Theorem: CategoryTheory.Paths.ext_functor
File: Mathlib/CategoryTheory/PathCategory.lean

tactic: fapply Functor.ext
state_before: V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
⊢ F = G
state_after: case h_obj
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
⊢ ∀ (X : Paths V), F.obj X = G.obj X

case h_map
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
⊢ autoParam (∀ (X Y : Paths V) (f : X ⟶ Y), F.map f = eqToHom ⋯ ≫ G.map f ≫ eqToHom ⋯) _auto✝

tactic: intro X
state_before: case h_obj
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
⊢ ∀ (X : Paths V), F.obj X = G.obj X
state_after: case h_obj
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
X : Paths V
⊢ F.obj X = G.obj X

tactic: rw [h_obj]
state_before: case h_obj
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
X : Paths V
⊢ F.obj X = G.obj X
state_after: no goals

tactic: intro X Y f
state_before: case h_map
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
⊢ autoParam (∀ (X Y : Paths V) (f : X ⟶ Y), F.map f = eqToHom ⋯ ≫ G.map f ≫ eqToHom ⋯) _auto✝
state_after: case h_map
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
X Y : Paths V
f : X ⟶ Y
⊢ F.map f = eqToHom ⋯ ≫ G.map f ≫ eqToHom ⋯

tactic: induction' f with Y' Z' g e ih
state_before: case h_map
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
X Y : Paths V
f : X ⟶ Y
⊢ F.map f = eqToHom ⋯ ≫ G.map f ≫ eqToHom ⋯
state_after: case h_map.nil
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
X Y : Paths V
⊢ F.map Quiver.Path.nil = eqToHom ⋯ ≫ G.map Quiver.Path.nil ≫ eqToHom ⋯

case h_map.cons
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
X Y Y' Z' : Paths V
g : Quiver.Path X Y'
e : Y' ⟶ Z'
ih : F.map g = eqToHom ⋯ ≫ G.map g ≫ eqToHom ⋯
⊢ F.map (g.cons e) = eqToHom ⋯ ≫ G.map (g.cons e) ≫ eqToHom ⋯

tactic: erw [F.map_id, G.map_id, Category.id_comp, eqToHom_trans, eqToHom_refl]
state_before: case h_map.nil
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
X Y : Paths V
⊢ F.map Quiver.Path.nil = eqToHom ⋯ ≫ G.map Quiver.Path.nil ≫ eqToHom ⋯
state_after: no goals

tactic: erw [F.map_comp g (Quiver.Hom.toPath e), G.map_comp g (Quiver.Hom.toPath e), ih, h]
state_before: case h_map.cons
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
X Y Y' Z' : Paths V
g : Quiver.Path X Y'
e : Y' ⟶ Z'
ih : F.map g = eqToHom ⋯ ≫ G.map g ≫ eqToHom ⋯
⊢ F.map (g.cons e) = eqToHom ⋯ ≫ G.map (g.cons e) ≫ eqToHom ⋯
state_after: case h_map.cons
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
X Y Y' Z' : Paths V
g : Quiver.Path X Y'
e : Y' ⟶ Z'
ih : F.map g = eqToHom ⋯ ≫ G.map g ≫ eqToHom ⋯
⊢ (eqToHom ⋯ ≫ G.map g ≫ eqToHom ⋯) ≫ eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯ =
    eqToHom ⋯ ≫ (G.map g ≫ G.map e.toPath) ≫ eqToHom ⋯

tactic: simp only [Category.id_comp, eqToHom_refl, eqToHom_trans_assoc, Category.assoc]
state_before: case h_map.cons
V : Type u₁
inst✝¹ : Quiver V
C : Type u_1
inst✝ : Category.{u_2, u_1} C
F G : Paths V ⥤ C
h_obj : F.obj = G.obj
h : ∀ (a b : V) (e : a ⟶ b), F.map e.toPath = eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯
X Y Y' Z' : Paths V
g : Quiver.Path X Y'
e : Y' ⟶ Z'
ih : F.map g = eqToHom ⋯ ≫ G.map g ≫ eqToHom ⋯
⊢ (eqToHom ⋯ ≫ G.map g ≫ eqToHom ⋯) ≫ eqToHom ⋯ ≫ G.map e.toPath ≫ eqToHom ⋯ =
    eqToHom ⋯ ≫ (G.map g ≫ G.map e.toPath) ≫ eqToHom ⋯
state_after: no goals