": 310,
"column": 4
} | {
"line": 313,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝² : Ring R\ninst✝¹ : TopologicalSpace R\nM : ModuleCat R\nI : Type u_1\nX : I → TopModuleCat R\nf : (i : I) → M ⟶ (X i).toModuleCat\nJ : Type u_2\ninst✝ : Category.{v_1, u_2} J\nF : J ⥤ TopModuleCat R\nc : Cone (F ⋙ forget₂ (TopModuleCat R) (ModuleCat R))\nhc : IsLimit c\ns : Cone F\nm... | ext x
refine congr($(hc.uniq ((forget₂ _ _).mapCone s) ((forget₂ _ _).map m) fun j ↦ ?_).hom x)
ext y
exact congr($(H j).hom y) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Topology.Basic | {
"line": 310,
"column": 4
} | {
"line": 313,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝² : Ring R\ninst✝¹ : TopologicalSpace R\nM : ModuleCat R\nI : Type u_1\nX : I → TopModuleCat R\nf : (i : I) → M ⟶ (X i).toModuleCat\nJ : Type u_2\ninst✝ : Category.{v_1, u_2} J\nF : J ⥤ TopModuleCat R\nc : Cone (F ⋙ forget₂ (TopModuleCat R) (ModuleCat R))\nhc : IsLimit c\ns : Cone F\nm... | ext x
refine congr($(hc.uniq ((forget₂ _ _).mapCone s) ((forget₂ _ _).map m) fun j ↦ ?_).hom x)
ext y
exact congr($(H j).hom y) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Subobject.FactorThru | {
"line": 68,
"column": 8
} | {
"line": 69,
"column": 79
} | [
{
"pp": "case a.mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nX Y : C\nP✝ : Subobject Y\nf : X ⟶ Y\nP Q : MonoOver Y\nh : P ≅ Q\n⊢ (fun P ↦ P.Factors f) P → (fun P ↦ P.Factors f) Q",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.ass... | rintro ⟨i, w⟩
exact ⟨i ≫ h.hom.hom.left, by rw [Category.assoc, Over.w h.hom.hom, w]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Subobject.FactorThru | {
"line": 68,
"column": 8
} | {
"line": 69,
"column": 79
} | [
{
"pp": "case a.mp\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nX Y : C\nP✝ : Subobject Y\nf : X ⟶ Y\nP Q : MonoOver Y\nh : P ≅ Q\n⊢ (fun P ↦ P.Factors f) P → (fun P ↦ P.Factors f) Q",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.ass... | rintro ⟨i, w⟩
exact ⟨i ≫ h.hom.hom.left, by rw [Category.assoc, Over.w h.hom.hom, w]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Subobject.Basic | {
"line": 605,
"column": 62
} | {
"line": 611,
"column": 47
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nX Y : C\ninst✝ : HasPullbacks C\nf : X ⟶ Y\ny : Subobject Y\n⊢ ∃ φ, IsPullback φ ((pullback f).obj y).arrow y.arrow f",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"CategoryTheory.Subobject.arrow",
"CategoryTheory... | by
obtain ⟨A, i, ⟨_, rfl⟩⟩ := mk_surjective y
rw [pullback_obj]
exists (underlyingIso (pullback.snd (mk i).arrow f)).hom ≫ pullback.fst (mk i).arrow f
exact IsPullback.of_iso (IsPullback.of_hasPullback (mk i).arrow f)
(underlyingIso (pullback.snd (mk i).arrow f)).symm (Iso.refl _) (Iso.refl _) (Iso.refl... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Subobject.MonoOver | {
"line": 368,
"column": 6
} | {
"line": 368,
"column": 15
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nf : X ⟶ Y\ninst✝ : Mono f\ng h : MonoOver X\ne : (map f).obj g ⟶ (map f).obj h\n⊢ e.hom.left ≫ h.arrow ≫ f = g.arrow ≫ f",
"usedConstants": [
"CategoryTheory.Over",
"CategoryTheory.instCateg... | apply w e | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 65
} | [
{
"pp": "case pos\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nX₁ : ι → V\nd₁ : (i j : ι) → X₁ i ⟶ X₁ j\ns₁ : ∀ (i j : ι), ¬c.Rel i j → d₁ i j = 0\nh₁ : ∀ (i j k : ι), c.Rel i j → c.Rel j k → d₁ i j ≫ d₁ j k = 0\nd₂ : (i j : ι) → X₁ i ⟶ X₁ j\ns₂ : ∀ (i j... | · simpa only [comp_id, id_comp, eqToHom_refl] using h_d i j hij | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 371,
"column": 73
} | {
"line": 373,
"column": 35
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni j j' : ι\nrij : c.Rel i j\nrij' : c.Rel i j'\n⊢ C.d i j' ≫ eqToHom ⋯ = C.d i j",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",... | by
obtain rfl := c.next_eq rij rij'
simp only [eqToHom_refl, comp_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 95,
"column": 2
} | {
"line": 96,
"column": 41
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.HomologyData\n⊢ S.Exact ↔ IsZero h.left.H",
"usedConstants": [
"CategoryTheory.ShortComplex.HomologyData.left",
"CategoryTheory.ShortComplex.HasHomology.mk'",
"CategoryTheory.ShortC... | haveI := HasHomology.mk' h
exact LeftHomologyData.exact_iff h.left | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 95,
"column": 2
} | {
"line": 96,
"column": 41
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.HomologyData\n⊢ S.Exact ↔ IsZero h.left.H",
"usedConstants": [
"CategoryTheory.ShortComplex.HomologyData.left",
"CategoryTheory.ShortComplex.HasHomology.mk'",
"CategoryTheory.ShortC... | haveI := HasHomology.mk' h
exact LeftHomologyData.exact_iff h.left | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 100,
"column": 2
} | {
"line": 101,
"column": 43
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.HomologyData\n⊢ S.Exact ↔ IsZero h.right.H",
"usedConstants": [
"CategoryTheory.ShortComplex.RightHomologyData.exact_iff",
"CategoryTheory.ShortComplex.HasHomology.mk'",
"CategoryTh... | haveI := HasHomology.mk' h
exact RightHomologyData.exact_iff h.right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 100,
"column": 2
} | {
"line": 101,
"column": 43
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.HomologyData\n⊢ S.Exact ↔ IsZero h.right.H",
"usedConstants": [
"CategoryTheory.ShortComplex.RightHomologyData.exact_iff",
"CategoryTheory.ShortComplex.HasHomology.mk'",
"CategoryTh... | haveI := HasHomology.mk' h
exact RightHomologyData.exact_iff h.right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 64
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} D\ninst✝⁴ : HasZeroMorphisms C\ninst✝³ : HasZeroMorphisms D\nS : ShortComplex C\nh : S.Exact\nF : C ⥤ D\ninst✝² : F.PreservesZeroMorphisms\ninst✝¹ : F.PreservesRightHomologyOf S\ninst✝ : (S.map F).HasHomology\nthis... | rw [S.rightHomologyData.exact_iff, IsZero.iff_id_eq_zero] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Functor | {
"line": 107,
"column": 2
} | {
"line": 108,
"column": 5
} | [
{
"pp": "T : Type u_1\ninst✝² : Category.{v_1, u_1} T\nV : Type u_2\ninst✝¹ : Category.{v_2, u_2} V\ninst✝ : Abelian V\nι : Type u_3\nc : ComplexShape ι\nK₁ K₂ : HomologicalComplex (T ⥤ V) c\nf : K₁ ⟶ K₂\ni : ι\n⊢ QuasiIsoAt f i ↔ ∀ (t : T), QuasiIsoAt ((((evaluation T V).obj t).mapHomologicalComplex c).map f) ... | simp only [quasiIsoAt_iff, ShortComplex.quasiIso_iff_evaluation]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Functor | {
"line": 107,
"column": 2
} | {
"line": 108,
"column": 5
} | [
{
"pp": "T : Type u_1\ninst✝² : Category.{v_1, u_1} T\nV : Type u_2\ninst✝¹ : Category.{v_2, u_2} V\ninst✝ : Abelian V\nι : Type u_3\nc : ComplexShape ι\nK₁ K₂ : HomologicalComplex (T ⥤ V) c\nf : K₁ ⟶ K₂\ni : ι\n⊢ QuasiIsoAt f i ↔ ∀ (t : T), QuasiIsoAt ((((evaluation T V).obj t).mapHomologicalComplex c).map f) ... | simp only [quasiIsoAt_iff, ShortComplex.quasiIso_iff_evaluation]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 853,
"column": 2
} | {
"line": 854,
"column": 16
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\n⊢ Mono (kernelSequence f).f",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.Mono",
"CategoryTheory.CategoryStruct.toQuive... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 853,
"column": 2
} | {
"line": 854,
"column": 16
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\n⊢ Mono (kernelSequence f).f",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.Mono",
"CategoryTheory.CategoryStruct.toQuive... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 857,
"column": 2
} | {
"line": 858,
"column": 16
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\n⊢ Epi (cokernelSequence f).g",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.ShortComplex.cokernelSequence",
"CategoryThe... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 857,
"column": 2
} | {
"line": 858,
"column": 16
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\n⊢ Epi (cokernelSequence f).g",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.ShortComplex.cokernelSequence",
"CategoryThe... | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 377,
"column": 24
} | {
"line": 377,
"column": 51
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nι : Type u_2\nc : ComplexShape ι\nK L M : HomologicalComplex C c\nφ : K ⟶ L\nψ : L ⟶ M\ni : ι\ninst✝² : K.HasHomology i\ninst✝¹ : L.HasHomology i\ninst✝ : M.HasHomology i\n⊢ ShortComplex.cyclesMap ((shortComplexFunctor C c i).ma... | ShortComplex.cyclesMap_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 832,
"column": 2
} | {
"line": 833,
"column": 34
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nι : Type u_2\nc : ComplexShape ι\nK : HomologicalComplex C c\ni j k : ι\nhi : c.prev j = i\nhk : c.next j = k\ninst✝¹ : K.HasHomology j\ninst✝ : (K.sc' i j k).HasHomology\n⊢ ShortComplex.cyclesMap ((natIsoSc' C c i j k hi hk).ho... | simp only [ShortComplex.cyclesMap_i, shortComplexFunctor_obj_X₂, shortComplexFunctor'_obj_X₂,
natIsoSc'_hom_app_τ₂, comp_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Action.Basic | {
"line": 62,
"column": 61
} | {
"line": 62,
"column": 87
} | [
{
"pp": "V : Type u_1\ninst✝¹ : Category.{v_1, u_1} V\nG : Type u_2\ninst✝ : Group G\nA : Action V G\ng : G\n⊢ A.ρ (g⁻¹ * g) = 𝟙 A.V",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"MonoidHom.instFunLike",
"inv_mul_cancel",
"HMul.hMul",
"DivInv... | rw [inv_mul_cancel, ρ_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Action.Basic | {
"line": 62,
"column": 61
} | {
"line": 62,
"column": 87
} | [
{
"pp": "V : Type u_1\ninst✝¹ : Category.{v_1, u_1} V\nG : Type u_2\ninst✝ : Group G\nA : Action V G\ng : G\n⊢ A.ρ (g⁻¹ * g) = 𝟙 A.V",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"MonoidHom.instFunLike",
"inv_mul_cancel",
"HMul.hMul",
"DivInv... | rw [inv_mul_cancel, ρ_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Action.Basic | {
"line": 62,
"column": 61
} | {
"line": 62,
"column": 87
} | [
{
"pp": "V : Type u_1\ninst✝¹ : Category.{v_1, u_1} V\nG : Type u_2\ninst✝ : Group G\nA : Action V G\ng : G\n⊢ A.ρ (g⁻¹ * g) = 𝟙 A.V",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"MonoidHom.instFunLike",
"inv_mul_cancel",
"HMul.hMul",
"DivInv... | rw [inv_mul_cancel, ρ_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.CompactlyCoherentSpace | {
"line": 89,
"column": 2
} | {
"line": 92,
"column": 47
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactlyCoherentSpace X\ns : Set X\n⊢ IsOpen s ↔ ∀ (K : Type u) [inst : TopologicalSpace K] [CompactSpace K] (f : K → X), Continuous f → IsOpen (f ⁻¹' s)",
"usedConstants": [
"Iff.mpr",
"Continuous",
"isCompact_iff_compactSpace",
... | refine ⟨fun hs _ _ _ _ hf ↦ hs.preimage hf, fun hs ↦ isOpen_iff |>.mpr ?_⟩
intro K hK
have : CompactSpace K := isCompact_iff_compactSpace.mp hK
exact hs K Subtype.val continuous_subtype_val | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.CompactlyCoherentSpace | {
"line": 89,
"column": 2
} | {
"line": 92,
"column": 47
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactlyCoherentSpace X\ns : Set X\n⊢ IsOpen s ↔ ∀ (K : Type u) [inst : TopologicalSpace K] [CompactSpace K] (f : K → X), Continuous f → IsOpen (f ⁻¹' s)",
"usedConstants": [
"Iff.mpr",
"Continuous",
"isCompact_iff_compactSpace",
... | refine ⟨fun hs _ _ _ _ hf ↦ hs.preimage hf, fun hs ↦ isOpen_iff |>.mpr ?_⟩
intro K hK
have : CompactSpace K := isCompact_iff_compactSpace.mp hK
exact hs K Subtype.val continuous_subtype_val | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.CompactOpen | {
"line": 494,
"column": 2
} | {
"line": 495,
"column": 53
} | [
{
"pp": "X : Type u_2\nY : Type u_3\nT : Type u_5\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace T\ns : Set T\nf : T → X → Y\ng : C(X, Y)\nf_cont : ContinuousOn (Function.uncurry f) (s ×ˢ univ)\n⊢ ContinuousOn (fun x ↦ mkD (f x) g) s",
"usedConstants": [
"Set.instS... | have (x) (hx : x ∈ s) : Continuous (f x) := f_cont.comp_continuous
(Continuous.prodMk_right x) fun _ ↦ ⟨hx, trivial⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.UniformSpace.Equiv | {
"line": 167,
"column": 29
} | {
"line": 167,
"column": 33
} | [
{
"pp": "α : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type u_3\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ninst✝ : UniformSpace δ\nf : α ≃ᵤ β\ng : β → α\nhg : Function.RightInverse g ⇑f\nx : β\n⊢ f.symm (f (g x)) = f.symm x",
"usedConstants": [
"Eq.mpr",
"UniformEq... | hg x | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Homotopy | {
"line": 820,
"column": 16
} | {
"line": 820,
"column": 93
} | [
{
"pp": "ι✝ : Type u_1\nV : Type u\ninst✝⁶ : Category.{v, u} V\ninst✝⁵ : Preadditive V\nc✝ : ComplexShape ι✝\nC✝ D E : HomologicalComplex V c✝\nf✝ g✝ : C✝ ⟶ D\nh✝ k : D ⟶ E\ni✝ : ι✝\nC : Type u_2\ninst✝⁴ : Category.{v_1, u_2} C\ninst✝³ : Preadditive C\nι : Type ?u.237649\nc : ComplexShape ι\ninst✝² : DecidableR... | by rw [← homologyMap_comp, h.homotopyInvHomId.homologyMap_eq, homologyMap_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1204,
"column": 9
} | {
"line": 1204,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\np : Filter ι\ns : Set β\nF : ι → α → β\nf : α → β\nt : Set α\nhF : ∀ᶠ (i : ι) in p, ∀ x ∈ t, F i x ∈ s\nhf : ∀ x ∈ t, f x ∈ s\ng : β → γ\nhg : UniformContinuousOn g s\nh : TendstoUniformlyOn F f p t... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1204,
"column": 9
} | {
"line": 1204,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\np : Filter ι\ns : Set β\nF : ι → α → β\nf : α → β\nt : Set α\nhF : ∀ᶠ (i : ι) in p, ∀ x ∈ t, F i x ∈ s\nhf : ∀ x ∈ t, f x ∈ s\ng : β → γ\nhg : UniformContinuousOn g s\nh : TendstoUniformlyOn F f p t... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1204,
"column": 9
} | {
"line": 1204,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\np : Filter ι\ns : Set β\nF : ι → α → β\nf : α → β\nt : Set α\nhF : ∀ᶠ (i : ι) in p, ∀ x ∈ t, F i x ∈ s\nhf : ∀ x ∈ t, f x ∈ s\ng : β → γ\nhg : UniformContinuousOn g s\nh : TendstoUniformlyOn F f p t... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.RankNullity | {
"line": 226,
"column": 6
} | {
"line": 226,
"column": 34
} | [
{
"pp": "R : Type u_2\nM : Type u\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : HasRankNullity.{u, u_2} R\ninst✝⁵ : StrongRankCondition R\ninst✝⁴ : IsDomain R\ninst✝³ : IsTorsionFree R M\nN : Type u_1\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nL : Submodule R M\ninst✝ : Module.Fin... | ← Submodule.finrank_eq_rank, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.ErdosKaplansky | {
"line": 134,
"column": 2
} | {
"line": 135,
"column": 33
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nh : ℵ₀ ≤ Module.rank K V\n⊢ lift.{u, v} (Module.rank K V) < Module.rank K (V →ₗ[K] K)",
"usedConstants": [
"Eq.mpr",
"Algebra.to_smulCommClass",
"Preorder.toLT",
"NonUnitalCommRing.toNonUn... | rw [rank_dual_eq_card_dual_of_aleph0_le_rank h, ← rank_dual_eq_card_dual_of_aleph0_le_rank' h]
exact lift_rank_lt_rank_dual' h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dimension.ErdosKaplansky | {
"line": 134,
"column": 2
} | {
"line": 135,
"column": 33
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nh : ℵ₀ ≤ Module.rank K V\n⊢ lift.{u, v} (Module.rank K V) < Module.rank K (V →ₗ[K] K)",
"usedConstants": [
"Eq.mpr",
"Algebra.to_smulCommClass",
"Preorder.toLT",
"NonUnitalCommRing.toNonUn... | rw [rank_dual_eq_card_dual_of_aleph0_le_rank h, ← rank_dual_eq_card_dual_of_aleph0_le_rank' h]
exact lift_rank_lt_rank_dual' h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.DivisionRing | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 47
} | [
{
"pp": "K : Type u\nV✝ V₁ V₂ V₃ : Type v\nι : Type w\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V✝\ninst✝² : Module K V✝\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K V₁\nV : Type u₀\nx✝¹ : AddCommGroup V\nx✝ : Module K V\nb : Module.Basis (Module.Free.ChooseBasisIndex K V) K V := Module.Free.chooseBasis K ... | Module.Free.rank_eq_card_chooseBasisIndex | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 280,
"column": 2
} | {
"line": 284,
"column": 49
} | [
{
"pp": "K : Type u\ninst✝¹ : DivisionRing K\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → ι → K\nhb : LinearIndependent K b\n⊢ ⇑(basisOfPiSpaceOfLinearIndependent hb) = b",
"usedConstants": [
"dite_cond_eq_true",
"Eq.mpr",
"Pi.Function.module",
"basisOfLinearIndependentOfCardEqFinrank",... | by_cases hι : Nonempty ι
· simp [hι, basisOfPiSpaceOfLinearIndependent]
· rw [basisOfPiSpaceOfLinearIndependent, dif_neg hι]
ext i
exact ((not_nonempty_iff.mp hι).false i).elim | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 280,
"column": 2
} | {
"line": 284,
"column": 49
} | [
{
"pp": "K : Type u\ninst✝¹ : DivisionRing K\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → ι → K\nhb : LinearIndependent K b\n⊢ ⇑(basisOfPiSpaceOfLinearIndependent hb) = b",
"usedConstants": [
"dite_cond_eq_true",
"Eq.mpr",
"Pi.Function.module",
"basisOfLinearIndependentOfCardEqFinrank",... | by_cases hι : Nonempty ι
· simp [hι, basisOfPiSpaceOfLinearIndependent]
· rw [basisOfPiSpaceOfLinearIndependent, dif_neg hι]
ext i
exact ((not_nonempty_iff.mp hι).false i).elim | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Products | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 47
} | [
{
"pp": "case hf\nR : Type u\ninst✝¹ : Ring R\nι : Type v\nZ : ι → ModuleCat R\ninst✝ : DecidableEq ι\ns : Cocone (Discrete.functor Z)\nf : (coproductCocone Z).pt ⟶ s.pt\nh : ∀ (j : Discrete ι), (coproductCocone Z).ι.app j ≫ f = s.ι.app j\n⊢ Hom.hom f = Hom.hom (↟(toModule R ι ↑s.1 fun i ↦ Hom.hom (s.ι.app { as... | refine DirectSum.linearMap_ext _ fun i ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Category.Grp.EpiMono | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 19
} | [
{
"pp": "case H\nA B : GrpCat\nf : A ⟶ B\nb1 b2 : ↑B\nx✝ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\n⊢ { toFun := fun x ↦ (b1 * b2) • x, invFun := fun x ↦ (b1 * b2)⁻¹ • x, left_inv := ⋯, right_inv := ⋯ } x✝ =\n ({ toFun := fun x ↦ b1 • x, invFun := fun x ... | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.ShortComplex.ShortExact | {
"line": 168,
"column": 49
} | {
"line": 170,
"column": 28
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Balanced C\nS : ShortComplex C\nhS : S.ShortExact\n⊢ IsColimit (CokernelCofork.ofπ S.g ⋯)",
"usedConstants": [
"CategoryTheory.ShortComplex.Exact.gIsCokernel",
"Ca... | by
have := hS.epi_g
exact hS.exact.gIsCokernel | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.Grp.EpiMono | {
"line": 240,
"column": 98
} | {
"line": 249,
"column": 43
} | [
{
"pp": "A B : GrpCat\nf : A ⟶ B\nx : ↑B\nhx : x ∈ (Hom.hom f).range\nb : ↑B\nhb : b ∉ (Hom.hom f).range\n⊢ (h x) (fromCoset ⟨b • ↑(Hom.hom f).range, ⋯⟩) = fromCoset ⟨(x * b) • ↑(Hom.hom f).range, ⋯⟩",
"usedConstants": [
"GrpCat.SurjectiveOfEpiAuxs.g",
"Eq.mpr",
"MulOne.toOne",
"Mono... | by
change ((τ).symm.trans (g x)).trans τ _ = _
simp only [tau, Equiv.coe_trans, Function.comp_apply]
rw [Equiv.symm_swap,
@Equiv.swap_apply_of_ne_of_ne X' _ (fromCoset ⟨f.hom.range, 1, one_leftCoset _⟩) ∞
(fromCoset ⟨b • ↑f.hom.range, b, rfl⟩) (fromCoset_ne_of_nin_range _ hb) (by simp)]
simp only [g_a... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Preadditive.LeftExact | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 84
} | [
{
"pp": "case preserves\nC : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\ninst✝⁵ : Preadditive C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\ninst✝³ : Preadditive D\nF : C ⥤ D\ninst✝² : F.PreservesZeroMorphisms\ninst✝¹ : HasBinaryBiproducts C\ninst✝ : ∀ {X Y : C} (f : X ⟶ Y), PreservesColimit (parallelPair f 0) F\nX Y... | let c' := isColimitCokernelCoforkOfCofork (i.ofIsoColimit (Cofork.isoCoforkOfπ c)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 43
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np : Submodule R M\nx : M\nhx : ¬p.mkQ x = 0\nhp' : Projective R (M ⧸ p)\n⊢ ∃ f, f (p.mkQ x) ≠ 0",
"usedConstants": [
"Submodule",
"Submodule.Quotient.addCommGroup",
"AddCommGroup.toAddCommMon... | exact Projective.exists_dual_ne_zero R hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Abelian.Exact | {
"line": 104,
"column": 2
} | {
"line": 113,
"column": 38
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Abelian C\nS : ShortComplex C\ncg : KernelFork S.g\nhg : IsLimit cg\ncf : CokernelCofork S.f\nhf : IsColimit cf\n⊢ S.Exact ↔ Fork.ι cg ≫ Cofork.π cf = 0",
"usedConstants": [
"CategoryTheory.Preadditive.IsIso.comp_left_eq_zero",
"Categor... | rw [exact_iff_kernel_ι_comp_cokernel_π_zero]
let e₁ := IsLimit.conePointUniqueUpToIso (kernelIsKernel S.g) hg
let e₂ := IsColimit.coconePointUniqueUpToIso (cokernelIsCokernel S.f) hf
have : cg.ι ≫ cf.π = e₁.inv ≫ kernel.ι S.g ≫ cokernel.π S.f ≫ e₂.hom := by
have eq₁ := IsLimit.conePointUniqueUpToIso_inv_comp ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.Exact | {
"line": 104,
"column": 2
} | {
"line": 113,
"column": 38
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Abelian C\nS : ShortComplex C\ncg : KernelFork S.g\nhg : IsLimit cg\ncf : CokernelCofork S.f\nhf : IsColimit cf\n⊢ S.Exact ↔ Fork.ι cg ≫ Cofork.π cf = 0",
"usedConstants": [
"CategoryTheory.Preadditive.IsIso.comp_left_eq_zero",
"Categor... | rw [exact_iff_kernel_ι_comp_cokernel_π_zero]
let e₁ := IsLimit.conePointUniqueUpToIso (kernelIsKernel S.g) hg
let e₂ := IsColimit.coconePointUniqueUpToIso (cokernelIsCokernel S.f) hf
have : cg.ι ≫ cf.π = e₁.inv ≫ kernel.ι S.g ≫ cokernel.π S.f ≫ e₂.hom := by
have eq₁ := IsLimit.conePointUniqueUpToIso_inv_comp ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Functor.KanExtension.Adjunction | {
"line": 397,
"column": 6
} | {
"line": 398,
"column": 60
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} D\nL✝ : C ⥤ D\nH : Type u_3\ninst✝⁴ : Category.{v_3, u_3} H\ninst✝³ : ∀ (F : C ⥤ H), L✝.HasRightKanExtension F\nL : C ⥤ D\ninst✝² : ∀ (G : C ⥤ H), L.HasRightKanExtension G\ninst✝¹ : HasLimitsOfShape C H\ninst✝ : Ha... | rw [assoc, assoc, limMap_π, limitIsoOfIsRightKanExtension_hom_π_assoc,
limitIsoOfIsRightKanExtension_hom_π, limMap_π_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Abelian.Subobject | {
"line": 50,
"column": 33
} | {
"line": 50,
"column": 91
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nA : Cᵒᵖ\nf : A ⟶ op X\nhf : Mono f\n⊢ ({ hom := (epiDesc f.unop (cokernel.π (kernel.ι f.unop)) ⋯).op, inv := (cokernel.desc (kernel.ι f.unop) f.unop ⋯).op,\n hom_inv_id := ⋯, inv_hom_id := ⋯ }.hom ≫\n f).unop =\n (co... | by simp only [unop_comp, Quiver.Hom.unop_op, comp_epiDesc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 206,
"column": 2
} | {
"line": 213,
"column": 44
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : ObjectProperty C\ninst✝ : Balanced C\nhP : P.IsSeparating\n⊢ P.IsDetecting",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"CategoryTheory.Category.assoc",
"CategoryTheory.IsIso",
"CategoryTheory.Epi",
"CategoryTheory.Mono... | intro X Y f hf
refine
(isIso_iff_mono_and_epi _).2 ⟨⟨fun g h hgh => hP _ _ fun G hG i => ?_⟩, ⟨fun g h hgh => ?_⟩⟩
· obtain ⟨t, -, ht⟩ := hf G hG (i ≫ g ≫ f)
rw [ht (i ≫ g) (Category.assoc _ _ _), ht (i ≫ h) (hgh.symm ▸ Category.assoc _ _ _)]
· refine hP _ _ fun G hG i => ?_
obtain ⟨t, rfl, -⟩ := hf G... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 206,
"column": 2
} | {
"line": 213,
"column": 44
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : ObjectProperty C\ninst✝ : Balanced C\nhP : P.IsSeparating\n⊢ P.IsDetecting",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"CategoryTheory.Category.assoc",
"CategoryTheory.IsIso",
"CategoryTheory.Epi",
"CategoryTheory.Mono... | intro X Y f hf
refine
(isIso_iff_mono_and_epi _).2 ⟨⟨fun g h hgh => hP _ _ fun G hG i => ?_⟩, ⟨fun g h hgh => ?_⟩⟩
· obtain ⟨t, -, ht⟩ := hf G hG (i ≫ g ≫ f)
rw [ht (i ≫ g) (Category.assoc _ _ _), ht (i ≫ h) (hgh.symm ▸ Category.assoc _ _ _)]
· refine hP _ _ fun G hG i => ?_
obtain ⟨t, rfl, -⟩ := hf G... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Subobject.Comma | {
"line": 95,
"column": 49
} | {
"line": 95,
"column": 88
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nS : D\nT : C ⥤ D\ninst✝¹ : HasFiniteLimits C\ninst✝ : PreservesFiniteLimits T\nA P : StructuredArrow S T\nf : P ⟶ A\nhf : Mono f\nq : (Functor.fromPUnit S).obj A.left ⟶ T.obj (Subobject.underlying.obj (projectSubobjec... | by dsimp; simpa [← T.map_comp] using hq | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 660,
"column": 2
} | {
"line": 660,
"column": 80
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasCoproduct fun x ↦ G\n⊢ (∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ (h : G ⟶ X), h ≫ f = h ≫ g) → f = g) ↔ ∀ (A : C), Epi (Sigma.desc fun f ↦ f)",
"usedConstants": [
"CategoryTheory.Epi",
"CategoryTheory.CategoryStruct.toQuiver",... | refine ⟨fun h A => ⟨fun u v huv => h _ _ fun i => ?_⟩, fun h X Y f g hh => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 670,
"column": 2
} | {
"line": 670,
"column": 80
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasProduct fun x ↦ G\n⊢ (∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ (h : Y ⟶ G), f ≫ h = g ≫ h) → f = g) ↔ ∀ (A : C), Mono (Pi.lift fun f ↦ f)",
"usedConstants": [
"CategoryTheory.Mono",
"CategoryTheory.CategoryStruct.toQuiver",
... | refine ⟨fun h A => ⟨fun u v huv => h _ _ fun i => ?_⟩, fun h X Y f g hh => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Subobject.Comma | {
"line": 177,
"column": 19
} | {
"line": 177,
"column": 57
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nS : C ⥤ D\nT : D\nA : CostructuredArrow S T\nP : (CostructuredArrow S T)ᵒᵖ\nf : P ⟶ op A\ninst✝ : Mono f.unop.left.op\n⊢ ((Subobject.underlyingIso f.unop.left.op).hom ≫ f.unop.left.op).unop = (Subobject.mk f.unop.left... | Subobject.underlyingIso_hom_comp_eq_mk | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | {
"line": 307,
"column": 16
} | {
"line": 307,
"column": 53
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\ninst✝ : Nonempty J\nt : Trident f\nlift : (s : Trident f) → s.pt ⟶ t.pt\nfac : ∀ (s : Trident f), lift s ≫ t.ι = s.ι\nuniq : ∀ (s : Trident f) (m : s.pt ⟶ t.pt), (∀ (j : WalkingParallelFamily J), m ≫ t.π.app j = s.π.app j) → ... | ← t.w (line (Classical.arbitrary J)), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Subobject.Comma | {
"line": 228,
"column": 6
} | {
"line": 228,
"column": 42
} | [
{
"pp": "case h.refine_2.a.h\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nS : C ⥤ D\nT : D\ninst✝¹ : HasFiniteColimits C\ninst✝ : PreservesFiniteColimits S\nA : CostructuredArrow S T\nP Q : (CostructuredArrow S T)ᵒᵖ\nf : P ⟶ op A\ng : Q ⟶ op A\nhf : Mono f\nhg : Mono g\... | exact unop_left_comp_ofMkLEMk_unop _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 743,
"column": 48
} | {
"line": 743,
"column": 58
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : LocallySmall.{w, v₁, u₁} C\nF : C ⥤ Type w\nX : C\nG : F.Elementsᵒᵖ ⥤ Type w := (CategoryOfElements.π F).op ⋙ shrinkYoneda.{w, v₁, u₁}.obj X\nc : G.CoconeTypes := G.coconeTypesEquiv.symm (coconeπOpCompShrinkYonedaObj F X)\nthis :\n ∀ (u : G.ColimitTyp... | rw [← hx₁] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 815,
"column": 4
} | {
"line": 815,
"column": 20
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : LocallySmall.{w, v₁, u₁} C\nF : C ⥤ Type w\nX : C\n⊢ ∀ (Y : (shrinkYoneda.{w, v₁, u₁}.flip.obj (op X)).Elements)\n (m : (shrinkYoneda.{w, v₁, u₁}.flip.obj (op X)).elementsMk X (shrinkYonedaObjObjEquiv.symm (𝟙 X)) ⟶ Y),\n m = (fun u ↦ ⟨shrinkYone... | rintro u ⟨m, hm⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Ring.Periodic | {
"line": 318,
"column": 52
} | {
"line": 318,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nc : α\ninst✝¹ : NonAssocRing α\ninst✝ : SubtractionMonoid β\nh : Antiperiodic f c\nhi : f 0 = 0\nn : ℕ\n⊢ f (-(↑(n + 1) * c)) = 0",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
... | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Injective | {
"line": 282,
"column": 76
} | {
"line": 289,
"column": 86
} | [
{
"pp": "R : Type u\ninst✝⁷ : Ring R\nQ : Type v\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ⇑i)\nh : Baer R Q\ny : N\nr : R\n... | by
have : r ∈ ideal i f y := by
change (r • y) ∈ (extensionOfMax i f).toLinearPMap.domain
rw [eq1]
apply Submodule.zero_mem _
rw [ExtensionOfMaxAdjoin.extendIdealTo_is_extension i f h y r this]
dsimp [ExtensionOfMaxAdjoin.idealTo]
simp only [eq1, ← ZeroMemClass.zero_def, (extensionOfMax i f).toLinea... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Totient | {
"line": 205,
"column": 6
} | {
"line": 206,
"column": 66
} | [
{
"pp": "p : ℕ\nhp : Prime p\nn : ℕ\nh1 : Function.Injective fun x ↦ x * p\nh2 : image (fun x ↦ x * p) (range (p ^ n)) ⊆ range (p ^ (n + 1))\n⊢ #(range (p ^ (n + 1)) \\ image (fun x ↦ x * p) (range (p ^ n))) = p ^ n * (p - 1)",
"usedConstants": [
"Nat.pow_succ'",
"instPowNat",
"Eq.mpr",
... | rw [card_sdiff_of_subset h2, Finset.card_image_of_injective _ h1, card_range, card_range, ←
one_mul (p ^ n), pow_succ', ← tsub_mul, one_mul, mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 823,
"column": 41
} | {
"line": 823,
"column": 82
} | [
{
"pp": "X : Type u_3\nY : Type u_5\nα : Type u_6\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\nuα : UniformSpace α\nA : Set Y\nu : Y → X → α\nS : Set X\nhA : EquicontinuousOn (u ∘ Subtype.val) S\nhu : Continuous (S.restrict ∘ u)\nx : X\nhx : x ∈ S\n⊢ Continuous (eval x ∘ u)",
"usedConstants": [
... | exact continuous_apply ⟨x, hx⟩ |>.comp hu | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 823,
"column": 41
} | {
"line": 823,
"column": 82
} | [
{
"pp": "X : Type u_3\nY : Type u_5\nα : Type u_6\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\nuα : UniformSpace α\nA : Set Y\nu : Y → X → α\nS : Set X\nhA : EquicontinuousOn (u ∘ Subtype.val) S\nhu : Continuous (S.restrict ∘ u)\nx : X\nhx : x ∈ S\n⊢ Continuous (eval x ∘ u)",
"usedConstants": [
... | exact continuous_apply ⟨x, hx⟩ |>.comp hu | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 823,
"column": 41
} | {
"line": 823,
"column": 82
} | [
{
"pp": "X : Type u_3\nY : Type u_5\nα : Type u_6\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\nuα : UniformSpace α\nA : Set Y\nu : Y → X → α\nS : Set X\nhA : EquicontinuousOn (u ∘ Subtype.val) S\nhu : Continuous (S.restrict ∘ u)\nx : X\nhx : x ∈ S\n⊢ Continuous (eval x ∘ u)",
"usedConstants": [
... | exact continuous_apply ⟨x, hx⟩ |>.comp hu | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.ToIntervalMod | {
"line": 689,
"column": 15
} | {
"line": 689,
"column": 52
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\nhα : Archimedean α\np : α\nhp : 0 < p\na b : α\n⊢ toIcoDiv hp a b = toIocDiv hp a b ∨ toIocDiv hp a b + 1 = toIcoDiv hp a b",
"usedConstants": [
"Eq.mpr",
"AddCommGroup.ModEq",
"congrArg",... | ← not_modEq_iff_toIcoDiv_eq_toIocDiv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 141,
"column": 7
} | {
"line": 141,
"column": 44
} | [
{
"pp": "case h.e'_2.h.h.e'_3\n𝕜 : Type u_1\ninst✝⁵ : AddCommGroup 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsOrderedAddMonoid 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\nhx : ¬x ≡ a [PMOD p]\nx✝ : 𝕜\n⊢ toIcoDiv hp a x = toIocDiv hp a x",
... | ← not_modEq_iff_toIcoDiv_eq_toIocDiv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 160,
"column": 16
} | {
"line": 160,
"column": 53
} | [
{
"pp": "case h.e'_2.h.h.e'_3\n𝕜 : Type u_1\ninst✝⁵ : AddCommGroup 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsOrderedAddMonoid 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\nhx : ¬x ≡ a [PMOD p]\nx✝ : 𝕜\n⊢ toIcoDiv hp a x = toIocDiv hp a x",
... | ← not_modEq_iff_toIcoDiv_eq_toIocDiv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 84
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : AddCommGroup 𝕜\np : 𝕜\nn : ℕ\nh0 : n ≠ 0\nhn : IsSMulRegular 𝕜 n\ny : 𝕜\nhx✝ : ↑y ∈ {x | n • x = 0}\nhx : n • y ∈ zmultiples p\nm' : ℤ\nhm : (fun x ↦ x • p) m' = n • y\nthis : NeZero n\n⊢ ∃ k y_1, ↑y_1 = ↑⟨↑y, hx✝⟩ ∧ n • y_1 = ↑k • p",
"usedConstants": [
"instHSMul"... | rw [← (Int.divModEquiv n).symm_apply_apply m', Int.divModEquiv_symm_apply] at hm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monoidal.Grp_ | {
"line": 304,
"column": 82
} | {
"line": 307,
"column": 45
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : CartesianMonoidalCategory C\nG✝ X✝ : C\ninst✝² : GrpObj G✝\nA B G : C\ninst✝¹ : CartesianMonoidalCategory C\ninst✝ : MonObj G\nh : (X : C) → (f : X ⟶ G) → Invertible f\nX Y : Cᵒᵖ\nf : X ⟶ Y\n⊢ (yoneda.obj G).map f ≫ (fun X f ↦ ⅟f) Y = (fun X f ↦ ⅟f) X... | by
ext g
simp_rw [types_comp_apply, yoneda_obj_map, invOf_eq_iff_left]
rw [← comp_mul, invOf_mul_self, comp_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.Refinements | {
"line": 224,
"column": 4
} | {
"line": 224,
"column": 88
} | [
{
"pp": "case mp\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : ∀ ⦃A : C⦄ (y : A ⟶ S₂.homology), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ homologyMap φ\nA : C\ny₂ : A ⟶ S₂.X₂\nhy₂ : y₂ ≫ S₂.g = 0\nA₁ : C\nπ₁ : A₁ ⟶ A\nhπ₁ : Epi π₁\nγ : A₁ ⟶ S₁.homology\... | obtain ⟨A₂, π₂, hπ₂, x₂, hx₂, fac⟩ := S₁.eq_liftCycles_homologyπ_up_to_refinements γ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Homology.ExactSequence | {
"line": 257,
"column": 2
} | {
"line": 273,
"column": 24
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nn : ℕ\nS : ComposableArrows C (n + 2)\n⊢ S.Exact ↔ (mk₂ (S.map' 0 1 isComplex₂_iff._proof_2 ⋯) (S.map' 1 2 isComplex₂_iff._proof_4 ⋯)).Exact ∧ S.δ₀.Exact",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Algebra.Ho... | constructor
· intro h
constructor
· rw [exact₂_iff]; swap
· rw [isComplex₂_iff]
exact h.toIsComplex.zero 0
exact h.exact 0 (by lia)
· exact Exact.mk (IsComplex.mk (fun i hi => h.toIsComplex.zero (i + 1)))
(fun i hi => h.exact (i + 1))
· rintro ⟨h, h₀⟩
refine Exact.mk (IsC... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ExactSequence | {
"line": 257,
"column": 2
} | {
"line": 273,
"column": 24
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nn : ℕ\nS : ComposableArrows C (n + 2)\n⊢ S.Exact ↔ (mk₂ (S.map' 0 1 isComplex₂_iff._proof_2 ⋯) (S.map' 1 2 isComplex₂_iff._proof_4 ⋯)).Exact ∧ S.δ₀.Exact",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Algebra.Ho... | constructor
· intro h
constructor
· rw [exact₂_iff]; swap
· rw [isComplex₂_iff]
exact h.toIsComplex.zero 0
exact h.exact 0 (by lia)
· exact Exact.mk (IsComplex.mk (fun i hi => h.toIsComplex.zero (i + 1)))
(fun i hi => h.exact (i + 1))
· rintro ⟨h, h₀⟩
refine Exact.mk (IsC... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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