mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.SmallObject.Iteration.ExtendToSucc

Goal state before:
pp: case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
J : Type u
inst✝¹ : LinearOrder J
inst✝ : SuccOrder J
j : J
hj : ¬IsMax j
F : ↑(Set.Iic j) ⥤ C
X : C
τ : F.obj ⟨j, ⋯⟩ ⟶ X
i₁ i₂ i₃ : J
h₁₂ : i₁ ≤ i₂
h₂₃ : i₂ ≤ i₃
h : i₃ ≤ Order.succ j
h₁ : i₃ ≤ j
⊢ (objIso F X ⟨i₁, ⋯⟩).hom ≫ F.map (homOfLE ⋯) ≫ (objIso F X ⟨i₃, h₁⟩).inv =
    (objIso F X ⟨i₁, ⋯⟩).hom ≫
      (F.map (homOfLE h₁₂) ≫ (objIso F X ⟨i₂, ⋯⟩).inv) ≫
        (objIso F X ⟨i₂, ⋯⟩).hom ≫ F.map (homOfLE h₂₃) ≫ (objIso F X ⟨i₃, h₁⟩).inv

Tactic:
assoc,

Goal state after:
case pos
C : Type u_1
inst✝² : Category.{v_1, u_1} C
J : Type u
inst✝¹ : LinearOrder J
inst✝ : SuccOrder J
j : J
hj : ¬IsMax j
F : ↑(Set.Iic j) ⥤ C
X : C
τ : F.obj ⟨j, ⋯⟩ ⟶ X
i₁ i₂ i₃ : J
h₁₂ : i₁ ≤ i₂
h₂₃ : i₂ ≤ i₃
h : i₃ ≤ Order.succ j
h₁ : i₃ ≤ j
⊢ (objIso F X ⟨i₁, ⋯⟩).hom ≫ F.map (homOfLE ⋯) ≫ (objIso F X ⟨i₃, h₁⟩).inv =
    (objIso F X ⟨i₁, ⋯⟩).hom ≫
      F.map (homOfLE h₁₂) ≫
        (objIso F X ⟨i₂, ⋯⟩).inv ≫ (objIso F X ⟨i₂, ⋯⟩).hom ≫ F.map (homOfLE h₂₃) ≫ (objIso F X ⟨i₃, h₁⟩).inv
Module: Mathlib.CategoryTheory.SmallObject.Construction

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S X : C
πX : X ⟶ S
inst✝¹ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
X' S' Z' : C
πX' : X' ⟶ S'
ι' : X' ⟶ Z'
πZ' : Z' ⟶ S'
fac' : ι' ≫ πZ' = πX'
eX : X' ≅ X
eS : S' ≅ S
eZ : Z' ≅ functorObj f πX
commι : ι' ≫ eZ.hom = eX.hom ≫ ιFunctorObj f πX
commπ : πZ' ≫ eS.hom = eZ.hom ≫ πFunctorObj f πX
i : I
t : A i ⟶ X'
b : B i ⟶ S'
fac : t ≫ πX' = f i ≫ b
⊢ (t ≫ eX.hom) ≫ πX = f i ≫ b ≫ eS.hom

Tactic:
assoc,

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S X : C
πX : X ⟶ S
inst✝¹ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
X' S' Z' : C
πX' : X' ⟶ S'
ι' : X' ⟶ Z'
πZ' : Z' ⟶ S'
fac' : ι' ≫ πZ' = πX'
eX : X' ≅ X
eS : S' ≅ S
eZ : Z' ≅ functorObj f πX
commι : ι' ≫ eZ.hom = eX.hom ≫ ιFunctorObj f πX
commπ : πZ' ≫ eS.hom = eZ.hom ≫ πFunctorObj f πX
i : I
t : A i ⟶ X'
b : B i ⟶ S'
fac : t ≫ πX' = f i ≫ b
⊢ t ≫ eX.hom ≫ πX = f i ≫ b ≫ eS.hom
Module: Mathlib.CategoryTheory.SmallObject.Iteration.FunctorOfCocone

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
J : Type u
inst✝ : LinearOrder J
j : J
F : ↑(Set.Iio j) ⥤ C
c : Cocone F
i₁ i₂ : J
hi : i₁ ≤ i₂
hi₂ : i₂ < j
⊢ ((ofCoconeObjIso c i₁ ⋯).hom ≫ F.map (homOfLE hi) ≫ (ofCoconeObjIso c i₂ hi₂).inv) ≫ (ofCoconeObjIso c i₂ hi₂).hom =
    (ofCoconeObjIso c i₁ ⋯).hom ≫ F.map (homOfLE hi)

Tactic:
assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
J : Type u
inst✝ : LinearOrder J
j : J
F : ↑(Set.Iio j) ⥤ C
c : Cocone F
i₁ i₂ : J
hi : i₁ ≤ i₂
hi₂ : i₂ < j
⊢ (ofCoconeObjIso c i₁ ⋯).hom ≫ (F.map (homOfLE hi) ≫ (ofCoconeObjIso c i₂ hi₂).inv) ≫ (ofCoconeObjIso c i₂ hi₂).hom =
    (ofCoconeObjIso c i₁ ⋯).hom ≫ F.map (homOfLE hi)
Module: Mathlib.CategoryTheory.SmallObject.Iteration.FunctorOfCocone

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
J : Type u
inst✝ : LinearOrder J
j : J
F : ↑(Set.Iio j) ⥤ C
c : Cocone F
i₁ i₂ : J
hi : i₁ ≤ i₂
hi₂ : i₂ < j
⊢ (ofCoconeObjIso c i₁ ⋯).hom ≫ (F.map (homOfLE hi) ≫ (ofCoconeObjIso c i₂ hi₂).inv) ≫ (ofCoconeObjIso c i₂ hi₂).hom =
    (ofCoconeObjIso c i₁ ⋯).hom ≫ F.map (homOfLE hi)

Tactic:
assoc,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
J : Type u
inst✝ : LinearOrder J
j : J
F : ↑(Set.Iio j) ⥤ C
c : Cocone F
i₁ i₂ : J
hi : i₁ ≤ i₂
hi₂ : i₂ < j
⊢ (ofCoconeObjIso c i₁ ⋯).hom ≫ F.map (homOfLE hi) ≫ (ofCoconeObjIso c i₂ hi₂).inv ≫ (ofCoconeObjIso c i₂ hi₂).hom =
    (ofCoconeObjIso c i₁ ⋯).hom ≫ F.map (homOfLE hi)
Module: Mathlib.CategoryTheory.SmallObject.Iteration.ExtendToSucc

Goal state before:
pp: case neg.inl
C : Type u_1
inst✝² : Category.{v_1, u_1} C
J : Type u
inst✝¹ : LinearOrder J
inst✝ : SuccOrder J
j : J
hj : ¬IsMax j
F : ↑(Set.Iic j) ⥤ C
X : C
τ : F.obj ⟨j, ⋯⟩ ⟶ X
i₁ i₂ : J
h₁₂ : i₁ ≤ i₂
h₂₃ : i₂ ≤ Order.succ j
h : Order.succ j ≤ Order.succ j
h₁ : ¬Order.succ j ≤ j
h₂ : i₂ ≤ j
⊢ (objIso F X ⟨i₁, ⋯⟩).hom ≫ F.map (homOfLE ⋯) ≫ τ ≫ (objSuccIso hj F X).inv ≫ 𝟙 (obj F X ⟨Order.succ j, ⋯⟩) =
    ((objIso F X ⟨i₁, ⋯⟩).hom ≫ F.map (homOfLE h₁₂) ≫ (objIso F X ⟨i₂, h₂⟩).inv) ≫
      (objIso F X ⟨i₂, h₂⟩).hom ≫ F.map (homOfLE h₂) ≫ τ ≫ (objSuccIso hj F X).inv ≫ 𝟙 (obj F X ⟨Order.succ j, ⋯⟩)

Tactic:
assoc,

Goal state after:
case neg.inl
C : Type u_1
inst✝² : Category.{v_1, u_1} C
J : Type u
inst✝¹ : LinearOrder J
inst✝ : SuccOrder J
j : J
hj : ¬IsMax j
F : ↑(Set.Iic j) ⥤ C
X : C
τ : F.obj ⟨j, ⋯⟩ ⟶ X
i₁ i₂ : J
h₁₂ : i₁ ≤ i₂
h₂₃ : i₂ ≤ Order.succ j
h : Order.succ j ≤ Order.succ j
h₁ : ¬Order.succ j ≤ j
h₂ : i₂ ≤ j
⊢ (objIso F X ⟨i₁, ⋯⟩).hom ≫ F.map (homOfLE ⋯) ≫ τ ≫ (objSuccIso hj F X).inv ≫ 𝟙 (obj F X ⟨Order.succ j, ⋯⟩) =
    (objIso F X ⟨i₁, ⋯⟩).hom ≫
      (F.map (homOfLE h₁₂) ≫ (objIso F X ⟨i₂, h₂⟩).inv) ≫
        (objIso F X ⟨i₂, h₂⟩).hom ≫ F.map (homOfLE h₂) ≫ τ ≫ (objSuccIso hj F X).inv ≫ 𝟙 (obj F X ⟨Order.succ j, ⋯⟩)
Module: Mathlib.CategoryTheory.SmallObject.Iteration.ExtendToSucc

Goal state before:
pp: case neg.inl
C : Type u_1
inst✝² : Category.{v_1, u_1} C
J : Type u
inst✝¹ : LinearOrder J
inst✝ : SuccOrder J
j : J
hj : ¬IsMax j
F : ↑(Set.Iic j) ⥤ C
X : C
τ : F.obj ⟨j, ⋯⟩ ⟶ X
i₁ i₂ : J
h₁₂ : i₁ ≤ i₂
h₂₃ : i₂ ≤ Order.succ j
h : Order.succ j ≤ Order.succ j
h₁ : ¬Order.succ j ≤ j
h₂ : i₂ ≤ j
⊢ (objIso F X ⟨i₁, ⋯⟩).hom ≫ F.map (homOfLE ⋯) ≫ τ ≫ (objSuccIso hj F X).inv ≫ 𝟙 (obj F X ⟨Order.succ j, ⋯⟩) =
    (objIso F X ⟨i₁, ⋯⟩).hom ≫
      (F.map (homOfLE h₁₂) ≫ (objIso F X ⟨i₂, h₂⟩).inv) ≫
        (objIso F X ⟨i₂, h₂⟩).hom ≫ F.map (homOfLE h₂) ≫ τ ≫ (objSuccIso hj F X).inv ≫ 𝟙 (obj F X ⟨Order.succ j, ⋯⟩)

Tactic:
assoc,

Goal state after:
case neg.inl
C : Type u_1
inst✝² : Category.{v_1, u_1} C
J : Type u
inst✝¹ : LinearOrder J
inst✝ : SuccOrder J
j : J
hj : ¬IsMax j
F : ↑(Set.Iic j) ⥤ C
X : C
τ : F.obj ⟨j, ⋯⟩ ⟶ X
i₁ i₂ : J
h₁₂ : i₁ ≤ i₂
h₂₃ : i₂ ≤ Order.succ j
h : Order.succ j ≤ Order.succ j
h₁ : ¬Order.succ j ≤ j
h₂ : i₂ ≤ j
⊢ (objIso F X ⟨i₁, ⋯⟩).hom ≫ F.map (homOfLE ⋯) ≫ τ ≫ (objSuccIso hj F X).inv ≫ 𝟙 (obj F X ⟨Order.succ j, ⋯⟩) =
    (objIso F X ⟨i₁, ⋯⟩).hom ≫
      F.map (homOfLE h₁₂) ≫
        (objIso F X ⟨i₂, h₂⟩).inv ≫
          (objIso F X ⟨i₂, h₂⟩).hom ≫ F.map (homOfLE h₂) ≫ τ ≫ (objSuccIso hj F X).inv ≫ 𝟙 (obj F X ⟨Order.succ j, ⋯⟩)
Module: Mathlib.CategoryTheory.SmallObject.Iteration.Nonempty

Goal state before:
pp: case e_F
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
Φ : SuccStruct C
J : Type u
inst✝⁴ : LinearOrder J
inst✝³ : OrderBot J
inst✝² : SuccOrder J
inst✝¹ : WellFoundedLT J
inst✝ : HasIterationOfShape J C
j : J
hj : ¬IsMax j
iter : Φ.Iteration j
i : J
hi : Order.IsSuccLimit i
hij : i ≤ Order.succ j
k : J
hk : k < i
hij' : i ≤ j
⊢ ∀ ⦃X Y : ↑(Set.Iio i)⦄ (f : X ⟶ Y),
    (restrictionLT iter.F hij').mapArrow.obj (Arrow.mk f) =
      (restrictionLT (extendToSucc hj iter.F (Φ.toSucc (iter.F.obj ⟨j, ⋯⟩))) hij).mapArrow.obj (Arrow.mk f)

Tactic:
rintro ⟨k₁, h₁⟩ ⟨k₂, h₂⟩ f

Goal state after:
case e_F
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
Φ : SuccStruct C
J : Type u
inst✝⁴ : LinearOrder J
inst✝³ : OrderBot J
inst✝² : SuccOrder J
inst✝¹ : WellFoundedLT J
inst✝ : HasIterationOfShape J C
j : J
hj : ¬IsMax j
iter : Φ.Iteration j
i : J
hi : Order.IsSuccLimit i
hij : i ≤ Order.succ j
k : J
hk : k < i
hij' : i ≤ j
k₁ : J
h₁ : k₁ ∈ Set.Iio i
k₂ : J
h₂ : k₂ ∈ Set.Iio i
f : ⟨k₁, h₁⟩ ⟶ ⟨k₂, h₂⟩
⊢ (restrictionLT iter.F hij').mapArrow.obj (Arrow.mk f) =
    (restrictionLT (extendToSucc hj iter.F (Φ.toSucc (iter.F.obj ⟨j, ⋯⟩))) hij).mapArrow.obj (Arrow.mk f)
Module: Mathlib.CategoryTheory.SmallObject.TransfiniteIteration

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
Φ : SuccStruct C
J : Type w
inst✝⁴ : LinearOrder J
inst✝³ : OrderBot J
inst✝² : SuccOrder J
inst✝¹ : WellFoundedLT J
inst✝ : HasIterationOfShape J C
i : J
hi : Order.IsSuccLimit i
⊢ ∀ {X Y : ↑(Set.Iio i)} (f : X ⟶ Y),
    (⋯.functor ⋙ Φ.iterationFunctor J).map f ≫ (eqToIso ⋯).hom = (eqToIso ⋯).hom ≫ (restrictionLT (Φ.iter i).F ⋯).map f

Tactic:
rintro ⟨k₁, h₁⟩ ⟨k₂, h₂⟩ f

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
Φ : SuccStruct C
J : Type w
inst✝⁴ : LinearOrder J
inst✝³ : OrderBot J
inst✝² : SuccOrder J
inst✝¹ : WellFoundedLT J
inst✝ : HasIterationOfShape J C
i : J
hi : Order.IsSuccLimit i
k₁ : J
h₁ : k₁ ∈ Set.Iio i
k₂ : J
h₂ : k₂ ∈ Set.Iio i
f : ⟨k₁, h₁⟩ ⟶ ⟨k₂, h₂⟩
⊢ (⋯.functor ⋙ Φ.iterationFunctor J).map f ≫ (eqToIso ⋯).hom = (eqToIso ⋯).hom ≫ (restrictionLT (Φ.iter i).F ⋯).map f
Module: Mathlib.CategoryTheory.SmallObject.Construction

Goal state before:
pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S X : C
πX : X ⟶ S
inst✝¹ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
X' S' Z' : C
πX' : X' ⟶ S'
ι' : X' ⟶ Z'
πZ' : Z' ⟶ S'
fac' : ι' ≫ πZ' = πX'
eX : X' ≅ X
eS : S' ≅ S
eZ : Z' ≅ functorObj f πX
commι : ι' ≫ eZ.hom = eX.hom ≫ ιFunctorObj f πX
commπ : πZ' ≫ eS.hom = eZ.hom ≫ πFunctorObj f πX
i : I
t : A i ⟶ X'
b : B i ⟶ S'
fac : t ≫ πX' = f i ≫ b
l : B i ⟶ functorObj f πX
hl₁ : f i ≫ l = (t ≫ eX.hom) ≫ ιFunctorObj f πX
hl₂ : l ≫ πFunctorObj f πX = b ≫ eS.hom
⊢ ((l ≫ eZ.inv) ≫ πZ') ≫ eS.hom = b ≫ eS.hom

Tactic:
assoc,

Goal state after:
case refine_2
C : Type u
inst✝² : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S X : C
πX : X ⟶ S
inst✝¹ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
X' S' Z' : C
πX' : X' ⟶ S'
ι' : X' ⟶ Z'
πZ' : Z' ⟶ S'
fac' : ι' ≫ πZ' = πX'
eX : X' ≅ X
eS : S' ≅ S
eZ : Z' ≅ functorObj f πX
commι : ι' ≫ eZ.hom = eX.hom ≫ ιFunctorObj f πX
commπ : πZ' ≫ eS.hom = eZ.hom ≫ πFunctorObj f πX
i : I
t : A i ⟶ X'
b : B i ⟶ S'
fac : t ≫ πX' = f i ≫ b
l : B i ⟶ functorObj f πX
hl₁ : f i ≫ l = (t ≫ eX.hom) ≫ ιFunctorObj f πX
hl₂ : l ≫ πFunctorObj f πX = b ≫ eS.hom
⊢ (l ≫ eZ.inv) ≫ πZ' ≫ eS.hom = b ≫ eS.hom
Module: Mathlib.CategoryTheory.SmallObject.Iteration.ExtendToSucc

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
J : Type u
inst✝¹ : LinearOrder J
inst✝ : SuccOrder J
j : J
hj : ¬IsMax j
F : ↑(Set.Iic j) ⥤ C
X : C
τ : F.obj ⟨j, ⋯⟩ ⟶ X
i₁ i₂ : J
hi : i₁ ≤ i₂
hi₂ : i₂ ≤ j
⊢ ((extendToSucc.objIso F X ⟨i₁, ⋯⟩).hom ≫ F.map (homOfLE ⋯) ≫ (extendToSucc.objIso F X ⟨i₂, hi₂⟩).inv) ≫
      (extendToSucc.objIso F X ⟨i₂, hi₂⟩).hom =
    (extendToSucc.objIso F X ⟨i₁, ⋯⟩).hom ≫ F.map (homOfLE hi)

Tactic:
assoc,

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
J : Type u
inst✝¹ : LinearOrder J
inst✝ : SuccOrder J
j : J
hj : ¬IsMax j
F : ↑(Set.Iic j) ⥤ C
X : C
τ : F.obj ⟨j, ⋯⟩ ⟶ X
i₁ i₂ : J
hi : i₁ ≤ i₂
hi₂ : i₂ ≤ j
⊢ (extendToSucc.objIso F X ⟨i₁, ⋯⟩).hom ≫
      (F.map (homOfLE ⋯) ≫ (extendToSucc.objIso F X ⟨i₂, hi₂⟩).inv) ≫ (extendToSucc.objIso F X ⟨i₂, hi₂⟩).hom =
    (extendToSucc.objIso F X ⟨i₁, ⋯⟩).hom ≫ F.map (homOfLE hi)
Module: Mathlib.CategoryTheory.SmallObject.Construction

Goal state before:
pp: case refine_2
C : Type u
inst✝² : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S X : C
πX : X ⟶ S
inst✝¹ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
X' S' Z' : C
πX' : X' ⟶ S'
ι' : X' ⟶ Z'
πZ' : Z' ⟶ S'
fac' : ι' ≫ πZ' = πX'
eX : X' ≅ X
eS : S' ≅ S
eZ : Z' ≅ functorObj f πX
commι : ι' ≫ eZ.hom = eX.hom ≫ ιFunctorObj f πX
commπ : πZ' ≫ eS.hom = eZ.hom ≫ πFunctorObj f πX
i : I
t : A i ⟶ X'
b : B i ⟶ S'
fac : t ≫ πX' = f i ≫ b
l : B i ⟶ functorObj f πX
hl₁ : f i ≫ l = (t ≫ eX.hom) ≫ ιFunctorObj f πX
hl₂ : l ≫ πFunctorObj f πX = b ≫ eS.hom
⊢ (l ≫ eZ.inv) ≫ πZ' ≫ eS.hom = b ≫ eS.hom

Tactic:
assoc,

Goal state after:
case refine_2
C : Type u
inst✝² : Category.{v, u} C
I : Type w
A B : I → C
f : (i : I) → A i ⟶ B i
S X : C
πX : X ⟶ S
inst✝¹ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C
inst✝ : HasPushout (functorObjTop f πX) (functorObjLeft f πX)
X' S' Z' : C
πX' : X' ⟶ S'
ι' : X' ⟶ Z'
πZ' : Z' ⟶ S'
fac' : ι' ≫ πZ' = πX'
eX : X' ≅ X
eS : S' ≅ S
eZ : Z' ≅ functorObj f πX
commι : ι' ≫ eZ.hom = eX.hom ≫ ιFunctorObj f πX
commπ : πZ' ≫ eS.hom = eZ.hom ≫ πFunctorObj f πX
i : I
t : A i ⟶ X'
b : B i ⟶ S'
fac : t ≫ πX' = f i ≫ b
l : B i ⟶ functorObj f πX
hl₁ : f i ≫ l = (t ≫ eX.hom) ≫ ιFunctorObj f πX
hl₂ : l ≫ πFunctorObj f πX = b ≫ eS.hom
⊢ l ≫ eZ.inv ≫ πZ' ≫ eS.hom = b ≫ eS.hom
Module: Mathlib.CategoryTheory.SmallObject.Iteration.ExtendToSucc

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
J : Type u
inst✝¹ : LinearOrder J
inst✝ : SuccOrder J
j : J
hj : ¬IsMax j
F : ↑(Set.Iic j) ⥤ C
X : C
τ : F.obj ⟨j, ⋯⟩ ⟶ X
i₁ i₂ : J
hi : i₁ ≤ i₂
hi₂ : i₂ ≤ j
⊢ (extendToSucc.objIso F X ⟨i₁, ⋯⟩).hom ≫
      (F.map (homOfLE ⋯) ≫ (extendToSucc.objIso F X ⟨i₂, hi₂⟩).inv) ≫ (extendToSucc.objIso F X ⟨i₂, hi₂⟩).hom =
    (extendToSucc.objIso F X ⟨i₁, ⋯⟩).hom ≫ F.map (homOfLE hi)

Tactic:
assoc,

Goal state after:
C : Type u_1
inst✝² : Category.{v_1, u_1} C
J : Type u
inst✝¹ : LinearOrder J
inst✝ : SuccOrder J
j : J
hj : ¬IsMax j
F : ↑(Set.Iic j) ⥤ C
X : C
τ : F.obj ⟨j, ⋯⟩ ⟶ X
i₁ i₂ : J
hi : i₁ ≤ i₂
hi₂ : i₂ ≤ j
⊢ (extendToSucc.objIso F X ⟨i₁, ⋯⟩).hom ≫
      F.map (homOfLE ⋯) ≫ (extendToSucc.objIso F X ⟨i₂, hi₂⟩).inv ≫ (extendToSucc.objIso F X ⟨i₂, hi₂⟩).hom =
    (extendToSucc.objIso F X ⟨i₁, ⋯⟩).hom ≫ F.map (homOfLE hi)
Module: Mathlib.CategoryTheory.SmallObject.TransfiniteIteration

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
Φ : SuccStruct C
J : Type w
inst✝⁴ : LinearOrder J
inst✝³ : OrderBot J
inst✝² : SuccOrder J
inst✝¹ : WellFoundedLT J
inst✝ : HasIterationOfShape J C
x✝¹ x✝ : J
f : x✝¹ ⟶ x✝
⊢ (Φ.iterationFunctorObjBotIso J).inv ≫ (Φ.iterationFunctor J).map (homOfLE ⋯) =
    ((Φ.iterationFunctorObjBotIso J).inv ≫ (Φ.iterationFunctor J).map (homOfLE ⋯)) ≫ (Φ.iterationFunctor J).map f

Tactic:
assoc,

Goal state after:
C : Type u
inst✝⁵ : Category.{v, u} C
Φ : SuccStruct C
J : Type w
inst✝⁴ : LinearOrder J
inst✝³ : OrderBot J
inst✝² : SuccOrder J
inst✝¹ : WellFoundedLT J
inst✝ : HasIterationOfShape J C
x✝¹ x✝ : J
f : x✝¹ ⟶ x✝
⊢ (Φ.iterationFunctorObjBotIso J).inv ≫ (Φ.iterationFunctor J).map (homOfLE ⋯) =
    (Φ.iterationFunctorObjBotIso J).inv ≫ (Φ.iterationFunctor J).map (homOfLE ⋯) ≫ (Φ.iterationFunctor J).map f
Module: Mathlib.CategoryTheory.SmallObject.TransfiniteCompositionLifting

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
J : Type w
inst✝¹ : LinearOrder J
inst✝ : OrderBot J
F : J ⥤ C
c : Cocone F
X Y : C
p : X ⟶ Y
f : F.obj ⊥ ⟶ X
g : c.pt ⟶ Y
j : J
sq' : SqStruct c p f g j
⊢ (F.map (homOfLE ⋯) ≫ sq'.f') ≫ p = c.ι.app ⊥ ≫ g

Tactic:
assoc,

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
J : Type w
inst✝¹ : LinearOrder J
inst✝ : OrderBot J
F : J ⥤ C
c : Cocone F
X Y : C
p : X ⟶ Y
f : F.obj ⊥ ⟶ X
g : c.pt ⟶ Y
j : J
sq' : SqStruct c p f g j
⊢ F.map (homOfLE ⋯) ≫ sq'.f' ≫ p = c.ι.app ⊥ ≫ g
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
f : Arrow C
j : κ.ord.ToType
this✝¹ : HasIterationOfShape κ.ord.ToType C
this✝ : NoMaxOrder κ.ord.ToType
this :
  (((succStruct I κ).iterationFunctor κ.ord.ToType).map (homOfLE ⋯)).app f =
    ((succStruct I κ).toSucc (((succStruct I κ).iterationFunctor κ.ord.ToType).obj j)).app f ≫
      ((succStruct I κ).iterationFunctorObjSuccIso j ⋯).inv.app f
⊢ Arrow.homMk (ιFunctorObj I.homFamily ((((succStruct I κ).iterationFunctor κ.ord.ToType).obj j).obj f).hom)
      (𝟙 ((((succStruct I κ).iterationFunctor κ.ord.ToType).obj j).obj f).right) ⋯ =
    (((succStruct I κ).toSucc (((succStruct I κ).iterationFunctor κ.ord.ToType).obj j)).app f ≫
        ((succStruct I κ).iterationFunctorObjSuccIso j ⋯).inv.app f) ≫
      ((succStruct I κ).iterationFunctorObjSuccIso j ⋯).hom.app f

Tactic:
assoc,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
f : Arrow C
j : κ.ord.ToType
this✝¹ : HasIterationOfShape κ.ord.ToType C
this✝ : NoMaxOrder κ.ord.ToType
this :
  (((succStruct I κ).iterationFunctor κ.ord.ToType).map (homOfLE ⋯)).app f =
    ((succStruct I κ).toSucc (((succStruct I κ).iterationFunctor κ.ord.ToType).obj j)).app f ≫
      ((succStruct I κ).iterationFunctorObjSuccIso j ⋯).inv.app f
⊢ Arrow.homMk (ιFunctorObj I.homFamily ((((succStruct I κ).iterationFunctor κ.ord.ToType).obj j).obj f).hom)
      (𝟙 ((((succStruct I κ).iterationFunctor κ.ord.ToType).obj j).obj f).right) ⋯ =
    ((succStruct I κ).toSucc (((succStruct I κ).iterationFunctor κ.ord.ToType).obj j)).app f ≫
      ((succStruct I κ).iterationFunctorObjSuccIso j ⋯).inv.app f ≫
        ((succStruct I κ).iterationFunctorObjSuccIso j ⋯).hom.app f
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y : C
f : X ⟶ Y
j : κ.ord.ToType
h₁ :
  Arrow.Hom.left (Arrow.Hom.right (iterationFunctorMapSuccAppArrowIso I κ (Arrow.mk f) j).hom) ≫
      πFunctorObj I.homFamily (((iterationFunctor I κ).obj j).obj (Arrow.mk f)).hom =
    (((iterationFunctor I κ).obj (Order.succ j)).obj (Arrow.mk f)).hom ≫
      Arrow.Hom.right (Arrow.Hom.right (iterationFunctorMapSuccAppArrowIso I κ (Arrow.mk f) j).hom)
h₂ :
  (iterationFunctor I κ).map (homOfLE ⋯) ≫
      (transfiniteCompositionOfShapeSuccStructPropιIteration I κ).incl.app (Order.succ j) =
    (transfiniteCompositionOfShapeSuccStructPropιIteration I κ).incl.app j
⊢ (πFunctorObj I.homFamily (((iterationFunctor I κ).obj j).obj (Arrow.mk f)).hom ≫
        (iterationFunctorObjObjRightIso I κ (Arrow.mk f) j).hom) ≫
      Arrow.Hom.right ((ιIteration I κ).app (Arrow.mk f)) =
    (((relativeCellComplexιObjFObjSuccIso I κ f j).inv ≫
          (relativeCellComplexιObj I κ f).incl.app (Order.succ j) ≫
            πObj I κ f ≫ (iterationFunctorObjObjRightIso I κ (Arrow.mk f) j).inv) ≫
        (iterationFunctorObjObjRightIso I κ (Arrow.mk f) j).hom) ≫
      Arrow.Hom.right ((ιIteration I κ).app (Arrow.mk f))

Tactic:
assoc,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y : C
f : X ⟶ Y
j : κ.ord.ToType
h₁ :
  Arrow.Hom.left (Arrow.Hom.right (iterationFunctorMapSuccAppArrowIso I κ (Arrow.mk f) j).hom) ≫
      πFunctorObj I.homFamily (((iterationFunctor I κ).obj j).obj (Arrow.mk f)).hom =
    (((iterationFunctor I κ).obj (Order.succ j)).obj (Arrow.mk f)).hom ≫
      Arrow.Hom.right (Arrow.Hom.right (iterationFunctorMapSuccAppArrowIso I κ (Arrow.mk f) j).hom)
h₂ :
  (iterationFunctor I κ).map (homOfLE ⋯) ≫
      (transfiniteCompositionOfShapeSuccStructPropιIteration I κ).incl.app (Order.succ j) =
    (transfiniteCompositionOfShapeSuccStructPropιIteration I κ).incl.app j
⊢ πFunctorObj I.homFamily (((iterationFunctor I κ).obj j).obj (Arrow.mk f)).hom ≫
      (iterationFunctorObjObjRightIso I κ (Arrow.mk f) j).hom ≫ Arrow.Hom.right ((ιIteration I κ).app (Arrow.mk f)) =
    (((relativeCellComplexιObjFObjSuccIso I κ f j).inv ≫
          (relativeCellComplexιObj I κ f).incl.app (Order.succ j) ≫
            πObj I κ f ≫ (iterationFunctorObjObjRightIso I κ (Arrow.mk f) j).inv) ≫
        (iterationFunctorObjObjRightIso I κ (Arrow.mk f) j).hom) ≫
      Arrow.Hom.right ((ιIteration I κ).app (Arrow.mk f))
Module: Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y : C
f : X ⟶ Y
j : κ.ord.ToType
h₁ :
  Arrow.Hom.left (Arrow.Hom.right (iterationFunctorMapSuccAppArrowIso I κ (Arrow.mk f) j).hom) ≫
      πFunctorObj I.homFamily (((iterationFunctor I κ).obj j).obj (Arrow.mk f)).hom =
    (((iterationFunctor I κ).obj (Order.succ j)).obj (Arrow.mk f)).hom ≫
      Arrow.Hom.right (Arrow.Hom.right (iterationFunctorMapSuccAppArrowIso I κ (Arrow.mk f) j).hom)
h₂ :
  (iterationFunctor I κ).map (homOfLE ⋯) ≫
      (transfiniteCompositionOfShapeSuccStructPropιIteration I κ).incl.app (Order.succ j) =
    (transfiniteCompositionOfShapeSuccStructPropιIteration I κ).incl.app j
⊢ πFunctorObj I.homFamily (((iterationFunctor I κ).obj j).obj (Arrow.mk f)).hom ≫
      (iterationFunctorObjObjRightIso I κ (Arrow.mk f) j).hom ≫ Arrow.Hom.right ((ιIteration I κ).app (Arrow.mk f)) =
    (((relativeCellComplexιObjFObjSuccIso I κ f j).inv ≫
          (relativeCellComplexιObj I κ f).incl.app (Order.succ j) ≫
            πObj I κ f ≫ (iterationFunctorObjObjRightIso I κ (Arrow.mk f) j).inv) ≫
        (iterationFunctorObjObjRightIso I κ (Arrow.mk f) j).hom) ≫
      Arrow.Hom.right ((ιIteration I κ).app (Arrow.mk f))

Tactic:
assoc,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
I : MorphismProperty C
κ : Cardinal.{w}
inst✝² : Fact κ.IsRegular
inst✝¹ : OrderBot κ.ord.ToType
inst✝ : I.IsCardinalForSmallObjectArgument κ
X Y : C
f : X ⟶ Y
j : κ.ord.ToType
h₁ :
  Arrow.Hom.left (Arrow.Hom.right (iterationFunctorMapSuccAppArrowIso I κ (Arrow.mk f) j).hom) ≫
      πFunctorObj I.homFamily (((iterationFunctor I κ).obj j).obj (Arrow.mk f)).hom =
    (((iterationFunctor I κ).obj (Order.succ j)).obj (Arrow.mk f)).hom ≫
      Arrow.Hom.right (Arrow.Hom.right (iterationFunctorMapSuccAppArrowIso I κ (Arrow.mk f) j).hom)
h₂ :
  (iterationFunctor I κ).map (homOfLE ⋯) ≫
      (transfiniteCompositionOfShapeSuccStructPropιIteration I κ).incl.app (Order.succ j) =
    (transfiniteCompositionOfShapeSuccStructPropιIteration I κ).incl.app j
⊢ πFunctorObj I.homFamily (((iterationFunctor I κ).obj j).obj (Arrow.mk f)).hom ≫
      (iterationFunctorObjObjRightIso I κ (Arrow.mk f) j).hom ≫ Arrow.Hom.right ((ιIteration I κ).app (Arrow.mk f)) =
    ((relativeCellComplexιObjFObjSuccIso I κ f j).inv ≫
        (relativeCellComplexιObj I κ f).incl.app (Order.succ j) ≫
          πObj I κ f ≫ (iterationFunctorObjObjRightIso I κ (Arrow.mk f) j).inv) ≫
      (iterationFunctorObjObjRightIso I κ (Arrow.mk f) j).hom ≫ Arrow.Hom.right ((ιIteration I κ).app (Arrow.mk f))