mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Galois.Topology

Goal state before:
pp: C : Type u₁
inst✝² : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
inst✝¹ : GaloisCategory C
inst✝ : FiberFunctor F
U : Set ((X : C) → Aut (F.obj X))
hUopen : IsOpen U
h1 : 1 ∈ ⇑(autEmbedding F) ⁻¹' U
h✝ : IsOpen (⇑(autEmbedding F) ⁻¹' U)
I : Finset C
u : (a : C) → Set (Aut (F.obj a))
ho : ∀ a ∈ I, IsOpen (u a) ∧ 1 a ∈ u a
ha✝ : (↑I).pi u ⊆ U
fι : ↥I → Type
ff : (X : ↥I) → fι X → C
fc : (X : ↥I) → (i : fι X) → ff X i ⟶ ↑X
h4 : (X : ↥I) → Limits.IsColimit (Limits.Cofan.mk (↑X) (fc X))
h5 : ∀ (X : ↥I) (i : fι X), IsConnected (ff X i)
h6 : ∀ (X : ↥I), Finite (fι X)
σ : Aut F
h : ∀ (X : ↑(⋃ X, Set.range (ff X))), σ.hom.app ↑X = 𝟙 (F.obj ↑X)
X : C
XinI : X ∈ ↑I
this : Fintype (fι ⟨X, XinI⟩)
x : (F.obj X).obj
j : fι ⟨X, XinI⟩
a : ((Discrete.functor (ff ⟨X, XinI⟩) ⋙ F).obj { as := j }).obj
ha : (ConcreteCategory.hom (F.map ((Limits.Cofan.mk (↑⟨X, XinI⟩) (fc ⟨X, XinI⟩)).ι.app { as := j }))) a = x
⊢ (ConcreteCategory.hom (F.map ((Limits.Cofan.mk (↑⟨X, XinI⟩) (fc ⟨X, XinI⟩)).ι.app { as := j })))
      ((ConcreteCategory.hom (σ.hom.app ((Discrete.functor (ff ⟨X, XinI⟩)).obj { as := j }))) a) =
    (ConcreteCategory.hom (F.map ((Limits.Cofan.mk (↑⟨X, XinI⟩) (fc ⟨X, XinI⟩)).ι.app { as := j }))) a

Tactic:
simp [h ⟨(ff _) j, ⟨Set.range (ff ⟨X, XinI⟩), ⟨⟨_, rfl⟩, ⟨j, rfl⟩⟩⟩⟩]

Goal state after:
Module: Mathlib.CategoryTheory.Galois.Prorepresentability

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{u₂, u₁} C
inst✝² : GaloisCategory C
F : C ⥤ FintypeCat
inst✝¹ : FiberFunctor F
X : C
inst✝ : IsConnected X
⊢ MulAction.IsPretransitive (Aut F) (F.obj X).obj

Tactic:
obtain ⟨A, f, hgal⟩ := exists_hom_from_galois_of_connected F X

Goal state after:
C : Type u₁
inst✝³ : Category.{u₂, u₁} C
inst✝² : GaloisCategory C
F : C ⥤ FintypeCat
inst✝¹ : FiberFunctor F
X : C
inst✝ : IsConnected X
A : C
f : A ⟶ X
hgal : IsGalois A
⊢ MulAction.IsPretransitive (Aut F) (F.obj X).obj
Module: Mathlib.CategoryTheory.Galois.IsFundamentalgroup

Goal state before:
pp: C : Type u₁
inst✝⁷ : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
G : Type u_1
inst✝⁶ : Group G
inst✝⁵ : (X : C) → MulAction G (F.obj X).obj
inst✝⁴ : IsNaturalSMul F G
inst✝³ : GaloisCategory C
inst✝² : FiberFunctor F
t : Aut F
ι : Type u_2
inst✝¹ : Finite ι
X : ι → C
inst✝ : ∀ (i : ι), IsGalois (X i)
h : ∀ (X : C) [IsGalois X], MulAction.IsPretransitive G (F.obj X).obj
x : (i : ι) → (F.obj (X i)).obj := fun i ↦ ⋯.some
P : C := ∏ᶜ X
is₁ : F.obj P ≅ ∏ᶜ fun i ↦ F.obj (X i) := PreservesProduct.iso F X
is₂ : (∏ᶜ fun i ↦ F.obj (X i)).obj ≃ ((i : ι) → (F.obj (X i)).obj) := FintypeCat.productEquiv fun i ↦ F.obj (X i)
px : (F.obj P).obj := (ConcreteCategory.hom is₁.inv) (is₂.symm x)
hpx : ∀ (i : ι), (ConcreteCategory.hom (F.map (Pi.π X i))) px = x i
A : C
f : A ⟶ P
a : (F.obj A).obj
left✝ : IsGalois A
hfa : (ConcreteCategory.hom (F.map f)) a = px
g : G
hg : ∀ (x : (F.obj A).obj), g • x = (ConcreteCategory.hom (t.hom.app A)) x
i : ι
y : (F.obj (X i)).obj
⊢ g • (ConcreteCategory.hom (F.map (Pi.π X i))) px =
    (ConcreteCategory.hom (t.hom.app (X i))) ((ConcreteCategory.hom (F.map (Pi.π X i))) px)

Tactic:
← IsNaturalSMul.naturality,

Goal state after:
C : Type u₁
inst✝⁷ : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
G : Type u_1
inst✝⁶ : Group G
inst✝⁵ : (X : C) → MulAction G (F.obj X).obj
inst✝⁴ : IsNaturalSMul F G
inst✝³ : GaloisCategory C
inst✝² : FiberFunctor F
t : Aut F
ι : Type u_2
inst✝¹ : Finite ι
X : ι → C
inst✝ : ∀ (i : ι), IsGalois (X i)
h : ∀ (X : C) [IsGalois X], MulAction.IsPretransitive G (F.obj X).obj
x : (i : ι) → (F.obj (X i)).obj := fun i ↦ ⋯.some
P : C := ∏ᶜ X
is₁ : F.obj P ≅ ∏ᶜ fun i ↦ F.obj (X i) := PreservesProduct.iso F X
is₂ : (∏ᶜ fun i ↦ F.obj (X i)).obj ≃ ((i : ι) → (F.obj (X i)).obj) := FintypeCat.productEquiv fun i ↦ F.obj (X i)
px : (F.obj P).obj := (ConcreteCategory.hom is₁.inv) (is₂.symm x)
hpx : ∀ (i : ι), (ConcreteCategory.hom (F.map (Pi.π X i))) px = x i
A : C
f : A ⟶ P
a : (F.obj A).obj
left✝ : IsGalois A
hfa : (ConcreteCategory.hom (F.map f)) a = px
g : G
hg : ∀ (x : (F.obj A).obj), g • x = (ConcreteCategory.hom (t.hom.app A)) x
i : ι
y : (F.obj (X i)).obj
⊢ (ConcreteCategory.hom (F.map (Pi.π X i))) (g • px) =
    (ConcreteCategory.hom (t.hom.app (X i))) ((ConcreteCategory.hom (F.map (Pi.π X i))) px)
Module: Mathlib.CategoryTheory.Galois.IsFundamentalgroup

Goal state before:
pp: C : Type u₁
inst✝⁷ : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
G : Type u_1
inst✝⁶ : Group G
inst✝⁵ : (X : C) → MulAction G (F.obj X).obj
inst✝⁴ : IsNaturalSMul F G
inst✝³ : GaloisCategory C
inst✝² : FiberFunctor F
t : Aut F
ι : Type u_2
inst✝¹ : Finite ι
X : ι → C
inst✝ : ∀ (i : ι), IsGalois (X i)
h : ∀ (X : C) [IsGalois X], MulAction.IsPretransitive G (F.obj X).obj
x : (i : ι) → (F.obj (X i)).obj := fun i ↦ ⋯.some
P : C := ∏ᶜ X
is₁ : F.obj P ≅ ∏ᶜ fun i ↦ F.obj (X i) := PreservesProduct.iso F X
is₂ : (∏ᶜ fun i ↦ F.obj (X i)).obj ≃ ((i : ι) → (F.obj (X i)).obj) := FintypeCat.productEquiv fun i ↦ F.obj (X i)
px : (F.obj P).obj := (ConcreteCategory.hom is₁.inv) (is₂.symm x)
hpx : ∀ (i : ι), (ConcreteCategory.hom (F.map (Pi.π X i))) px = x i
A : C
f : A ⟶ P
a : (F.obj A).obj
left✝ : IsGalois A
hfa : (ConcreteCategory.hom (F.map f)) a = px
g : G
hg : ∀ (x : (F.obj A).obj), g • x = (ConcreteCategory.hom (t.hom.app A)) x
i : ι
y : (F.obj (X i)).obj
⊢ (ConcreteCategory.hom (F.map (Pi.π X i))) (g • (ConcreteCategory.hom (F.map f)) a) =
    (ConcreteCategory.hom (F.map (Pi.π X i)))
      ((ConcreteCategory.hom (F.map f)) ((ConcreteCategory.hom (t.hom.app A)) a))

Tactic:
← IsNaturalSMul.naturality,

Goal state after:
C : Type u₁
inst✝⁷ : Category.{u₂, u₁} C
F : C ⥤ FintypeCat
G : Type u_1
inst✝⁶ : Group G
inst✝⁵ : (X : C) → MulAction G (F.obj X).obj
inst✝⁴ : IsNaturalSMul F G
inst✝³ : GaloisCategory C
inst✝² : FiberFunctor F
t : Aut F
ι : Type u_2
inst✝¹ : Finite ι
X : ι → C
inst✝ : ∀ (i : ι), IsGalois (X i)
h : ∀ (X : C) [IsGalois X], MulAction.IsPretransitive G (F.obj X).obj
x : (i : ι) → (F.obj (X i)).obj := fun i ↦ ⋯.some
P : C := ∏ᶜ X
is₁ : F.obj P ≅ ∏ᶜ fun i ↦ F.obj (X i) := PreservesProduct.iso F X
is₂ : (∏ᶜ fun i ↦ F.obj (X i)).obj ≃ ((i : ι) → (F.obj (X i)).obj) := FintypeCat.productEquiv fun i ↦ F.obj (X i)
px : (F.obj P).obj := (ConcreteCategory.hom is₁.inv) (is₂.symm x)
hpx : ∀ (i : ι), (ConcreteCategory.hom (F.map (Pi.π X i))) px = x i
A : C
f : A ⟶ P
a : (F.obj A).obj
left✝ : IsGalois A
hfa : (ConcreteCategory.hom (F.map f)) a = px
g : G
hg : ∀ (x : (F.obj A).obj), g • x = (ConcreteCategory.hom (t.hom.app A)) x
i : ι
y : (F.obj (X i)).obj
⊢ (ConcreteCategory.hom (F.map (Pi.π X i))) ((ConcreteCategory.hom (F.map f)) (g • a)) =
    (ConcreteCategory.hom (F.map (Pi.π X i)))
      ((ConcreteCategory.hom (F.map f)) ((ConcreteCategory.hom (t.hom.app A)) a))
Module: Mathlib.CategoryTheory.Groupoid.Subgroupoid

Goal state before:
pp: case mp
C : Type u
inst✝ : Groupoid C
S : Subgroupoid C
⊢ S.IsTotallyDisconnected → ∀ (c d : C), (S.arrows c d).Nonempty → c = d

Tactic:
rintro h c d ⟨f, fS⟩
    exact congr_arg Subtype.val <| h ⟨c, mem_objs_of_src S fS⟩ ⟨d, mem_objs_of_tgt S fS⟩ ⟨f, fS⟩

Goal state after:
Module: Mathlib.CategoryTheory.Limits.FormalCoproducts.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
A : Type u₁
inst✝¹ : Category.{v₁, u₁} A
inst✝ : HasCoproducts A
J : Type w
f : J → FormalCoproduct C
F : C ⥤ A
s : Cocone (Discrete.functor f ⋙ (eval C A).obj F)
m : (((eval C A).obj F).mapCocone (cofan J f)).pt ⟶ s.pt
h : ∀ (j : Discrete J), (((eval C A).obj F).mapCocone (cofan J f)).ι.app j ≫ m = s.ι.app j
x✝ : (cofan J f).pt.I
i₁ : J
i₂ : (f i₁).I
⊢ Sigma.ι (fun i ↦ F.obj ((cofan J f).pt.obj i)) ⟨i₁, i₂⟩ ≫ m =
    Sigma.ι (fun i ↦ F.obj ((cofan J f).pt.obj i)) ⟨i₁, i₂⟩ ≫
      Sigma.desc fun i ↦ Sigma.ι (F.obj ∘ (f i.fst).obj) i.snd ≫ s.ι.app { as := i.fst }

Tactic:
by simp [← h, cofan, Function.comp_def]

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.ChosenPullback

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
X₁ X₂ X₃ S : C
f₁ : X₁ ⟶ S
f₂ : X₂ ⟶ S
f₃ : X₃ ⟶ S
h₁₂ : ChosenPullback f₁ f₂
h₂₃ : ChosenPullback f₂ f₃
h₁₃ : ChosenPullback f₁ f₃
h : ChosenPullback₃ h₁₂ h₂₃ h₁₃
x✝ : C
a : x✝ ⟶ h₁₃.pullback
b : x✝ ⟶ h₂₃.pullback
w : a ≫ h₁₃.p₂ = b ≫ h₂₃.p₂
⊢ (b ≫ h₂₃.p₁) ≫ f₂ = (a ≫ h₁₃.p₁) ≫ f₁

Tactic:
simpa using w.symm =≫ f₃

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.BicartesianSq

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
X : C
s : Cocone (span 0 (𝟙 X))
⊢ ∀ (j : WalkingSpan), (PushoutCocone.mk 0 0 ⋯).ι.app j ≫ 0 = s.ι.app j

Tactic:
have c :=
              @PushoutCocone.coequalizer_ext _ _ _ _ _ _ _ s _ 0 (𝟙 _)
                (by simp [eq_iff_true_of_subsingleton]) (by simpa using PushoutCocone.condition s)

Goal state after:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : HasZeroObject C
inst✝ : HasZeroMorphisms C
X : C
s : Cocone (span 0 (𝟙 X))
c : ∀ (j : WalkingSpan), s.ι.app j ≫ 0 = s.ι.app j ≫ 𝟙 s.pt
⊢ ∀ (j : WalkingSpan), (PushoutCocone.mk 0 0 ⋯).ι.app j ≫ 0 = s.ι.app j
Module: Mathlib.CategoryTheory.ObjectProperty.FunctorCategory.PreservesLimits

Goal state before:
pp: J : Type u_1
J' : Type u_2
C : Type u_3
D : Type u_4
K : Type u_5
K' : Type u_6
inst✝⁵ : Category.{v_1, u_5} K
inst✝⁴ : Category.{v_2, u_6} K'
inst✝³ : Category.{v_3, u_1} J
inst✝² : Category.{v_4, u_2} J'
inst✝¹ : Category.{v_5, u_3} C
inst✝ : Category.{v_6, u_4} D
⊢ (preservesColimitsOfShape K).IsClosedUnderIsomorphisms

Tactic:
rw [preservesColimitsOfShape_eq_iSup]

Goal state after:
J : Type u_1
J' : Type u_2
C : Type u_3
D : Type u_4
K : Type u_5
K' : Type u_6
inst✝⁵ : Category.{v_1, u_5} K
inst✝⁴ : Category.{v_2, u_6} K'
inst✝³ : Category.{v_3, u_1} J
inst✝² : Category.{v_4, u_2} J'
inst✝¹ : Category.{v_5, u_3} C
inst✝ : Category.{v_6, u_4} D
⊢ (⨅ F, preservesColimit F).IsClosedUnderIsomorphisms
Module: Mathlib.CategoryTheory.Limits.WeakLimits.Basic

Goal state before:
pp: J : Type u_1
inst✝³ : Category.{v_1, u_1} J
K : Type u_2
inst✝² : Category.{v_2, u_2} K
C : Type u_3
inst✝¹ : Category.{v_3, u_3} C
F : J ⥤ C
D : Type u_4
inst✝ : Category.{v_4, u_4} D
G : K ⥤ D
t t' : Cone F
l : IsLimit t
l' : IsWeakLimit t'
x✝ : J
⊢ (l'.lift t ≫ l.lift t') ≫ t.π.app x✝ = 𝟙 t.pt ≫ t.π.app x✝

Tactic:
by rw [assoc, id_comp, l.fac t', l'.fac t]

Goal state after:
Module: Mathlib.CategoryTheory.Localization.Pi

Goal state before:
pp: case h_empty
⊢ ∀ {C : PEmpty.{w + 1} → Type u₁} {D : PEmpty.{w + 1} → Type u₂}
    [inst : (j : PEmpty.{w + 1}) → Category.{v₁, u₁} (C j)] [inst_1 : (j : PEmpty.{w + 1}) → Category.{v₂, u₂} (D j)]
    (L : (j : PEmpty.{w + 1}) → C j ⥤ D j) (W : (j : PEmpty.{w + 1}) → MorphismProperty (C j))
    [∀ (j : PEmpty.{w + 1}), (W j).ContainsIdentities] [∀ (j : PEmpty.{w + 1}), (L j).IsLocalization (W j)],
    (Functor.pi L).IsLocalization (MorphismProperty.pi W)

Tactic:
intro C D _ _ L W _ _

Goal state after:
case h_empty
C : PEmpty.{w + 1} → Type u₁
D : PEmpty.{w + 1} → Type u₂
inst✝³ : (j : PEmpty.{w + 1}) → Category.{v₁, u₁} (C j)
inst✝² : (j : PEmpty.{w + 1}) → Category.{v₂, u₂} (D j)
L : (j : PEmpty.{w + 1}) → C j ⥤ D j
W : (j : PEmpty.{w + 1}) → MorphismProperty (C j)
inst✝¹ : ∀ (j : PEmpty.{w + 1}), (W j).ContainsIdentities
inst✝ : ∀ (j : PEmpty.{w + 1}), (L j).IsLocalization (W j)
⊢ (Functor.pi L).IsLocalization (MorphismProperty.pi W)
Module: Mathlib.CategoryTheory.Monoidal.Action.End

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝³ : Category.{v_1, u_1} C
inst✝² : MonoidalCategory C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : MonoidalRightAction C D
c₁ c₂ c₃ : C
d : D
⊢ ({ app := fun x ↦ (αᵣ x c₁ c₂).inv, naturality := ⋯ } ▷ (curriedAction C D).obj c₃ ≫
          { app := fun x ↦ (αᵣ x (c₁ ⊗ c₂) c₃).inv, naturality := ⋯ } ≫ (curriedAction C D).map (α_ c₁ c₂ c₃).hom).app
      d =
    ((α_ ((curriedAction C D).obj c₁) ((curriedAction C D).obj c₂) ((curriedAction C D).obj c₃)).hom ≫
          (curriedAction C D).obj c₁ ◁ { app := fun x ↦ (αᵣ x c₂ c₃).inv, naturality := ⋯ } ≫
            { app := fun x ↦ (αᵣ x c₁ (c₂ ⊗ c₃)).inv, naturality := ⋯ }).app
      d

Tactic:
simpa [-actionHom_associator] using
      (IsIso.inv_eq_inv.mpr <| actionHom_associator c₁ c₂ c₃ d).symm =≫
        d ⊴ᵣ (α_ c₁ c₂ c₃).hom

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Arrow

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasPushouts C
inst✝³ : HasInitial C
inst✝² : CartesianMonoidalCategory C
inst✝¹ : MonoidalClosed C
inst✝ : BraidedCategory C
X Y : Arrow C
f : X ⟶ Y
⊢ f ▷ 𝟙_ (Arrow C) ≫ (ρ_ Y).hom = (ρ_ X).hom ≫ f

Tactic:
by
  refine Arrow.hom_ext _ _ (pushout.hom_ext ?_ (by simp)) (by simp)
  apply (initialIsInitial.ofIso (zeroMul initialIsInitial).symm).hom_ext

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Mod

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : CartesianMonoidalCategory C
M : C
inst✝¹ : MonObj M
X : C
inst✝ : ModObj M X
Z Z' : C
g : Z' ⟶ Z
m : Z ⟶ M
x : Z ⟶ X
⊢ g ≫ (m • x) = g ≫ m • g ≫ x

Tactic:
Hom.smul_def,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : CartesianMonoidalCategory C
M : C
inst✝¹ : MonObj M
X : C
inst✝ : ModObj M X
Z Z' : C
g : Z' ⟶ Z
m : Z ⟶ M
x : Z ⟶ X
⊢ g ≫ lift m x ≫ γ = g ≫ m • g ≫ x
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Mod

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : CartesianMonoidalCategory C
M : C
inst✝¹ : MonObj M
X : C
inst✝ : ModObj M X
Z Z' : C
g : Z' ⟶ Z
m : Z ⟶ M
x : Z ⟶ X
⊢ g ≫ lift m x ≫ γ = g ≫ m • g ≫ x

Tactic:
Hom.smul_def,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : CartesianMonoidalCategory C
M : C
inst✝¹ : MonObj M
X : C
inst✝ : ModObj M X
Z Z' : C
g : Z' ⟶ Z
m : Z ⟶ M
x : Z ⟶ X
⊢ g ≫ lift m x ≫ γ = lift (g ≫ m) (g ≫ x) ≫ γ
Module: Mathlib.CategoryTheory.Monoidal.Cartesian.Normal

Goal state before:
pp: case refine_2
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : CartesianMonoidalCategory C
G H : C
inst✝³ : GrpObj G
inst✝² : GrpObj H
φ : H ⟶ G
inst✝¹ : IsMonHom φ
inst✝ : Mono φ
hnormal : ∀ (X : C), (monoidHom φ X).range.Normal
⊢ ∃ ψ, ψ ≫ φ = G ◁ φ ≫ conj G

Tactic:
have (X : C) (g : X ⟶ G) (h : X ⟶ H) : ∃ (h' : X ⟶ H), h' ≫ φ = g * h ≫ φ * g⁻¹ :=
      (hnormal X).conj_mem (h ≫ φ) (by simp) g

Goal state after:
case refine_2
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : CartesianMonoidalCategory C
G H : C
inst✝³ : GrpObj G
inst✝² : GrpObj H
φ : H ⟶ G
inst✝¹ : IsMonHom φ
inst✝ : Mono φ
hnormal : ∀ (X : C), (monoidHom φ X).range.Normal
this : ∀ (X : C) (g : X ⟶ G) (h : X ⟶ H), ∃ h', h' ≫ φ = g * h ≫ φ * g⁻¹
⊢ ∃ ψ, ψ ≫ φ = G ◁ φ ≫ conj G
Module: Mathlib.CategoryTheory.Monoidal.Closed.Ideal

Goal state before:
pp: C : Type u₁
D : Type u₂
inst✝⁵ : Category.{v₁, u₁} C
inst✝⁴ : Category.{v₁, u₂} D
i : D ⥤ C
inst✝³ : CartesianMonoidalCategory C
inst✝² : MonoidalClosed C
A : C
inst✝¹ : Reflective i
inst✝ : ExponentialIdeal i
⊢ i ⋙ ihom A ⋙ reflector i ⋙ i ≅ i ⋙ ihom A

Tactic:
by
  symm
  apply NatIso.ofComponents _ _
  · intro X
    haveI := Functor.essImage.unit_isIso (ExponentialIdeal.exp_closed (i.obj_mem_essImage X) A)
    apply asIso ((reflectorAdjunction i).unit.app (A ⟹ i.obj X))
  · simp [asIso]

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ (α_ (P.X ⊗ S.X) Q.X L.X).hom ≫
      (((α_ P.X S.X (Q.X ⊗ L.X)).hom ≫
            P.X ◁ S.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ≫
          P.X ◁ TensorBimod.actLeft Q L) ≫
        coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))
          ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫
            P.X ◁ TensorBimod.actLeft Q L) =
    ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
      (α_ P.X Q.X L.X).hom ≫
        P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
          coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))
            ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫
              P.X ◁ TensorBimod.actLeft Q L)

Tactic:
slice_lhs 3 4 =>
          rw [← whiskerLeft_comp, TensorBimod.whiskerLeft_π_actLeft, whiskerLeft_comp]

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ (α_ (P.X ⊗ S.X) Q.X L.X).hom ≫
      (α_ P.X S.X (Q.X ⊗ L.X)).hom ≫
        (P.X ◁ (α_ S.X Q.X L.X).inv ≫
            P.X ◁ (Q.actLeft ▷ L.X ≫ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))) ≫
          coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))
            ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫
              P.X ◁ TensorBimod.actLeft Q L) =
    ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
      (α_ P.X Q.X L.X).hom ≫
        P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
          coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))
            ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫
              P.X ◁ TensorBimod.actLeft Q L)
Module: Mathlib.CategoryTheory.MorphismProperty.LocalClosure

Goal state before:
pp: case refine_1.of_presieve
C : Type u
inst✝⁶ : Category.{v, u} C
K : Precoverage C
P : MorphismProperty C
inst✝⁵ : P.RespectsIso
inst✝⁴ : P.RespectsLeft K.morphismProperty
inst✝³ : K.HasIsos
inst✝² : K.IsStableUnderBaseChange
inst✝¹ : K.IsStableUnderComposition
inst✝ : K.HasPullbacks
X Y : C
f✝ : X ⟶ Y
X✝ Y✝ : C
f : X✝ ⟶ Y✝
E : K.ZeroHypercover X✝
h : ∀ (U : C) (g : U ⟶ X✝), Presieve.ofArrows E.X E.f g → sourceLocalClosure K P (g ≫ f)
ih :
  ∀ (U : C) (g : U ⟶ X✝),
    Presieve.ofArrows E.X E.f g → ∃ R ∈ K.coverings U, ∀ (U_1 : C) (g_1 : U_1 ⟶ U), R g_1 → P (g_1 ≫ g ≫ f)
S : (i : E.I₀) → Presieve (E.X i)
hS : ∀ (i : E.I₀), S i ∈ K.coverings (E.X i)
h' : ∀ (i : E.I₀) (U : C) (g : U ⟶ E.X i), S i g → P (g ≫ E.f i ≫ f)
⊢ ∃ R ∈ K.coverings X✝, ∀ (U : C) (g : U ⟶ X✝), R g → P (g ≫ f)

Tactic:
simp_rw [K.mem_iff_exists_zeroHypercover] at hS

Goal state after:
case refine_1.of_presieve
C : Type u
inst✝⁶ : Category.{v, u} C
K : Precoverage C
P : MorphismProperty C
inst✝⁵ : P.RespectsIso
inst✝⁴ : P.RespectsLeft K.morphismProperty
inst✝³ : K.HasIsos
inst✝² : K.IsStableUnderBaseChange
inst✝¹ : K.IsStableUnderComposition
inst✝ : K.HasPullbacks
X Y : C
f✝ : X ⟶ Y
X✝ Y✝ : C
f : X✝ ⟶ Y✝
E : K.ZeroHypercover X✝
h : ∀ (U : C) (g : U ⟶ X✝), Presieve.ofArrows E.X E.f g → sourceLocalClosure K P (g ≫ f)
ih :
  ∀ (U : C) (g : U ⟶ X✝),
    Presieve.ofArrows E.X E.f g → ∃ R ∈ K.coverings U, ∀ (U_1 : C) (g_1 : U_1 ⟶ U), R g_1 → P (g_1 ≫ g ≫ f)
S : (i : E.I₀) → Presieve (E.X i)
h' : ∀ (i : E.I₀) (U : C) (g : U ⟶ E.X i), S i g → P (g ≫ E.f i ≫ f)
hS : ∀ (i : E.I₀), ∃ 𝒰, S i = Presieve.ofArrows 𝒰.X 𝒰.f
⊢ ∃ R ∈ K.coverings X✝, ∀ (U : C) (g : U ⟶ X✝), R g → P (g ≫ f)
Module: Mathlib.CategoryTheory.MorphismProperty.LocalEpi

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝¹ : G.Faithful
inst✝ : G.Full
⊢ (isoClosure fun x ↦ x ∈ Set.range G.obj).localEpi = (MorphismProperty.epimorphisms D).inverseImage F

Tactic:
localEpi_isoClosure,

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
D : Type u_2
inst✝² : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
inst✝¹ : G.Faithful
inst✝ : G.Full
⊢ (localEpi fun x ↦ x ∈ Set.range G.obj) = (MorphismProperty.epimorphisms D).inverseImage F
Module: Mathlib.CategoryTheory.Preadditive.FreydCategory.RightFreyd

Goal state before:
pp: V : Type u_1
inst✝² : Category.{v_1, u_1} V
inst✝¹ : Preadditive V
u v : Arrow V
f✝ g✝ : u ⟶ v
inst✝ : HasZeroObject V
x✝¹ x✝ : V
f g : x✝¹ ⟶ x✝
eq : (functor V).map f = (functor V).map g
⊢ f = g

Tactic:
by
    dsimp at eq
    rw [quotient_map_eq_iff] at eq
    simpa [← sub_eq_zero] using! eq.some.comm

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

Goal state before:
pp: C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
⊢ (∑ j_1,
      (if h : i = i then if h' : h ▸ j = j_1 then eqToHom ⋯ else 0 else 0) ≫
        if h : i' = i then if h' : h ▸ j' = j_1 then eqToHom ⋯ else 0 else 0) =
    𝟙 { ι := (j : Fin n) × (f j).ι, fintype := Sigma.instFintype, X := fun p ↦ (f p.fst).X p.snd } ⟨i, j⟩ ⟨i', j'⟩

pp: case h₀
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
⊢ ∀ b ∈ Finset.univ,
    b ≠ i →
      (∑ j_1,
          (if h : i = b then if h' : h ▸ j = j_1 then eqToHom ⋯ else 0 else 0) ≫
            if h : i' = b then if h' : h ▸ j' = j_1 then eqToHom ⋯ else 0 else 0) =
        0

pp: case h₁
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
⊢ i ∉ Finset.univ →
    (∑ j_1,
        (if h : i = i then if h' : h ▸ j = j_1 then eqToHom ⋯ else 0 else 0) ≫
          if h : i' = i then if h' : h ▸ j' = j_1 then eqToHom ⋯ else 0 else 0) =
      0

Tactic:
rotate_left

Goal state after:
case h₀
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
⊢ ∀ b ∈ Finset.univ,
    b ≠ i →
      (∑ j_1,
          (if h : i = b then if h' : h ▸ j = j_1 then eqToHom ⋯ else 0 else 0) ≫
            if h : i' = b then if h' : h ▸ j' = j_1 then eqToHom ⋯ else 0 else 0) =
        0

case h₁
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
⊢ i ∉ Finset.univ →
    (∑ j_1,
        (if h : i = i then if h' : h ▸ j = j_1 then eqToHom ⋯ else 0 else 0) ≫
          if h : i' = i then if h' : h ▸ j' = j_1 then eqToHom ⋯ else 0 else 0) =
      0

C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Preadditive C
n : ℕ
f : Fin n → Mat_ C
i : Fin n
j : (f i).ι
i' : Fin n
j' : (f i').ι
⊢ (∑ j_1,
      (if h : i = i then if h' : h ▸ j = j_1 then eqToHom ⋯ else 0 else 0) ≫
        if h : i' = i then if h' : h ▸ j' = j_1 then eqToHom ⋯ else 0 else 0) =
    𝟙 { ι := (j : Fin n) × (f j).ι, fintype := Sigma.instFintype, X := fun p ↦ (f p.fst).X p.snd } ⟨i, j⟩ ⟨i', j'⟩