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
  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
          intro c d; 

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 : PFunctor.M (P F)
d : Cofix F
⊢ r (Quot.mk Mcongr c) d → Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h₁ d

Next tactic:
apply Quot.inductionOn (motive := _) d
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
          intro c d; apply Quot.inductionOn (motive := _) d; 

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 : PFunctor.M (P F)
d : Cofix F
⊢ ∀ (a : PFunctor.M (P F)),
    r (Quot.mk Mcongr c) (Quot.mk Mcongr a) →
      Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr a)

Next tactic:
clear d
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
          intro c d; apply Quot.inductionOn (motive := _) d; clear d
          

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 : PFunctor.M (P F)
⊢ ∀ (a : PFunctor.M (P F)),
    r (Quot.mk Mcongr c) (Quot.mk Mcongr a) →
      Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr a)

Next tactic:
intro d 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
          intro c; apply Quot.inductionOn (motive := _) c; clear c
          intro c d; apply Quot.inductionOn (motive := _) d; clear d
          intro d rcd; 

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 d : PFunctor.M (P F)
rcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d)
⊢ Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr c) = Quot.lift (Quot.mk r') h₁ (Quot.mk Mcongr d)

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
      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; 

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)
h₁ : ∀ (u v : PFunctor.M (P F)), Mcongr u v → Quot.mk r' u = Quot.mk r' v
c d : PFunctor.M (P F)
rcd : r (Quot.mk Mcongr c) (Quot.mk Mcongr d)
⊢ r' c d

Next tactic:
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
          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)
    

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
f : Quot r → Quot r' :=
  Quot.lift (Quot.lift (Quot.mk r') h₁)
    (_ : ∀ (c b : Cofix F), r c b → Quot.lift (Quot.mk r') h₁ c = Quot.lift (Quot.mk r') h₁ 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 : f ∘ Quot.mk r ∘ Quot.mk Mcongr = Quot.mk r' := rfl
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
          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
    

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
f : Quot r → Quot r' :=
  Quot.lift (Quot.lift (Quot.mk r') h₁)
    (_ : ∀ (c b : Cofix F), r c b → Quot.lift (Quot.mk r') h₁ c = Quot.lift (Quot.mk r') h₁ b)
this : f ∘ Quot.mk r ∘ Quot.mk Mcongr = Quot.mk r'
⊢ 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:
rw [← this, ← PFunctor.map_map _ _ f, ← PFunctor.map_map _ _ (Quot.mk r), abs_map, abs_map,
      abs_map, h₀]
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
          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]
    

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
f : Quot r → Quot r' :=
  Quot.lift (Quot.lift (Quot.mk r') h₁)
    (_ : ∀ (c b : Cofix F), r c b → Quot.lift (Quot.mk r') h₁ c = Quot.lift (Quot.mk r') h₁ b)
this : f ∘ Quot.mk r ∘ Quot.mk Mcongr = Quot.mk r'
⊢ f <$> Quot.mk r <$> Quot.mk Mcongr <$> abs (PFunctor.M.dest b) =
    abs (PFunctor.map (P F) (f ∘ Quot.mk r ∘ Quot.mk Mcongr) (PFunctor.M.dest b))

Next tactic:
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
    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]
  

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)
this : IsPrecongr r'
⊢ Mcongr x y

Next tactic:
refine' ⟨r', this, rxy⟩
theorem Cofix.bisim_rel (r : Cofix F → Cofix F → Prop)
    (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
  

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

Next tactic:
let r' (x y) := x = y ∨ r x y
theorem Cofix.bisim_rel (r : Cofix F → Cofix F → Prop)
    (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
  let r' (x y) := x = y ∨ r x y
  

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

Next tactic:
intro x y rxy
theorem Cofix.bisim_rel (r : Cofix F → Cofix F → Prop)
    (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
  let r' (x y) := x = y ∨ r x y
  intro x y rxy
  

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

Next tactic:
apply Cofix.bisim_aux r'
theorem Cofix.bisim_rel (r : Cofix F → Cofix F → Prop)
    (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
  let r' (x y) := x = y ∨ r x y
  intro x y rxy
  apply Cofix.bisim_aux r'
  · 

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

Next tactic:
intro x
theorem Cofix.bisim_rel (r : Cofix F → Cofix F → Prop)
    (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
  let r' (x y) := x = y ∨ r x y
  intro x y rxy
  apply Cofix.bisim_aux r'
  · intro x
    

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

Next tactic:
left
theorem Cofix.bisim_rel (r : Cofix F → Cofix F → Prop)
    (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
  let r' (x y) := x = y ∨ r x y
  intro x y rxy
  apply Cofix.bisim_aux r'
  · intro x
    left
    

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

Next tactic:
rfl
theorem Cofix.bisim_rel (r : Cofix F → Cofix F → Prop)
    (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
  let r' (x y) := x = y ∨ r x y
  intro x y rxy
  apply Cofix.bisim_aux r'
  · intro x
    left
    rfl
  · 

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

Next tactic:
intro x y r'xy
theorem Cofix.bisim_rel (r : Cofix F → Cofix F → Prop)
    (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
  let r' (x y) := x = y ∨ r x y
  intro x y rxy
  apply Cofix.bisim_aux r'
  · intro x
    left
    rfl
  · intro x y r'xy
    

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

Next tactic:
cases' r'xy with r'xy r'xy
theorem Cofix.bisim_rel (r : Cofix F → Cofix F → Prop)
    (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
  let r' (x y) := x = y ∨ r x y
  intro x y rxy
  apply Cofix.bisim_aux r'
  · intro x
    left
    rfl
  · intro x y r'xy
    cases' r'xy with r'xy r'xy
    · 

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

Next tactic:
rw [r'xy]
theorem Cofix.bisim_rel (r : Cofix F → Cofix F → Prop)
    (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
  let r' (x y) := x = y ∨ r x y
  intro x y rxy
  apply Cofix.bisim_aux r'
  · intro x
    left
    rfl
  · intro x y r'xy
    cases' r'xy with r'xy r'xy
    · rw [r'xy]
    

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

Next tactic:
have : ∀ x y, r x y → r' x y := fun x y h => Or.inr h
theorem Cofix.bisim_rel (r : Cofix F → Cofix F → Prop)
    (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
  let r' (x y) := x = y ∨ r x y
  intro x y rxy
  apply Cofix.bisim_aux r'
  · intro x
    left
    rfl
  · intro x y r'xy
    cases' r'xy with r'xy r'xy
    · rw [r'xy]
    have : ∀ x y, r x y → r' x y := fun x y h => Or.inr h
    

Proof state:
case h.inr
F : Type u → Type u
inst✝ : Functor F
q : QPF F
r : Cofix F → Cofix F → Prop
h : ∀ (x y : Cofix F), r x y → Quot.mk r <$> dest x = Quot.mk r <$> dest y
r' : Cofix F → Cofix F → Prop := fun x y => x = y ∨ r x y
x✝ y✝ : Cofix F
rxy : r x✝ y✝
x y : Cofix F
r'xy : r x y
this : ∀ (x y : Cofix F), r x y → r' x y
⊢ Quot.mk r' <$> dest x = Quot.mk r' <$> dest y

Next tactic:
rw [← Quot.factor_mk_eq _ _ this]