-- Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean
theorem Cofix.bisim_rel {α : TypeVec n} (r : Cofix F α → Cofix F α → Prop)
    (h : ∀ x y, r x y →
      appendFun id (Quot.mk r) <$$> Cofix.dest x = appendFun id (Quot.mk r) <$$> Cofix.dest y) :
    ∀ x y, r x y → x = y

Proof state:
case h.inr
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
r' : Cofix F α → Cofix F α → Prop := fun x y => x = y ∨ r x y
x✝ y✝ : Cofix F α
rxy : r x✝ y✝
x y : Cofix F α
r'xy : r x y
this : ∀ (x y : Cofix F α), r x y → r' x y
⊢ (TypeVec.id ::: Quot.mk r') <$$> x.dest = (TypeVec.id ::: Quot.mk r') <$$> y.dest

Tactic:
rw [← Quot.factor_mk_eq _ _ this]

Result:
case h.inr
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
r' : Cofix F α → Cofix F α → Prop := fun x y => x = y ∨ r x y
x✝ y✝ : Cofix F α
rxy : r x✝ y✝
x y : Cofix F α
r'xy : r x y
this : ∀ (x y : Cofix F α), r x y → r' x y
⊢ (TypeVec.id ::: Quot.factor r r' this ∘ Quot.mk r) <$$> x.dest =
    (TypeVec.id ::: Quot.factor r r' this ∘ Quot.mk r) <$$> y.dest

Proof state:
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
r' : Cofix F α → Cofix F α → Prop := fun x y => x = y ∨ r x y
⊢ ∀ (x y : Cofix F α), r x y → x = y

Tactic:
intro x y rxy

Result:
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
r' : Cofix F α → Cofix F α → Prop := fun x y => x = y ∨ r x y
x y : Cofix F α
rxy : r x y
⊢ x = y

Proof state:
case a.h
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
r' : Cofix F α → Cofix F α → Prop := fun x y => x = y ∨ r x y
x y : Cofix F α
rxy : r x y
⊢ r x y

Tactic:
exact rxy

Result:
no goals

Proof state:
case h.inl
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h✝ : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
r' : Cofix F α → Cofix F α → Prop := fun x y => x = y ∨ r x y
x✝ y✝ : Cofix F α
rxy : r x✝ y✝
x y : Cofix F α
h : x = y
⊢ (TypeVec.id ::: Quot.mk r') <$$> x.dest = (TypeVec.id ::: Quot.mk r') <$$> y.dest

Tactic:
rw [h]

Result:
no goals

Proof state:
case a
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
r' : Cofix F α → Cofix F α → Prop := fun x y => x = y ∨ r x y
x y : Cofix F α
rxy : r x y
⊢ r' x y

Tactic:
right

Result:
case a.h
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
r' : Cofix F α → Cofix F α → Prop := fun x y => x = y ∨ r x y
x y : Cofix F α
rxy : r x y
⊢ r x y

Proof state:
case h'.h
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
r' : Cofix F α → Cofix F α → Prop := fun x y => x = y ∨ r x y
x✝ y : Cofix F α
rxy : r x✝ y
x : Cofix F α
⊢ x = x

Tactic:
rfl

Result:
no goals

Proof state:
case h.inr
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
r' : Cofix F α → Cofix F α → Prop := fun x y => x = y ∨ r x y
x✝ y✝ : Cofix F α
rxy : r x✝ y✝
x y : Cofix F α
r'xy : r x y
this : ∀ (x y : Cofix F α), r x y → r' x y
⊢ (TypeVec.id ::: Quot.factor r r' this ∘ Quot.mk r) <$$> x.dest =
    (TypeVec.id ::: Quot.factor r r' this ∘ Quot.mk r) <$$> y.dest

Tactic:
dsimp [r']

Result:
case h.inr
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
r' : Cofix F α → Cofix F α → Prop := fun x y => x = y ∨ r x y
x✝ y✝ : Cofix F α
rxy : r x✝ y✝
x y : Cofix F α
r'xy : r x y
this : ∀ (x y : Cofix F α), r x y → r' x y
⊢ (TypeVec.id ::: Quot.factor r (fun x y => x = y ∨ r x y) this ∘ Quot.mk r) <$$> x.dest =
    (TypeVec.id ::: Quot.factor r (fun x y => x = y ∨ r x y) this ∘ Quot.mk r) <$$> y.dest

Proof state:
case h'
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
r' : Cofix F α → Cofix F α → Prop := fun x y => x = y ∨ r x y
x y : Cofix F α
rxy : r x y
⊢ ∀ (x : Cofix F α), r' x x

Tactic:
intro x

Result:
case h'
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
r' : Cofix F α → Cofix F α → Prop := fun x y => x = y ∨ r x y
x✝ y : Cofix F α
rxy : r x✝ y
x : Cofix F α
⊢ r' x x

Proof state:
case h.inr
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
r' : Cofix F α → Cofix F α → Prop := fun x y => x = y ∨ r x y
x✝ y✝ : Cofix F α
rxy : r x✝ y✝
x y : Cofix F α
r'xy : r x y
this : ∀ (x y : Cofix F α), r x y → r' x y
⊢ ((TypeVec.id ::: Quot.factor r (fun x y => x = y ∨ r x y) this) ⊚ (TypeVec.id ::: Quot.mk r)) <$$> x.dest =
    ((TypeVec.id ::: Quot.factor r (fun x y => x = y ∨ r x y) this) ⊚ (TypeVec.id ::: Quot.mk r)) <$$> y.dest

Tactic:
rw [@comp_map _ _ q _ _ _ (appendFun id (Quot.mk r)),
  @comp_map _ _ q _ _ _ (appendFun id (Quot.mk r))]

Result:
case h.inr
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
r : Cofix F α → Cofix F α → Prop
h : ∀ (x y : Cofix F α), r x y → (TypeVec.id ::: Quot.mk r) <$$> x.dest = (TypeVec.id ::: Quot.mk r) <$$> y.dest
r' : Cofix F α → Cofix F α → Prop := fun x y => x = y ∨ r x y
x✝ y✝ : Cofix F α
rxy : r x✝ y✝
x y : Cofix F α
r'xy : r x y
this : ∀ (x y : Cofix F α), r x y → r' x y
⊢ (TypeVec.id ::: Quot.factor r (fun x y => x = y ∨ r x y) this) <$$> (TypeVec.id ::: Quot.mk r) <$$> x.dest =
    (TypeVec.id ::: Quot.factor r (fun x y => x = y ∨ r x y) this) <$$> (TypeVec.id ::: Quot.mk r) <$$> y.dest