mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h : S₁.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
⊢ φ.τ₁ ≫ S₂.f = h.f' ≫ h.i ≫ φ.τ₂

Tactic:
f'_i_assoc,

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h : S₁.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
⊢ φ.τ₁ ≫ S₂.f = S₁.f ≫ φ.τ₂
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : S₂.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
wi : i ≫ S₁.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₁.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₁.f ⋯)
hf' : f' ≫ i = S₁.f
⊢ S₁.f = φ.τ₁ ≫ h.f' ≫ h.i ≫ inv φ.τ₂

Tactic:
f'_i_assoc,

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : S₂.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
wi : i ≫ S₁.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₁.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₁.f ⋯)
hf' : f' ≫ i = S₁.f
⊢ S₁.f = φ.τ₁ ≫ S₂.f ≫ inv φ.τ₂
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S S₁ S₂ S₃ : ShortComplex C
φ : S₁ ⟶ S₂
h : S₂.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
i : h.K ⟶ S₁.X₂ := h.i ≫ inv φ.τ₂
wi : i ≫ S₁.g = 0
hi : IsLimit (KernelFork.ofι i wi)
f' : (KernelFork.ofι S₁.f ⋯).pt ⟶ (KernelFork.ofι i wi).pt := hi.lift (KernelFork.ofι S₁.f ⋯)
hf' : f' ≫ i = S₁.f
hf'' : f' = φ.τ₁ ≫ h.f'
wπ : f' ≫ h.π = 0
Z'✝ : C
x : h.K ⟶ Z'✝
hx : f' ≫ x = 0
⊢ h.f' ≫ x = 0

Tactic:
rw [← cancel_epi φ.τ₁, ← reassoc_of% hf'', hx, comp_zero]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Homology

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
⊢ (homologyMap' φ S₁.homologyData S₂.homologyData).op =
    (S₂.op.leftHomologyIso.symm ≪≫ S₂.leftHomologyOpIso ≪≫ S₂.rightHomologyIso.symm.op).inv ≫
      homologyMap' (opMap φ) S₂.op.homologyData S₁.op.homologyData ≫
        (S₁.op.leftHomologyIso.symm ≪≫ S₁.leftHomologyOpIso ≪≫ S₁.rightHomologyIso.symm.op).hom

Tactic:
rw [homologyMap'_op]

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
⊢ S₂.homologyData.iso.inv.op ≫
      homologyMap' (opMap φ) S₂.homologyData.op S₁.homologyData.op ≫ S₁.homologyData.iso.hom.op =
    (S₂.op.leftHomologyIso.symm ≪≫ S₂.leftHomologyOpIso ≪≫ S₂.rightHomologyIso.symm.op).inv ≫
      homologyMap' (opMap φ) S₂.op.homologyData S₁.op.homologyData ≫
        (S₁.op.leftHomologyIso.symm ≪≫ S₁.leftHomologyOpIso ≪≫ S₁.rightHomologyIso.symm.op).hom
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h : S₂.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
⊢ S₁.f = φ.τ₁ ≫ h.f' ≫ h.i ≫ inv φ.τ₂

Tactic:
f'_i_assoc,

Goal state after:
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h : S₂.LeftHomologyData
inst✝² : Epi φ.τ₁
inst✝¹ : IsIso φ.τ₂
inst✝ : Mono φ.τ₃
⊢ S₁.f = φ.τ₁ ≫ S₂.f ≫ inv φ.τ₂
Module: Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
h : S.RightHomologyData
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : h.IsPreservedBy F
this✝ : PreservesColimit (parallelPair S.f 0) F
this : PreservesLimit (parallelPair h.g' 0) F
wp : F.map S.f ≫ F.map h.p = 0
hp : IsColimit ((CokernelCofork.ofπ h.p ⋯).map F)
g' : F.obj h.Q ⟶ F.obj S.X₃ := hp.desc (CokernelCofork.ofπ (S.map F).g ⋯)
hg' : g' = F.map h.g'
⊢ (S.map F).RightHomologyData

Tactic:
have wι : F.map h.ι ≫ g' = 0 := by rw [hg', ← F.map_comp, ι_g', F.map_zero]

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
inst✝³ : HasZeroMorphisms C
inst✝² : HasZeroMorphisms D
S S₁ S₂ : ShortComplex C
h : S.RightHomologyData
F : C ⥤ D
inst✝¹ : F.PreservesZeroMorphisms
inst✝ : h.IsPreservedBy F
this✝ : PreservesColimit (parallelPair S.f 0) F
this : PreservesLimit (parallelPair h.g' 0) F
wp : F.map S.f ≫ F.map h.p = 0
hp : IsColimit ((CokernelCofork.ofπ h.p ⋯).map F)
g' : F.obj h.Q ⟶ F.obj S.X₃ := hp.desc (CokernelCofork.ofπ (S.map F).g ⋯)
hg' : g' = F.map h.g'
wι : F.map h.ι ≫ g' = 0
⊢ (S.map F).RightHomologyData
Module: Mathlib.Algebra.Homology.ShortComplex.Preadditive

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
S₁ S₂ : ShortComplex C
φ φ' : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
γ' : RightHomologyMapData φ' h₁ h₂
⊢ rightHomologyMap' (φ + φ') h₁ h₂ = rightHomologyMap' φ h₁ h₂ + rightHomologyMap' φ' h₁ h₂

Tactic:
simp only [γ.rightHomologyMap'_eq, γ'.rightHomologyMap'_eq,
    (γ.add γ').rightHomologyMap'_eq, RightHomologyMapData.add_φH]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Ab

Goal state before:
pp: S : ShortComplex Ab
⊢ ∀ (a b : ↑S.X₁),
    ⟨(ConcreteCategory.hom S.f) (a + b), ⋯⟩ = ⟨(ConcreteCategory.hom S.f) a, ⋯⟩ + ⟨(ConcreteCategory.hom S.f) b, ⋯⟩

Tactic:
by aesop

Goal state after:
Module: Mathlib.CategoryTheory.Preadditive.LeftExact

Goal state before:
pp: C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
inst✝³ : Preadditive D
F : C ⥤ D
inst✝² : F.PreservesZeroMorphisms
inst✝¹ : HasBinaryBiproducts C
inst✝ : ∀ {X Y : C} (f : X ⟶ Y), PreservesLimit (parallelPair f 0) F
X Y : C
f g : X ⟶ Y
this✝ : PreservesBinaryBiproducts F := preservesBinaryBiproducts_of_preservesBinaryProducts F
this : F.Additive
c : Cone (parallelPair f g)
i : IsLimit c
c' : IsLimit (KernelFork.ofι (Fork.ι c) ⋯) := isLimitKernelForkOfFork (i.ofIsoLimit (Fork.isoForkOfι c))
iFc : IsLimit (KernelFork.ofι (F.map (Fork.ι c)) ⋯) := isLimitForkMapOfIsLimit' F ⋯ c'
p : parallelPair (F.map (f - g)) 0 ≅ parallelPair (F.map f - F.map g) 0 := parallelPair.eqOfHomEq ⋯ ⋯
⊢ IsLimit (Fork.ofι (F.map (Fork.ι c)) ⋯)

Tactic:
exact
    IsLimit.ofIsoLimit
      (isLimitForkOfKernelFork ((IsLimit.postcomposeHomEquiv p _).symm iFc))
      (Fork.ext (Iso.refl _) (by simp [p]))

Goal state after:
Module: Mathlib.CategoryTheory.Adjunction.Reflective

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝² : Category.{v₁, u₁} C
inst✝¹ : Category.{v₂, u₂} D
i : D ⥤ C
inst✝ : Reflective i
X : C
⊢ (reflectorAdjunction i).unit.app (i.obj ((reflector i).obj X)) =
    i.map ((reflector i).map ((reflectorAdjunction i).unit.app X))

Tactic:
rw [← cancel_mono (i.map ((reflectorAdjunction i).counit.app ((reflector i).obj X))),
    ← i.map_comp]
  simp

Goal state after:
Module: Mathlib.CategoryTheory.Subobject.Basic

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
X : C
p : Subobject X → Prop
h : ∀ ⦃A : C⦄ (f : A ⟶ X) [inst : Mono f], p (mk f)
P : Subobject X
⊢ p P

Tactic:
induction P using Quotient.inductionOn' with | _ a
  exact h a.arrow

Goal state after:
Module: Mathlib.CategoryTheory.Subobject.MonoOver

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X Y Z : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
e : C ≌ D
f : MonoOver (e.functor.obj X)
⊢ Mono ((Over.post e.inverse).obj ((forget (e.functor.obj X)).obj f)).hom

Tactic:
by
        dsimp
        infer_instance

Goal state after:
Module: Mathlib.CategoryTheory.Subobject.Limits

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
X✝ Y Z : C
inst✝² : HasZeroMorphisms C
f : X✝ ⟶ Y
inst✝¹ : HasKernel f
inst✝ : HasCokernels C
X : C
⊢ ∀ ⦃A B : C⦄ (f : A ⟶ X) (g : B ⟶ X) [inst : Mono f] [inst_1 : Mono g],
    Subobject.mk f ≤ Subobject.mk g →
      Subobject.lift (fun x f x_1 ↦ Subobject.mk (cokernel.π f).op) ⋯ (Subobject.mk f) ≤
        Subobject.lift (fun x f x_1 ↦ Subobject.mk (cokernel.π f).op) ⋯ (Subobject.mk g)

Tactic:
intro A B f g hf hg h
      dsimp only [Subobject.lift_mk]
      refine Subobject.mk_le_mk_of_comm (cokernel.desc f (cokernel.π g) ?_).op ?_
      · rw [← Subobject.ofMkLEMk_comp h, Category.assoc, cokernel.condition, comp_zero]
      · exact Quiver.Hom.unop_inj (cokernel.π_desc _ _ _)

Goal state after:
Module: Mathlib.CategoryTheory.Subobject.Lattice

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : HasPullbacks C
A : C
f g : Subobject A
⊢ IsPullback ((f ⊓ g).ofLE f ⋯) ((f ⊓ g).ofLE g ⋯) f.arrow g.arrow

Tactic:
refine ⟨⟨by simp⟩, ⟨PullbackCone.IsLimit.mk _ (fun s ↦ (f ⊓ g).factorThru (s.fst ≫ f.arrow) ?_)
    ?_ (fun s ↦ ?_) fun _ _ h _ ↦ ?_⟩⟩

Goal state after:
case refine_1
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : HasPullbacks C
A : C
f g : Subobject A
s : PullbackCone f.arrow g.arrow
⊢ (f ⊓ g).Factors (s.fst ≫ f.arrow)

case refine_2
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : HasPullbacks C
A : C
f g : Subobject A
⊢ ∀ (s : PullbackCone f.arrow g.arrow), (f ⊓ g).factorThru (s.fst ≫ f.arrow) ⋯ ≫ (f ⊓ g).ofLE f ⋯ = s.fst

case refine_3
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : HasPullbacks C
A : C
f g : Subobject A
s : PullbackCone f.arrow g.arrow
⊢ (f ⊓ g).factorThru (s.fst ≫ f.arrow) ⋯ ≫ (f ⊓ g).ofLE g ⋯ = s.snd

case refine_4
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : HasPullbacks C
A : C
f g : Subobject A
x✝² : PullbackCone f.arrow g.arrow
x✝¹ : x✝².pt ⟶ underlying.obj (f ⊓ g)
h : x✝¹ ≫ (f ⊓ g).ofLE f ⋯ = x✝².fst
x✝ : x✝¹ ≫ (f ⊓ g).ofLE g ⋯ = x✝².snd
⊢ x✝¹ = (f ⊓ g).factorThru (x✝².fst ≫ f.arrow) ⋯
Module: Mathlib.CategoryTheory.Abelian.Exact

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
Z : C
tfae_2_to_1 : cokernel.π f = 0 → Epi f
x✝ : Epi f
⊢ cokernel.π f = 0

Tactic:
← cancel_epi f,

Goal state after:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Abelian C
X Y : C
f : X ⟶ Y
Z : C
tfae_2_to_1 : cokernel.π f = 0 → Epi f
x✝ : Epi f
⊢ f ≫ cokernel.π f = f ≫ 0
Module: Mathlib.CategoryTheory.Subobject.Lattice

Goal state before:
pp: case mp
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X✝ Y✝ Z✝ : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X : C
Y : Subobject X
W Z : Subobject (underlying.obj Y)
⊢ ⟨mk (W.arrow ≫ Y.arrow), ⋯⟩ ≤ ⟨mk (Z.arrow ≫ Y.arrow), ⋯⟩ → W ≤ Z

pp: case mpr
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X✝ Y✝ Z✝ : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X : C
Y : Subobject X
W Z : Subobject (underlying.obj Y)
⊢ W ≤ Z → ⟨mk (W.arrow ≫ Y.arrow), ⋯⟩ ≤ ⟨mk (Z.arrow ≫ Y.arrow), ⋯⟩

Tactic:
· intro h
      exact le_of_comm (((underlyingIso _).inv ≫ ofLE _ _ (Subtype.mk_le_mk.mp h) ≫
        (underlyingIso _).hom)) (by cat_disch)

Goal state after:
case mpr
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
X✝ Y✝ Z✝ : C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
X : C
Y : Subobject X
W Z : Subobject (underlying.obj Y)
⊢ W ≤ Z → ⟨mk (W.arrow ≫ Y.arrow), ⋯⟩ ≤ ⟨mk (Z.arrow ≫ Y.arrow), ⋯⟩
Module: Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products

Goal state before:
pp: C : Type u₁
inst✝ : Category.{v₁, u₁} C
α : Type u_1
Z : α → C
c c' : Cofan Z
f : Fan fun x ↦ op (Z x)
hc : IsColimit c
hc' : IsColimit c'
hf : IsLimit f
⊢ ((opCoproductIsoProduct' hc hf).hom ≫ (opCoproductIsoProduct' hc' hf).inv).unop =
    (hc.coconePointUniqueUpToIso hc').op.inv.unop

Tactic:
apply hc'.hom_ext

Goal state after:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
α : Type u_1
Z : α → C
c c' : Cofan Z
f : Fan fun x ↦ op (Z x)
hc : IsColimit c
hc' : IsColimit c'
hf : IsLimit f
⊢ ∀ (j : Discrete α),
    c'.ι.app j ≫ ((opCoproductIsoProduct' hc hf).hom ≫ (opCoproductIsoProduct' hc' hf).inv).unop =
      c'.ι.app j ≫ (hc.coconePointUniqueUpToIso hc').op.inv.unop
Module: Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products

Goal state before:
pp: case e'_2.e'_6.e'_5
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
α : Type u_1
Z : α → C
inst✝ : HasCoproduct Z
X : C
π : (a : α) → Z a ⟶ X
e_1✝ : (op X ⟶ ∏ᶜ fun x ↦ op (Z x)) = (op (Cofan.mk X π).pt ⟶ (Fan.mk (∏ᶜ fun x ↦ op (Z x)) (Pi.π fun x ↦ op (Z x))).pt)
e_3✝¹ : op X = op (Cofan.mk X π).pt
e_4✝¹ : op (∐ Z) = op (Cofan.mk (∐ Z) (Sigma.ι Z)).pt
e_3✝ : ∐ Z = (Cofan.mk (∐ Z) (Sigma.ι Z)).pt
e_4✝ : X = (Cofan.mk X π).pt
⊢ Sigma.desc π = (coproductIsCoproduct Z).desc (Cofan.mk X π)

Tactic:
simp [Sigma.desc, coproductIsCoproduct]

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products

Goal state before:
pp: case h₁
C : Type u₁
inst✝² : Category.{v₁, u₁} C
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
X : Type v₂
A B : C
inst✝ : HasBinaryProduct A B
⊢ coprod.inl ≫ coprod.desc prod.fst.op prod.snd.op ≫ (prod.lift coprod.inl.unop coprod.inr.unop).op =
    coprod.inl ≫ 𝟙 (op A ⨿ op B)

Tactic:
simp only [colimit.ι_desc_assoc]

Goal state after:
case h₁
C : Type u₁
inst✝² : Category.{v₁, u₁} C
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
X : Type v₂
A B : C
inst✝ : HasBinaryProduct A B
⊢ (BinaryCofan.mk prod.fst.op prod.snd.op).ι.app { as := WalkingPair.left } ≫
      (prod.lift coprod.inl.unop coprod.inr.unop).op =
    coprod.inl ≫ 𝟙 (op A ⨿ op B)
Module: Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products

Goal state before:
pp: case h₂
C : Type u₁
inst✝² : Category.{v₁, u₁} C
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
X : Type v₂
A B : C
inst✝ : HasBinaryProduct A B
⊢ coprod.inr ≫ coprod.desc prod.fst.op prod.snd.op ≫ (prod.lift coprod.inl.unop coprod.inr.unop).op =
    coprod.inr ≫ 𝟙 (op A ⨿ op B)

Tactic:
simp only [colimit.ι_desc_assoc]

Goal state after:
case h₂
C : Type u₁
inst✝² : Category.{v₁, u₁} C
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
X : Type v₂
A B : C
inst✝ : HasBinaryProduct A B
⊢ (BinaryCofan.mk prod.fst.op prod.snd.op).ι.app { as := WalkingPair.right } ≫
      (prod.lift coprod.inl.unop coprod.inr.unop).op =
    coprod.inr ≫ 𝟙 (op A ⨿ op B)
Module: Mathlib.CategoryTheory.Functor.KanExtension.Adjunction

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
L : C ⥤ D
H : Type u_3
inst✝¹ : Category.{v_3, u_3} H
inst✝ : ∀ (F : C ⥤ H), L.HasRightKanExtension F
F : D ⥤ H
G₁ G₂ : C ⥤ H
β : ((whiskeringLeft C D H).obj L).obj F ⟶ G₁
f : G₁ ⟶ G₂
X : C
h : (L.ran.map f).app (L.obj X) ≫ (L.ranCounit.app G₂).app X = (L.ranCounit.app G₁).app X ≫ f.app X
⊢ β.app X ≫ f.app X =
    ((L.ran.obj G₁).liftOfIsRightKanExtension (L.ranCounit.app G₁) F β).app (L.obj X) ≫
      (L.ranCounit.app G₁).app X ≫ f.app X

Tactic:
liftOfIsRightKanExtension_fac_app_assoc

Goal state after:
C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : Category.{v_2, u_2} D
L : C ⥤ D
H : Type u_3
inst✝¹ : Category.{v_3, u_3} H
inst✝ : ∀ (F : C ⥤ H), L.HasRightKanExtension F
F : D ⥤ H
G₁ G₂ : C ⥤ H
β : ((whiskeringLeft C D H).obj L).obj F ⟶ G₁
f : G₁ ⟶ G₂
X : C
h : (L.ran.map f).app (L.obj X) ≫ (L.ranCounit.app G₂).app X = (L.ranCounit.app G₁).app X ≫ f.app X
⊢ β.app X ≫ f.app X = β.app X ≫ f.app X
Module: Mathlib.CategoryTheory.Limits.Shapes.Countable

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
J : Type u_2
inst✝² : Countable J
inst✝¹ : Preorder J
inst✝ : IsCofiltered J
d : J
⊢ IsConnected (CostructuredArrow (sequentialFunctor J) d)

Tactic:
obtain ⟨n, (g : (sequentialFunctor J).obj ⟨n⟩ ≤ d)⟩ := sequentialFunctor_initial_aux J d

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
J : Type u_2
inst✝² : Countable J
inst✝¹ : Preorder J
inst✝ : IsCofiltered J
d : J
n : ℕ
g : (sequentialFunctor J).obj (op n) ≤ d
⊢ IsConnected (CostructuredArrow (sequentialFunctor J) d)
Module: Mathlib.CategoryTheory.Limits.Shapes.Countable

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
J : Type u_2
inst✝² : Countable J
inst✝¹ : Preorder J
inst✝ : IsCofiltered J
d : J
⊢ IsConnected (CostructuredArrow (sequentialFunctor J) d)

Tactic:
obtain ⟨n, (g : (sequentialFunctor J).obj ⟨n⟩ ≤ d)⟩ := sequentialFunctor_initial_aux J d
    have : Nonempty (CostructuredArrow (sequentialFunctor J) d) :=
      ⟨CostructuredArrow.mk (homOfLE g)⟩
    apply isConnected_of_zigzag
    refine fun i j ↦ ⟨[j], ?_⟩
    simp only [List.isChain_cons_cons, Zag, List.isChain_singleton, and_true, ne_eq,
      not_false_eq_true, List.getLast_cons, List.getLast_singleton', reduceCtorEq]
    clear! C
    wlog! h : (unop i.left) ≤ (unop j.left)
    · exact or_comm.1 (this J d n g inferInstance j i (le_of_lt h))
    · right
      exact ⟨CostructuredArrow.homMk (homOfLE h).op rfl⟩

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Basic

Goal state before:
pp: C : Type u
inst✝¹¹ : Category.{v, u} C
D : Type u_1
J : Type u_2
inst✝¹⁰ : Category.{v_1, u_1} D
inst✝⁹ : Category.{v_2, u_2} J
inst✝⁸ : HasLimitsOfShape J C
inst✝⁷ : HasLimitsOfShape J D
inst✝⁶ : HasExactLimitsOfShape J D
F : C ⥤ D
inst✝⁵ : PreservesFiniteColimits F
inst✝⁴ : ReflectsFiniteColimits F
inst✝³ : HasFiniteColimits C
inst✝² : PreservesLimitsOfShape J F
I : Type
inst✝¹ : SmallCategory I
inst✝ : FinCategory I
G : I ⥤ J ⥤ C
c : Cocone G
hc : IsColimit c
⊢ IsColimit (lim.mapCocone c)

Tactic:
apply isColimitOfReflects F

Goal state after:
C : Type u
inst✝¹¹ : Category.{v, u} C
D : Type u_1
J : Type u_2
inst✝¹⁰ : Category.{v_1, u_1} D
inst✝⁹ : Category.{v_2, u_2} J
inst✝⁸ : HasLimitsOfShape J C
inst✝⁷ : HasLimitsOfShape J D
inst✝⁶ : HasExactLimitsOfShape J D
F : C ⥤ D
inst✝⁵ : PreservesFiniteColimits F
inst✝⁴ : ReflectsFiniteColimits F
inst✝³ : HasFiniteColimits C
inst✝² : PreservesLimitsOfShape J F
I : Type
inst✝¹ : SmallCategory I
inst✝ : FinCategory I
G : I ⥤ J ⥤ C
c : Cocone G
hc : IsColimit c
⊢ IsColimit (F.mapCocone (lim.mapCocone c))
Module: Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Basic

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
J : Type u_1
inst✝³ : Category.{v_1, u_1} J
D : Type u_2
inst✝² : Category.{v_2, u_2} D
e : C ≌ D
inst✝¹ : HasColimitsOfShape J C
inst✝ : HasExactColimitsOfShape J C
this : HasColimitsOfShape J D
x✝² : Type
x✝¹ : SmallCategory x✝²
x✝ : FinCategory x✝²
K : x✝² ⥤ J ⥤ D
⊢ e.symm.congrRight.functor.obj (e.congrRight.inverse ⋙ colim ⋙ e.functor) ≅ e.symm.congrRight.functor.obj colim

Tactic:
exact isoWhiskerLeft (_ ⋙ colim) e.unitIso.symm ≪≫ (preservesColimitNatIso e.inverse).symm

Goal state after:
Module: Mathlib.CategoryTheory.Functor.KanExtension.Basic

Goal state before:
pp: C : Type u_1
C' : Type u_2
H : Type u_3
D : Type u_4
D' : Type u_5
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} C'
inst✝³ : Category.{v_3, u_3} H
inst✝² : Category.{v_4, u_4} D
inst✝¹ : Category.{v_5, u_5} D'
F' : D ⥤ H
L : C ⥤ D
F : C ⥤ H
α : F ⟶ L ⋙ F'
inst✝ : F'.IsLeftKanExtension α
c : Cocone F
hc : IsColimit c
s : Cocone F'
m : (F'.coconeOfIsLeftKanExtension α c).pt ⟶ s.pt
hm : ∀ (j : D), (F'.coconeOfIsLeftKanExtension α c).ι.app j ≫ m = s.ι.app j
j : C
this : (F'.coconeOfIsLeftKanExtension α c).ι.app (L.obj j) ≫ m = s.ι.app (L.obj j)
⊢ c.ι.app j ≫ m = c.ι.app j ≫ hc.desc { pt := s.1, ι := α ≫ L.whiskerLeft s.ι }

Tactic:
nth_rw 1 [← F'.descOfIsLeftKanExtension_fac_app α ((const D).obj c.pt)]

Goal state after:
C : Type u_1
C' : Type u_2
H : Type u_3
D : Type u_4
D' : Type u_5
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} C'
inst✝³ : Category.{v_3, u_3} H
inst✝² : Category.{v_4, u_4} D
inst✝¹ : Category.{v_5, u_5} D'
F' : D ⥤ H
L : C ⥤ D
F : C ⥤ H
α : F ⟶ L ⋙ F'
inst✝ : F'.IsLeftKanExtension α
c : Cocone F
hc : IsColimit c
s : Cocone F'
m : (F'.coconeOfIsLeftKanExtension α c).pt ⟶ s.pt
hm : ∀ (j : D), (F'.coconeOfIsLeftKanExtension α c).ι.app j ≫ m = s.ι.app j
j : C
this : (F'.coconeOfIsLeftKanExtension α c).ι.app (L.obj j) ≫ m = s.ι.app (L.obj j)
⊢ (α.app j ≫ (F'.descOfIsLeftKanExtension α ((const D).obj c.pt) c.ι).app (L.obj j)) ≫ m =
    c.ι.app j ≫ hc.desc { pt := s.1, ι := α ≫ L.whiskerLeft s.ι }
Module: Mathlib.CategoryTheory.Abelian.Transfer

Goal state before:
pp: C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasFiniteProducts C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
inst✝² : Abelian D
F : C ⥤ D
G : D ⥤ C
inst✝¹ : G.PreservesZeroMorphisms
inst✝ : PreservesFiniteLimits G
i : F ⋙ G ≅ 𝟭 C
adj : G ⊣ F
this✝ : HasKernels C
this : HasCokernels C
⊢ ∀ {X Y : C} (f : X ⟶ Y), IsIso (Abelian.coimageImageComparison f)

Tactic:
intro X Y f

Goal state after:
C : Type u₁
inst✝⁶ : Category.{v₁, u₁} C
inst✝⁵ : Preadditive C
inst✝⁴ : HasFiniteProducts C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
inst✝² : Abelian D
F : C ⥤ D
G : D ⥤ C
inst✝¹ : G.PreservesZeroMorphisms
inst✝ : PreservesFiniteLimits G
i : F ⋙ G ≅ 𝟭 C
adj : G ⊣ F
this✝ : HasKernels C
this : HasCokernels C
X Y : C
f : X ⟶ Y
⊢ IsIso (Abelian.coimageImageComparison f)
Module: Mathlib.CategoryTheory.Limits.Presheaf

Goal state before:
pp: case mp
C : Type u₁
inst✝² : Category.{v₁, u₁} C
ℰ : Type u₂
inst✝¹ : Category.{v₂, u₂} ℰ
A : C ⥤ ℰ
inst✝ : uliftYoneda.{max w v₂, v₁, u₁}.HasPointwiseLeftKanExtension A
L : (Cᵒᵖ ⥤ Type (max w v₁ v₂)) ⥤ ℰ
α : A ⟶ uliftYoneda.{max w v₂, v₁, u₁} ⋙ L
a✝ : L.IsLeftKanExtension α
⊢ IsIso α ∧
    PreservesColimitsOfSize.{v₁, max w u₁ v₁ v₂, max (max (max u₁ v₁) v₂) w, v₂,
        max (max (max u₁ (v₁ + 1)) (v₂ + 1)) (w + 1), u₂}
      L

Tactic:
exact ⟨inferInstance, preservesColimits_of_natIso (Functor.leftKanExtensionUnique _
      (uliftYoneda.{max w v₂}.leftKanExtensionUnit A) _ α)⟩

Goal state after:
Module: Mathlib.CategoryTheory.Comma.LocallySmall

Goal state before:
pp: A : Type u₁
B : Type u₂
T : Type u₃
inst✝⁴ : Category.{v₁, u₁} A
inst✝³ : Category.{v₂, u₂} B
inst✝² : Category.{v₃, u₃} T
L : A ⥤ T
R : B ⥤ T
inst✝¹ : LocallySmall.{w, v₁, u₁} A
inst✝ : LocallySmall.{w, v₂, u₂} B
X Y : Comma L R
x✝² x✝¹ : X ⟶ Y
x✝ : (fun g ↦ (g.left, g.right)) x✝² = (fun g ↦ (g.left, g.right)) x✝¹
⊢ x✝² = x✝¹

Tactic:
by aesop

Goal state after:
Module: Mathlib.CategoryTheory.Generator.Basic

Goal state before:
pp: case mpr.refine_2
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
P : ObjectProperty C
hP : P.IsCodetecting
X Y : C
f : X ⟶ Y
inst✝ : Epi f
hf : ∀ (G : C), P G → Function.Surjective ⇑(ConcreteCategory.hom ((yoneda.obj G).map f.op))
A : C
hA : P A
g : X ⟶ A
l₁ l₂ : Y ⟶ A
h₁ : f ≫ l₁ = g
h₂ : f ≫ l₂ = g
⊢ l₁ = l₂

Tactic:
← cancel_epi f,

Goal state after:
case mpr.refine_2
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
P : ObjectProperty C
hP : P.IsCodetecting
X Y : C
f : X ⟶ Y
inst✝ : Epi f
hf : ∀ (G : C), P G → Function.Surjective ⇑(ConcreteCategory.hom ((yoneda.obj G).map f.op))
A : C
hA : P A
g : X ⟶ A
l₁ l₂ : Y ⟶ A
h₁ : f ≫ l₁ = g
h₂ : f ≫ l₂ = g
⊢ f ≫ l₁ = f ≫ l₂
Module: Mathlib.CategoryTheory.Limits.Types.Coproducts

Goal state before:
pp: C : Type u
F : C → Type v
⊢ ∀ (x : (sigma F).pt), ∃ i y, (sigma F).inj i y = x

Tactic:
by aesop

Goal state after:
Module: Mathlib.Algebra.Category.Grp.Injective

Goal state before:
pp: case inr
A : Type u
inst✝¹ : AddCommGroup A
inst✝ : DivisibleBy A ℤ
m : ℤ
g : ↥(ℤ ∙ m) →ₗ[ℤ] A
h0 : m ≠ 0
gₘ : A := g ⟨m, ⋯⟩
n : ℤ
hn : n • m ∈ ℤ ∙ m
⊢ n • gₘ = g ⟨n • m, hn⟩

Tactic:
← map_zsmul g,

Goal state after:
case inr
A : Type u
inst✝¹ : AddCommGroup A
inst✝ : DivisibleBy A ℤ
m : ℤ
g : ↥(ℤ ∙ m) →ₗ[ℤ] A
h0 : m ≠ 0
gₘ : A := g ⟨m, ⋯⟩
n : ℤ
hn : n • m ∈ ℤ ∙ m
⊢ g (n • ⟨m, ⋯⟩) = g ⟨n • m, hn⟩
Module: Mathlib.GroupTheory.Divisible

Goal state before:
pp: case ofNat
A : Type u_1
inst✝¹ : Group A
inst✝ : RootableBy A ℕ
a : A
a✝ : ℕ
hn : ofNat a✝ ≠ 0
⊢ (match ofNat a✝ with
      | ofNat n => RootableBy.root a n
      | -[n+1] => (RootableBy.root a (n + 1))⁻¹) ^
      ofNat a✝ =
    a

Tactic:
rw [Int.ofNat_eq_natCast, Nat.cast_ne_zero] at hn
      simp [RootableBy.root_cancel _ hn]

Goal state after:
Module: Mathlib.Algebra.Ring.NegOnePow

Goal state before:
pp: n : ℤ
⊢ (-n).negOnePow = n.negOnePow

Tactic:
dsimp [negOnePow]
  simp only [zpow_neg, ← inv_zpow, inv_neg, inv_one]

Goal state after: