l3lab/ntp-mathlib huggingface dataset:
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  

Proof state:
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
⊢ ∀ (a : PFunctor.M (P F)) (y : Cofix F), r (Quot.mk Mcongr a) y → Quot.mk Mcongr a = y

Next tactic:
intro x y
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  

Proof state:
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x : PFunctor.M (P F)
y : Cofix F
⊢ r (Quot.mk Mcongr x) y → Quot.mk Mcongr x = y

Next tactic:
apply Quot.inductionOn (motive := _) y
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  apply Quot.inductionOn (motive := _) y
  

Proof state:
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x : PFunctor.M (P F)
y : Cofix F
⊢ ∀ (a : PFunctor.M (P F)), r (Quot.mk Mcongr x) (Quot.mk Mcongr a) → Quot.mk Mcongr x = Quot.mk Mcongr a

Next tactic:
clear y
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  apply Quot.inductionOn (motive := _) y
  clear y
  

Proof state:
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x : PFunctor.M (P F)
⊢ ∀ (a : PFunctor.M (P F)), r (Quot.mk Mcongr x) (Quot.mk Mcongr a) → Quot.mk Mcongr x = Quot.mk Mcongr a

Next tactic:
intro y rxy
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  apply Quot.inductionOn (motive := _) y
  clear y
  intro y rxy
  

Proof state:
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x y : PFunctor.M (P F)
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
⊢ Quot.mk Mcongr x = Quot.mk Mcongr y

Next tactic:
apply Quot.sound
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  apply Quot.inductionOn (motive := _) y
  clear y
  intro y rxy
  apply Quot.sound
  

Proof state:
case a
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x y : PFunctor.M (P F)
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
⊢ Mcongr x y

Next tactic:
let r' x y := r (Quot.mk _ x) (Quot.mk _ y)
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  apply Quot.inductionOn (motive := _) y
  clear y
  intro y rxy
  apply Quot.sound
  let r' x y := r (Quot.mk _ x) (Quot.mk _ y)
  

Proof state:
case a
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x y : PFunctor.M (P F)
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
⊢ Mcongr x y

Next tactic:
have : IsPrecongr r' := by
    intro a b r'ab
    have h₀ :
      Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =
        Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=
      h _ _ r'ab
    have h₁ : ∀ u v : q.P.M, Mcongr u v → Quot.mk r' u = Quot.mk r' v := by
      intro u v cuv
      apply Quot.sound
      simp only
      rw [Quot.sound cuv]
      apply h'
    let f : Quot r → Quot r' :=
      Quot.lift (Quot.lift (Quot.mk r') h₁)
        (by
          intro c; apply Quot.inductionOn (motive := _) c; clear c
          intro c d; apply Quot.inductionOn (motive := _) d; clear d
          intro d rcd; apply Quot.sound; apply rcd)
    have : f ∘ Quot.mk r ∘ Quot.mk Mcongr = Quot.mk r' := rfl
    rw [← this, ← PFunctor.map_map _ _ f, ← PFunctor.map_map _ _ (Quot.mk r), abs_map, abs_map,
      abs_map, h₀]
    rw [← PFunctor.map_map _ _ f, ← PFunctor.map_map _ _ (Quot.mk r), abs_map, abs_map, abs_map]
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  apply Quot.inductionOn (motive := _) y
  clear y
  intro y rxy
  apply Quot.sound
  let r' x y := r (Quot.mk _ x) (Quot.mk _ y)
  have : IsPrecongr r' := by
    

Proof state:
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x y : PFunctor.M (P F)
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
⊢ IsPrecongr r'

Next tactic:
intro a b r'ab
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  apply Quot.inductionOn (motive := _) y
  clear y
  intro y rxy
  apply Quot.sound
  let r' x y := r (Quot.mk _ x) (Quot.mk _ y)
  have : IsPrecongr r' := by
    intro a b r'ab
    

Proof state:
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x y : PFunctor.M (P F)
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
a b : PFunctor.M (P F)
r'ab : r' a b
⊢ abs (PFunctor.map (P F) (Quot.mk r') (PFunctor.M.dest a)) = abs (PFunctor.map (P F) (Quot.mk r') (PFunctor.M.dest b))

Next tactic:
have h₀ :
      Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =
        Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=
      h _ _ r'ab
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  apply Quot.inductionOn (motive := _) y
  clear y
  intro y rxy
  apply Quot.sound
  let r' x y := r (Quot.mk _ x) (Quot.mk _ y)
  have : IsPrecongr r' := by
    intro a b r'ab
    have h₀ :
      Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =
        Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=
      h _ _ r'ab
    

Proof state:
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x y : PFunctor.M (P F)
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
a b : PFunctor.M (P F)
r'ab : r' a b
h₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)
⊢ abs (PFunctor.map (P F) (Quot.mk r') (PFunctor.M.dest a)) = abs (PFunctor.map (P F) (Quot.mk r') (PFunctor.M.dest b))

Next tactic:
have h₁ : ∀ u v : q.P.M, Mcongr u v → Quot.mk r' u = Quot.mk r' v := by
      intro u v cuv
      apply Quot.sound
      simp only
      rw [Quot.sound cuv]
      apply h'
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  apply Quot.inductionOn (motive := _) y
  clear y
  intro y rxy
  apply Quot.sound
  let r' x y := r (Quot.mk _ x) (Quot.mk _ y)
  have : IsPrecongr r' := by
    intro a b r'ab
    have h₀ :
      Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =
        Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=
      h _ _ r'ab
    have h₁ : ∀ u v : q.P.M, Mcongr u v → Quot.mk r' u = Quot.mk r' v := by
      

Proof state:
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x y : PFunctor.M (P F)
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
a b : PFunctor.M (P F)
r'ab : r' a b
h₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)
⊢ ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v

Next tactic:
intro u v cuv
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  apply Quot.inductionOn (motive := _) y
  clear y
  intro y rxy
  apply Quot.sound
  let r' x y := r (Quot.mk _ x) (Quot.mk _ y)
  have : IsPrecongr r' := by
    intro a b r'ab
    have h₀ :
      Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =
        Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=
      h _ _ r'ab
    have h₁ : ∀ u v : q.P.M, Mcongr u v → Quot.mk r' u = Quot.mk r' v := by
      intro u v cuv
      

Proof state:
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x y : PFunctor.M (P F)
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
a b : PFunctor.M (P F)
r'ab : r' a b
h₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)
u v : PFunctor.M (P F)
cuv : Mcongr u v
⊢ Quot.mk r' u = Quot.mk r' v

Next tactic:
apply Quot.sound
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  apply Quot.inductionOn (motive := _) y
  clear y
  intro y rxy
  apply Quot.sound
  let r' x y := r (Quot.mk _ x) (Quot.mk _ y)
  have : IsPrecongr r' := by
    intro a b r'ab
    have h₀ :
      Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =
        Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=
      h _ _ r'ab
    have h₁ : ∀ u v : q.P.M, Mcongr u v → Quot.mk r' u = Quot.mk r' v := by
      intro u v cuv
      apply Quot.sound
      

Proof state:
case a
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x y : PFunctor.M (P F)
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
a b : PFunctor.M (P F)
r'ab : r' a b
h₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)
u v : PFunctor.M (P F)
cuv : Mcongr u v
⊢ r' u v

Next tactic:
simp only
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  apply Quot.inductionOn (motive := _) y
  clear y
  intro y rxy
  apply Quot.sound
  let r' x y := r (Quot.mk _ x) (Quot.mk _ y)
  have : IsPrecongr r' := by
    intro a b r'ab
    have h₀ :
      Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =
        Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=
      h _ _ r'ab
    have h₁ : ∀ u v : q.P.M, Mcongr u v → Quot.mk r' u = Quot.mk r' v := by
      intro u v cuv
      apply Quot.sound
      simp only
      

Proof state:
case a
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x y : PFunctor.M (P F)
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
a b : PFunctor.M (P F)
r'ab : r' a b
h₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)
u v : PFunctor.M (P F)
cuv : Mcongr u v
⊢ r (Quot.mk Mcongr u) (Quot.mk Mcongr v)

Next tactic:
rw [Quot.sound cuv]
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  apply Quot.inductionOn (motive := _) y
  clear y
  intro y rxy
  apply Quot.sound
  let r' x y := r (Quot.mk _ x) (Quot.mk _ y)
  have : IsPrecongr r' := by
    intro a b r'ab
    have h₀ :
      Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =
        Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=
      h _ _ r'ab
    have h₁ : ∀ u v : q.P.M, Mcongr u v → Quot.mk r' u = Quot.mk r' v := by
      intro u v cuv
      apply Quot.sound
      simp only
      rw [Quot.sound cuv]
      

Proof state:
case a
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x y : PFunctor.M (P F)
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
a b : PFunctor.M (P F)
r'ab : r' a b
h₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)
u v : PFunctor.M (P F)
cuv : Mcongr u v
⊢ r (Quot.mk Mcongr v) (Quot.mk Mcongr v)

Next tactic:
apply h'
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  apply Quot.inductionOn (motive := _) y
  clear y
  intro y rxy
  apply Quot.sound
  let r' x y := r (Quot.mk _ x) (Quot.mk _ y)
  have : IsPrecongr r' := by
    intro a b r'ab
    have h₀ :
      Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =
        Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=
      h _ _ r'ab
    have h₁ : ∀ u v : q.P.M, Mcongr u v → Quot.mk r' u = Quot.mk r' v := by
      intro u v cuv
      apply Quot.sound
      simp only
      rw [Quot.sound cuv]
      apply h'
    

Proof state:
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x y : PFunctor.M (P F)
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
a b : PFunctor.M (P F)
r'ab : r' a b
h₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)
h₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v
⊢ abs (PFunctor.map (P F) (Quot.mk r') (PFunctor.M.dest a)) = abs (PFunctor.map (P F) (Quot.mk r') (PFunctor.M.dest b))

Next tactic:
let f : Quot r → Quot r' :=
      Quot.lift (Quot.lift (Quot.mk r') h₁)
        (by
          intro c; apply Quot.inductionOn (motive := _) c; clear c
          intro c d; apply Quot.inductionOn (motive := _) d; clear d
          intro d rcd; apply Quot.sound; apply rcd)
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  apply Quot.inductionOn (motive := _) y
  clear y
  intro y rxy
  apply Quot.sound
  let r' x y := r (Quot.mk _ x) (Quot.mk _ y)
  have : IsPrecongr r' := by
    intro a b r'ab
    have h₀ :
      Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =
        Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=
      h _ _ r'ab
    have h₁ : ∀ u v : q.P.M, Mcongr u v → Quot.mk r' u = Quot.mk r' v := by
      intro u v cuv
      apply Quot.sound
      simp only
      rw [Quot.sound cuv]
      apply h'
    let f : Quot r → Quot r' :=
      Quot.lift (Quot.lift (Quot.mk r') h₁)
        (by
          

Proof state:
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x y : PFunctor.M (P F)
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
a b : PFunctor.M (P F)
r'ab : r' a b
h₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)
h₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v
⊢ ∀ (a b : Cofix F), r a b → Quot.lift (Quot.mk r') h₁ a = Quot.lift (Quot.mk r') h₁ b

Next tactic:
intro c
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  apply Quot.inductionOn (motive := _) y
  clear y
  intro y rxy
  apply Quot.sound
  let r' x y := r (Quot.mk _ x) (Quot.mk _ y)
  have : IsPrecongr r' := by
    intro a b r'ab
    have h₀ :
      Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =
        Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=
      h _ _ r'ab
    have h₁ : ∀ u v : q.P.M, Mcongr u v → Quot.mk r' u = Quot.mk r' v := by
      intro u v cuv
      apply Quot.sound
      simp only
      rw [Quot.sound cuv]
      apply h'
    let f : Quot r → Quot r' :=
      Quot.lift (Quot.lift (Quot.mk r') h₁)
        (by
          intro c; 

Proof state:
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x y : PFunctor.M (P F)
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
a b : PFunctor.M (P F)
r'ab : r' a b
h₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)
h₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v
c : Cofix F
⊢ ∀ (b : Cofix F), r c b → Quot.lift (Quot.mk r') h₁ c = Quot.lift (Quot.mk r') h₁ b

Next tactic:
apply Quot.inductionOn (motive := _) c
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  apply Quot.inductionOn (motive := _) y
  clear y
  intro y rxy
  apply Quot.sound
  let r' x y := r (Quot.mk _ x) (Quot.mk _ y)
  have : IsPrecongr r' := by
    intro a b r'ab
    have h₀ :
      Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =
        Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=
      h _ _ r'ab
    have h₁ : ∀ u v : q.P.M, Mcongr u v → Quot.mk r' u = Quot.mk r' v := by
      intro u v cuv
      apply Quot.sound
      simp only
      rw [Quot.sound cuv]
      apply h'
    let f : Quot r → Quot r' :=
      Quot.lift (Quot.lift (Quot.mk r') h₁)
        (by
          intro c; apply Quot.inductionOn (motive := _) c; 

Proof state:
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x y : PFunctor.M (P F)
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
a b : PFunctor.M (P F)
r'ab : r' a b
h₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)
h₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v
c : Cofix F
⊢ ∀ (a : PFunctor.M (P F)) (b : Cofix F),
    r (Quot.mk Mcongr a) b → Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr a) = Quot.lift (Quot.mk r') h₁ b

Next tactic:
clear c
private theorem Cofix.bisim_aux (r : Cofix F → Cofix F → Prop) (h' : ∀ x, r x x)
    (h : ∀ x y, r x y → Quot.mk r <$> Cofix.dest x = Quot.mk r <$> Cofix.dest y) :
    ∀ x y, r x y → x = y := by
  intro x
  apply Quot.inductionOn (motive := _) x
  clear x
  intro x y
  apply Quot.inductionOn (motive := _) y
  clear y
  intro y rxy
  apply Quot.sound
  let r' x y := r (Quot.mk _ x) (Quot.mk _ y)
  have : IsPrecongr r' := by
    intro a b r'ab
    have h₀ :
      Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) =
        Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) :=
      h _ _ r'ab
    have h₁ : ∀ u v : q.P.M, Mcongr u v → Quot.mk r' u = Quot.mk r' v := by
      intro u v cuv
      apply Quot.sound
      simp only
      rw [Quot.sound cuv]
      apply h'
    let f : Quot r → Quot r' :=
      Quot.lift (Quot.lift (Quot.mk r') h₁)
        (by
          intro c; apply Quot.inductionOn (motive := _) c; clear c
          

Proof state:
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h' : ∀ (x : Cofix F), r x x
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
x y : PFunctor.M (P F)
rxy : r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
r' : PFunctor.M (P F) → PFunctor.M (P F) → Prop := fun x y => r (Quot.mk Mcongr x) (Quot.mk Mcongr y)
a b : PFunctor.M (P F)
r'ab : r' a b
h₀ : Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest a) = Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b)
h₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v
⊢ ∀ (a : PFunctor.M (P F)) (b : Cofix F),
    r (Quot.mk Mcongr a) b → Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr a) = Quot.lift (Quot.mk r') h₁ b

Next tactic:
intro c d