l3lab/ntp-mathlib huggingface dataset:
theorem agree_children {n : ℕ} (x : CofixA F (succ n)) (y : CofixA F (succ n + 1)) {i j}
    (h₀ : HEq i j) (h₁ : Agree x y) : Agree (children' x i) (children' y j) := by
  cases' h₁ with _ _ _ _ _ _ hagree; 

Proof state:
case intro
F : PFunctor.{u}
n : ℕ
a✝ : F.A
x✝ : B F a✝ → CofixA F n
x'✝ : B F a✝ → CofixA F (n + 1)
hagree : ∀ (i : B F a✝), Agree (x✝ i) (x'✝ i)
i : B F (head' (CofixA.intro a✝ x✝))
j : B F (head' (CofixA.intro a✝ x'✝))
h₀ : HEq i j
⊢ Agree (children' (CofixA.intro a✝ x✝) i) (children' (CofixA.intro a✝ x'✝) j)

Next tactic:
cases h₀
theorem agree_children {n : ℕ} (x : CofixA F (succ n)) (y : CofixA F (succ n + 1)) {i j}
    (h₀ : HEq i j) (h₁ : Agree x y) : Agree (children' x i) (children' y j) := by
  cases' h₁ with _ _ _ _ _ _ hagree; cases h₀
  

Proof state:
case intro.refl
F : PFunctor.{u}
n : ℕ
a✝ : F.A
x✝ : B F a✝ → CofixA F n
x'✝ : B F a✝ → CofixA F (n + 1)
hagree : ∀ (i : B F a✝), Agree (x✝ i) (x'✝ i)
i : B F (head' (CofixA.intro a✝ x✝))
⊢ Agree (children' (CofixA.intro a✝ x✝) i) (children' (CofixA.intro a✝ x'✝) i)

Next tactic:
apply hagree
theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n)) (h : Agree x y) :
    truncate y = x := by
  

Proof state:
F : PFunctor.{u}
n : ℕ
x : CofixA F n
y : CofixA F (succ n)
h : Agree x y
⊢ truncate y = x

Next tactic:
induction n
theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n)) (h : Agree x y) :
    truncate y = x := by
  induction n <;> 

Proof state:
case zero
F : PFunctor.{u}
x : CofixA F zero
y : CofixA F (succ zero)
h : Agree x y
⊢ truncate y = x

Next tactic:
cases x
theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n)) (h : Agree x y) :
    truncate y = x := by
  induction n <;> 

Proof state:
case succ
F : PFunctor.{u}
n✝ : ℕ
n_ih✝ : ∀ (x : CofixA F n✝) (y : CofixA F (succ n✝)), Agree x y → truncate y = x
x : CofixA F (succ n✝)
y : CofixA F (succ (succ n✝))
h : Agree x y
⊢ truncate y = x

Next tactic:
cases x
theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n)) (h : Agree x y) :
    truncate y = x := by
  induction n <;> cases x <;> 

Proof state:
case zero.continue
F : PFunctor.{u}
y : CofixA F (succ zero)
h : Agree CofixA.continue y
⊢ truncate y = CofixA.continue

Next tactic:
cases y
theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n)) (h : Agree x y) :
    truncate y = x := by
  induction n <;> cases x <;> 

Proof state:
case succ.intro
F : PFunctor.{u}
n✝ : ℕ
n_ih✝ : ∀ (x : CofixA F n✝) (y : CofixA F (succ n✝)), Agree x y → truncate y = x
y : CofixA F (succ (succ n✝))
a✝¹ : F.A
a✝ : B F a✝¹ → CofixA F n✝
h : Agree (CofixA.intro a✝¹ a✝) y
⊢ truncate y = CofixA.intro a✝¹ a✝

Next tactic:
cases y
theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n)) (h : Agree x y) :
    truncate y = x := by
  induction n <;> cases x <;> cases y
  · 

Proof state:
case zero.continue.intro
F : PFunctor.{u}
a✝¹ : F.A
a✝ : B F a✝¹ → CofixA F 0
h : Agree CofixA.continue (CofixA.intro a✝¹ a✝)
⊢ truncate (CofixA.intro a✝¹ a✝) = CofixA.continue

Next tactic:
rfl
theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n)) (h : Agree x y) :
    truncate y = x := by
  induction n <;> cases x <;> cases y
  · rfl
  · -- cases' h with _ _ _ _ _ h₀ h₁
    

Proof state:
case succ.intro.intro
F : PFunctor.{u}
n✝ : ℕ
n_ih✝ : ∀ (x : CofixA F n✝) (y : CofixA F (succ n✝)), Agree x y → truncate y = x
a✝³ : F.A
a✝² : B F a✝³ → CofixA F n✝
a✝¹ : F.A
a✝ : B F a✝¹ → CofixA F (succ n✝)
h : Agree (CofixA.intro a✝³ a✝²) (CofixA.intro a✝¹ a✝)
⊢ truncate (CofixA.intro a✝¹ a✝) = CofixA.intro a✝³ a✝²

Next tactic:
cases h
theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n)) (h : Agree x y) :
    truncate y = x := by
  induction n <;> cases x <;> cases y
  · rfl
  · -- cases' h with _ _ _ _ _ h₀ h₁
    cases h
    

Proof state:
case succ.intro.intro.intro
F : PFunctor.{u}
n✝ : ℕ
n_ih✝ : ∀ (x : CofixA F n✝) (y : CofixA F (succ n✝)), Agree x y → truncate y = x
a✝² : F.A
a✝¹ : B F a✝² → CofixA F n✝
x'✝ : B F a✝² → CofixA F (n✝ + 1)
a✝ : ∀ (i : B F a✝²), Agree (a✝¹ i) (x'✝ i)
⊢ truncate (CofixA.intro a✝² x'✝) = CofixA.intro a✝² a✝¹

Next tactic:
simp only [truncate, Function.comp, true_and_iff, eq_self_iff_true, heq_iff_eq]
theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n)) (h : Agree x y) :
    truncate y = x := by
  induction n <;> cases x <;> cases y
  · rfl
  · -- cases' h with _ _ _ _ _ h₀ h₁
    cases h
    simp only [truncate, Function.comp, true_and_iff, eq_self_iff_true, heq_iff_eq]
    -- porting note: used to be `ext y`
    

Proof state:
case succ.intro.intro.intro
F : PFunctor.{u}
n✝ : ℕ
n_ih✝ : ∀ (x : CofixA F n✝) (y : CofixA F (succ n✝)), Agree x y → truncate y = x
a✝² : F.A
a✝¹ : B F a✝² → CofixA F n✝
x'✝ : B F a✝² → CofixA F (n✝ + 1)
a✝ : ∀ (i : B F a✝²), Agree (a✝¹ i) (x'✝ i)
⊢ (CofixA.intro a✝² fun x => truncate (x'✝ x)) = CofixA.intro a✝² a✝¹

Next tactic:
rename_i n_ih a f y h₁
theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n)) (h : Agree x y) :
    truncate y = x := by
  induction n <;> cases x <;> cases y
  · rfl
  · -- cases' h with _ _ _ _ _ h₀ h₁
    cases h
    simp only [truncate, Function.comp, true_and_iff, eq_self_iff_true, heq_iff_eq]
    -- porting note: used to be `ext y`
    rename_i n_ih a f y h₁
    

Proof state:
case succ.intro.intro.intro
F : PFunctor.{u}
n✝ : ℕ
n_ih : ∀ (x : CofixA F n✝) (y : CofixA F (succ n✝)), Agree x y → truncate y = x
a : F.A
f : B F a → CofixA F n✝
y : B F a → CofixA F (n✝ + 1)
h₁ : ∀ (i : B F a), Agree (f i) (y i)
⊢ (CofixA.intro a fun x => truncate (y x)) = CofixA.intro a f

Next tactic:
suffices (fun x => truncate (y x)) = f
      by simp [this]; try (exact HEq.rfl;)
theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n)) (h : Agree x y) :
    truncate y = x := by
  induction n <;> cases x <;> cases y
  · rfl
  · -- cases' h with _ _ _ _ _ h₀ h₁
    cases h
    simp only [truncate, Function.comp, true_and_iff, eq_self_iff_true, heq_iff_eq]
    -- porting note: used to be `ext y`
    rename_i n_ih a f y h₁
    suffices (fun x => truncate (y x)) = f
      by 

Proof state:
F : PFunctor.{u}
n✝ : ℕ
n_ih : ∀ (x : CofixA F n✝) (y : CofixA F (succ n✝)), Agree x y → truncate y = x
a : F.A
f : B F a → CofixA F n✝
y : B F a → CofixA F (n✝ + 1)
h₁ : ∀ (i : B F a), Agree (f i) (y i)
this : (fun x => truncate (y x)) = f
⊢ (CofixA.intro a fun x => truncate (y x)) = CofixA.intro a f

Next tactic:
simp [this]
theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n)) (h : Agree x y) :
    truncate y = x := by
  induction n <;> cases x <;> cases y
  · rfl
  · -- cases' h with _ _ _ _ _ h₀ h₁
    cases h
    simp only [truncate, Function.comp, true_and_iff, eq_self_iff_true, heq_iff_eq]
    -- porting note: used to be `ext y`
    rename_i n_ih a f y h₁
    suffices (fun x => truncate (y x)) = f
      by simp [this]; 

Proof state:

Next tactic:
try (exact HEq.rfl;)
theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n)) (h : Agree x y) :
    truncate y = x := by
  induction n <;> cases x <;> cases y
  · rfl
  · -- cases' h with _ _ _ _ _ h₀ h₁
    cases h
    simp only [truncate, Function.comp, true_and_iff, eq_self_iff_true, heq_iff_eq]
    -- porting note: used to be `ext y`
    rename_i n_ih a f y h₁
    suffices (fun x => truncate (y x)) = f
      by simp [this]; try (

Proof state:

Next tactic:
exact HEq.rfl
theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n)) (h : Agree x y) :
    truncate y = x := by
  induction n <;> cases x <;> cases y
  · rfl
  · -- cases' h with _ _ _ _ _ h₀ h₁
    cases h
    simp only [truncate, Function.comp, true_and_iff, eq_self_iff_true, heq_iff_eq]
    -- porting note: used to be `ext y`
    rename_i n_ih a f y h₁
    suffices (fun x => truncate (y x)) = f
      by simp [this]; try (exact HEq.rfl;)
    

Proof state:
case succ.intro.intro.intro
F : PFunctor.{u}
n✝ : ℕ
n_ih : ∀ (x : CofixA F n✝) (y : CofixA F (succ n✝)), Agree x y → truncate y = x
a : F.A
f : B F a → CofixA F n✝
y : B F a → CofixA F (n✝ + 1)
h₁ : ∀ (i : B F a), Agree (f i) (y i)
⊢ (fun x => truncate (y x)) = f

Next tactic:
funext y
theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n)) (h : Agree x y) :
    truncate y = x := by
  induction n <;> cases x <;> cases y
  · rfl
  · -- cases' h with _ _ _ _ _ h₀ h₁
    cases h
    simp only [truncate, Function.comp, true_and_iff, eq_self_iff_true, heq_iff_eq]
    -- porting note: used to be `ext y`
    rename_i n_ih a f y h₁
    suffices (fun x => truncate (y x)) = f
      by simp [this]; try (exact HEq.rfl;)
    funext y

    

Proof state:
case succ.intro.intro.intro.h
F : PFunctor.{u}
n✝ : ℕ
n_ih : ∀ (x : CofixA F n✝) (y : CofixA F (succ n✝)), Agree x y → truncate y = x
a : F.A
f : B F a → CofixA F n✝
y✝ : B F a → CofixA F (n✝ + 1)
h₁ : ∀ (i : B F a), Agree (f i) (y✝ i)
y : B F a
⊢ truncate (y✝ y) = f y

Next tactic:
apply n_ih
theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n)) (h : Agree x y) :
    truncate y = x := by
  induction n <;> cases x <;> cases y
  · rfl
  · -- cases' h with _ _ _ _ _ h₀ h₁
    cases h
    simp only [truncate, Function.comp, true_and_iff, eq_self_iff_true, heq_iff_eq]
    -- porting note: used to be `ext y`
    rename_i n_ih a f y h₁
    suffices (fun x => truncate (y x)) = f
      by simp [this]; try (exact HEq.rfl;)
    funext y

    apply n_ih
    

Proof state:
case succ.intro.intro.intro.h.h
F : PFunctor.{u}
n✝ : ℕ
n_ih : ∀ (x : CofixA F n✝) (y : CofixA F (succ n✝)), Agree x y → truncate y = x
a : F.A
f : B F a → CofixA F n✝
y✝ : B F a → CofixA F (n✝ + 1)
h₁ : ∀ (i : B F a), Agree (f i) (y✝ i)
y : B F a
⊢ Agree (f y) (y✝ y)

Next tactic:
apply h₁
theorem P_corec (i : X) (n : ℕ) : Agree (sCorec f i n) (sCorec f i (succ n)) := by
  

Proof state:
F : PFunctor.{u}
X : Type w
f : X → ↑F X
i : X
n : ℕ
⊢ Agree (sCorec f i n) (sCorec f i (succ n))

Next tactic:
induction' n with n n_ih generalizing i
theorem P_corec (i : X) (n : ℕ) : Agree (sCorec f i n) (sCorec f i (succ n)) := by
  induction' n with n n_ih generalizing i
  

Proof state:
case zero
F : PFunctor.{u}
X : Type w
f : X → ↑F X
i : X
⊢ Agree (sCorec f i zero) (sCorec f i (succ zero))
case succ
F : PFunctor.{u}
X : Type w
f : X → ↑F X
n : ℕ
n_ih : ∀ (i : X), Agree (sCorec f i n) (sCorec f i (succ n))
i : X
⊢ Agree (sCorec f i (succ n)) (sCorec f i (succ (succ n)))

Next tactic:
constructor
theorem P_corec (i : X) (n : ℕ) : Agree (sCorec f i n) (sCorec f i (succ n)) := by
  induction' n with n n_ih generalizing i
  constructor
  

Proof state:
case succ
F : PFunctor.{u}
X : Type w
f : X → ↑F X
n : ℕ
n_ih : ∀ (i : X), Agree (sCorec f i n) (sCorec f i (succ n))
i : X
⊢ Agree (sCorec f i (succ n)) (sCorec f i (succ (succ n)))

Next tactic:
cases' h : f i with y g
theorem P_corec (i : X) (n : ℕ) : Agree (sCorec f i n) (sCorec f i (succ n)) := by
  induction' n with n n_ih generalizing i
  constructor
  cases' h : f i with y g
  

Proof state:
case succ.mk
F : PFunctor.{u}
X : Type w
f : X → ↑F X
n : ℕ
n_ih : ∀ (i : X), Agree (sCorec f i n) (sCorec f i (succ n))
i : X
y : F.A
g : B F y → X
h : f i = { fst := y, snd := g }
⊢ Agree (sCorec f i (succ n)) (sCorec f i (succ (succ n)))

Next tactic:
constructor
theorem P_corec (i : X) (n : ℕ) : Agree (sCorec f i n) (sCorec f i (succ n)) := by
  induction' n with n n_ih generalizing i
  constructor
  cases' h : f i with y g
  constructor
  

Proof state:
case succ.mk.a
F : PFunctor.{u}
X : Type w
f : X → ↑F X
n : ℕ
n_ih : ∀ (i : X), Agree (sCorec f i n) (sCorec f i (succ n))
i : X
y : F.A
g : B F y → X
h : f i = { fst := y, snd := g }
⊢ ∀ (i_1 : B F (f i).fst), Agree (sCorec f (Sigma.snd (f i) i_1) n) (sCorec f (Sigma.snd (f i) i_1) (succ n))

Next tactic:
introv
theorem P_corec (i : X) (n : ℕ) : Agree (sCorec f i n) (sCorec f i (succ n)) := by
  induction' n with n n_ih generalizing i
  constructor
  cases' h : f i with y g
  constructor
  introv
  

Proof state:
case succ.mk.a
F : PFunctor.{u}
X : Type w
f : X → ↑F X
n : ℕ
n_ih : ∀ (i : X), Agree (sCorec f i n) (sCorec f i (succ n))
i✝ : X
y : F.A
g : B F y → X
h : f i✝ = { fst := y, snd := g }
i : B F (f i✝).fst
⊢ Agree (sCorec f (Sigma.snd (f i✝) i) n) (sCorec f (Sigma.snd (f i✝) i) (succ n))

Next tactic:
apply n_ih
instance CofixA.instSubsingleton : Subsingleton (CofixA F 0) :=
  ⟨by 

Proof state:
F : PFunctor.{u}
X : Type w
f : X → ↑F X
⊢ ∀ (a b : CofixA F 0), a = b

Next tactic:
rintro ⟨⟩ ⟨⟩
instance CofixA.instSubsingleton : Subsingleton (CofixA F 0) :=
  ⟨by rintro ⟨⟩ ⟨⟩; 

Proof state:
case continue.continue
F : PFunctor.{u}
X : Type w
f : X → ↑F X
⊢ continue = continue

Next tactic:
rfl
theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
    head' (x (succ n)) = head' (x (succ m)) := by
  

Proof state:
F : PFunctor.{u}
X : Type w
f : X → ↑F X
n m : ℕ
x : (n : ℕ) → CofixA F n
Hconsistent : AllAgree x
⊢ head' (x (succ n)) = head' (x (succ m))

Next tactic:
suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
    head' (x (succ n)) = head' (x (succ m)) := by
  suffices ∀ n, head' (x (succ n)) = head' (x 1) by 

Proof state:
F : PFunctor.{u}
X : Type w
f : X → ↑F X
n m : ℕ
x : (n : ℕ) → CofixA F n
Hconsistent : AllAgree x
this : ∀ (n : ℕ), head' (x (succ n)) = head' (x 1)
⊢ head' (x (succ n)) = head' (x (succ m))

Next tactic:
simp [this]
theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
    head' (x (succ n)) = head' (x (succ m)) := by
  suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
  

Proof state:
F : PFunctor.{u}
X : Type w
f : X → ↑F X
n m : ℕ
x : (n : ℕ) → CofixA F n
Hconsistent : AllAgree x
⊢ ∀ (n : ℕ), head' (x (succ n)) = head' (x 1)

Next tactic:
clear m n
theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
    head' (x (succ n)) = head' (x (succ m)) := by
  suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
  clear m n
  

Proof state:
F : PFunctor.{u}
X : Type w
f : X → ↑F X
x : (n : ℕ) → CofixA F n
Hconsistent : AllAgree x
⊢ ∀ (n : ℕ), head' (x (succ n)) = head' (x 1)

Next tactic:
intro n
theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
    head' (x (succ n)) = head' (x (succ m)) := by
  suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
  clear m n
  intro n
  

Proof state:
F : PFunctor.{u}
X : Type w
f : X → ↑F X
x : (n : ℕ) → CofixA F n
Hconsistent : AllAgree x
n : ℕ
⊢ head' (x (succ n)) = head' (x 1)

Next tactic:
cases' h₀ : x (succ n) with _ i₀ f₀
theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
    head' (x (succ n)) = head' (x (succ m)) := by
  suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
  clear m n
  intro n
  cases' h₀ : x (succ n) with _ i₀ f₀
  

Proof state:
case intro
F : PFunctor.{u}
X : Type w
f : X → ↑F X
x : (n : ℕ) → CofixA F n
Hconsistent : AllAgree x
n : ℕ
i₀ : F.A
f₀ : B F i₀ → CofixA F n
h₀ : x (succ n) = CofixA.intro i₀ f₀
⊢ head' (CofixA.intro i₀ f₀) = head' (x 1)

Next tactic:
cases' h₁ : x 1 with _ i₁ f₁
theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
    head' (x (succ n)) = head' (x (succ m)) := by
  suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
  clear m n
  intro n
  cases' h₀ : x (succ n) with _ i₀ f₀
  cases' h₁ : x 1 with _ i₁ f₁
  

Proof state:
case intro.intro
F : PFunctor.{u}
X : Type w
f : X → ↑F X
x : (n : ℕ) → CofixA F n
Hconsistent : AllAgree x
n : ℕ
i₀ : F.A
f₀ : B F i₀ → CofixA F n
h₀ : x (succ n) = CofixA.intro i₀ f₀
i₁ : F.A
f₁ : B F i₁ → CofixA F 0
h₁ : x 1 = CofixA.intro i₁ f₁
⊢ head' (CofixA.intro i₀ f₀) = head' (CofixA.intro i₁ f₁)

Next tactic:
dsimp only [head']
theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
    head' (x (succ n)) = head' (x (succ m)) := by
  suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
  clear m n
  intro n
  cases' h₀ : x (succ n) with _ i₀ f₀
  cases' h₁ : x 1 with _ i₁ f₁
  dsimp only [head']
  

Proof state:
case intro.intro
F : PFunctor.{u}
X : Type w
f : X → ↑F X
x : (n : ℕ) → CofixA F n
Hconsistent : AllAgree x
n : ℕ
i₀ : F.A
f₀ : B F i₀ → CofixA F n
h₀ : x (succ n) = CofixA.intro i₀ f₀
i₁ : F.A
f₁ : B F i₁ → CofixA F 0
h₁ : x 1 = CofixA.intro i₁ f₁
⊢ i₀ = i₁

Next tactic:
induction' n with n n_ih
theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
    head' (x (succ n)) = head' (x (succ m)) := by
  suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
  clear m n
  intro n
  cases' h₀ : x (succ n) with _ i₀ f₀
  cases' h₁ : x 1 with _ i₁ f₁
  dsimp only [head']
  induction' n with n n_ih
  · 

Proof state:
case intro.intro.zero
F : PFunctor.{u}
X : Type w
f : X → ↑F X
x : (n : ℕ) → CofixA F n
Hconsistent : AllAgree x
i₀ i₁ : F.A
f₁ : B F i₁ → CofixA F 0
h₁ : x 1 = CofixA.intro i₁ f₁
f₀ : B F i₀ → CofixA F zero
h₀ : x (succ zero) = CofixA.intro i₀ f₀
⊢ i₀ = i₁

Next tactic:
rw [h₁] at h₀
theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
    head' (x (succ n)) = head' (x (succ m)) := by
  suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
  clear m n
  intro n
  cases' h₀ : x (succ n) with _ i₀ f₀
  cases' h₁ : x 1 with _ i₁ f₁
  dsimp only [head']
  induction' n with n n_ih
  · rw [h₁] at h₀
    

Proof state:
case intro.intro.zero
F : PFunctor.{u}
X : Type w
f : X → ↑F X
x : (n : ℕ) → CofixA F n
Hconsistent : AllAgree x
i₀ i₁ : F.A
f₁ : B F i₁ → CofixA F 0
h₁ : x 1 = CofixA.intro i₁ f₁
f₀ : B F i₀ → CofixA F zero
h₀ : CofixA.intro i₁ f₁ = CofixA.intro i₀ f₀
⊢ i₀ = i₁

Next tactic:
cases h₀
theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
    head' (x (succ n)) = head' (x (succ m)) := by
  suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
  clear m n
  intro n
  cases' h₀ : x (succ n) with _ i₀ f₀
  cases' h₁ : x 1 with _ i₁ f₁
  dsimp only [head']
  induction' n with n n_ih
  · rw [h₁] at h₀
    cases h₀
    

Proof state:
case intro.intro.zero.refl
F : PFunctor.{u}
X : Type w
f : X → ↑F X
x : (n : ℕ) → CofixA F n
Hconsistent : AllAgree x
i₀ : F.A
f₀ : B F i₀ → CofixA F zero
h₁ : x 1 = CofixA.intro i₀ f₀
⊢ i₀ = i₀

Next tactic:
trivial
theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
    head' (x (succ n)) = head' (x (succ m)) := by
  suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
  clear m n
  intro n
  cases' h₀ : x (succ n) with _ i₀ f₀
  cases' h₁ : x 1 with _ i₁ f₁
  dsimp only [head']
  induction' n with n n_ih
  · rw [h₁] at h₀
    cases h₀
    trivial
  · 

Proof state:
case intro.intro.succ
F : PFunctor.{u}
X : Type w
f : X → ↑F X
x : (n : ℕ) → CofixA F n
Hconsistent : AllAgree x
i₀ i₁ : F.A
f₁ : B F i₁ → CofixA F 0
h₁ : x 1 = CofixA.intro i₁ f₁
n : ℕ
n_ih : ∀ (f₀ : B F i₀ → CofixA F n), x (succ n) = CofixA.intro i₀ f₀ → i₀ = i₁
f₀ : B F i₀ → CofixA F (succ n)
h₀ : x (succ (succ n)) = CofixA.intro i₀ f₀
⊢ i₀ = i₁

Next tactic:
have H := Hconsistent (succ n)
theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
    head' (x (succ n)) = head' (x (succ m)) := by
  suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
  clear m n
  intro n
  cases' h₀ : x (succ n) with _ i₀ f₀
  cases' h₁ : x 1 with _ i₁ f₁
  dsimp only [head']
  induction' n with n n_ih
  · rw [h₁] at h₀
    cases h₀
    trivial
  · have H := Hconsistent (succ n)
    

Proof state:
case intro.intro.succ
F : PFunctor.{u}
X : Type w
f : X → ↑F X
x : (n : ℕ) → CofixA F n
Hconsistent : AllAgree x
i₀ i₁ : F.A
f₁ : B F i₁ → CofixA F 0
h₁ : x 1 = CofixA.intro i₁ f₁
n : ℕ
n_ih : ∀ (f₀ : B F i₀ → CofixA F n), x (succ n) = CofixA.intro i₀ f₀ → i₀ = i₁
f₀ : B F i₀ → CofixA F (succ n)
h₀ : x (succ (succ n)) = CofixA.intro i₀ f₀
H : Agree (x (succ n)) (x (succ (succ n)))
⊢ i₀ = i₁

Next tactic:
cases' h₂ : x (succ n) with _ i₂ f₂
theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
    head' (x (succ n)) = head' (x (succ m)) := by
  suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
  clear m n
  intro n
  cases' h₀ : x (succ n) with _ i₀ f₀
  cases' h₁ : x 1 with _ i₁ f₁
  dsimp only [head']
  induction' n with n n_ih
  · rw [h₁] at h₀
    cases h₀
    trivial
  · have H := Hconsistent (succ n)
    cases' h₂ : x (succ n) with _ i₂ f₂
    

Proof state:
case intro.intro.succ.intro
F : PFunctor.{u}
X : Type w
f : X → ↑F X
x : (n : ℕ) → CofixA F n
Hconsistent : AllAgree x
i₀ i₁ : F.A
f₁ : B F i₁ → CofixA F 0
h₁ : x 1 = CofixA.intro i₁ f₁
n : ℕ
n_ih : ∀ (f₀ : B F i₀ → CofixA F n), x (succ n) = CofixA.intro i₀ f₀ → i₀ = i₁
f₀ : B F i₀ → CofixA F (succ n)
h₀ : x (succ (succ n)) = CofixA.intro i₀ f₀
H : Agree (x (succ n)) (x (succ (succ n)))
i₂ : F.A
f₂ : B F i₂ → CofixA F n
h₂ : x (succ n) = CofixA.intro i₂ f₂
⊢ i₀ = i₁

Next tactic:
rw [h₀, h₂] at H
theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
    head' (x (succ n)) = head' (x (succ m)) := by
  suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
  clear m n
  intro n
  cases' h₀ : x (succ n) with _ i₀ f₀
  cases' h₁ : x 1 with _ i₁ f₁
  dsimp only [head']
  induction' n with n n_ih
  · rw [h₁] at h₀
    cases h₀
    trivial
  · have H := Hconsistent (succ n)
    cases' h₂ : x (succ n) with _ i₂ f₂
    rw [h₀, h₂] at H
    

Proof state:
case intro.intro.succ.intro
F : PFunctor.{u}
X : Type w
f : X → ↑F X
x : (n : ℕ) → CofixA F n
Hconsistent : AllAgree x
i₀ i₁ : F.A
f₁ : B F i₁ → CofixA F 0
h₁ : x 1 = CofixA.intro i₁ f₁
n : ℕ
n_ih : ∀ (f₀ : B F i₀ → CofixA F n), x (succ n) = CofixA.intro i₀ f₀ → i₀ = i₁
f₀ : B F i₀ → CofixA F (succ n)
h₀ : x (succ (succ n)) = CofixA.intro i₀ f₀
i₂ : F.A
f₂ : B F i₂ → CofixA F n
H : Agree (CofixA.intro i₂ f₂) (CofixA.intro i₀ f₀)
h₂ : x (succ n) = CofixA.intro i₂ f₂
⊢ i₀ = i₁

Next tactic:
apply n_ih (truncate ∘ f₀)
theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
    head' (x (succ n)) = head' (x (succ m)) := by
  suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
  clear m n
  intro n
  cases' h₀ : x (succ n) with _ i₀ f₀
  cases' h₁ : x 1 with _ i₁ f₁
  dsimp only [head']
  induction' n with n n_ih
  · rw [h₁] at h₀
    cases h₀
    trivial
  · have H := Hconsistent (succ n)
    cases' h₂ : x (succ n) with _ i₂ f₂
    rw [h₀, h₂] at H
    apply n_ih (truncate ∘ f₀)
    

Proof state:
case intro.intro.succ.intro
F : PFunctor.{u}
X : Type w
f : X → ↑F X
x : (n : ℕ) → CofixA F n
Hconsistent : AllAgree x
i₀ i₁ : F.A
f₁ : B F i₁ → CofixA F 0
h₁ : x 1 = CofixA.intro i₁ f₁
n : ℕ
n_ih : ∀ (f₀ : B F i₀ → CofixA F n), x (succ n) = CofixA.intro i₀ f₀ → i₀ = i₁
f₀ : B F i₀ → CofixA F (succ n)
h₀ : x (succ (succ n)) = CofixA.intro i₀ f₀
i₂ : F.A
f₂ : B F i₂ → CofixA F n
H : Agree (CofixA.intro i₂ f₂) (CofixA.intro i₀ f₀)
h₂ : x (succ n) = CofixA.intro i₂ f₂
⊢ x (succ n) = CofixA.intro i₀ (truncate ∘ f₀)

Next tactic:
rw [h₂]
theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
    head' (x (succ n)) = head' (x (succ m)) := by
  suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
  clear m n
  intro n
  cases' h₀ : x (succ n) with _ i₀ f₀
  cases' h₁ : x 1 with _ i₁ f₁
  dsimp only [head']
  induction' n with n n_ih
  · rw [h₁] at h₀
    cases h₀
    trivial
  · have H := Hconsistent (succ n)
    cases' h₂ : x (succ n) with _ i₂ f₂
    rw [h₀, h₂] at H
    apply n_ih (truncate ∘ f₀)
    rw [h₂]
    

Proof state:
case intro.intro.succ.intro
F : PFunctor.{u}
X : Type w
f : X → ↑F X
x : (n : ℕ) → CofixA F n
Hconsistent : AllAgree x
i₀ i₁ : F.A
f₁ : B F i₁ → CofixA F 0
h₁ : x 1 = CofixA.intro i₁ f₁
n : ℕ
n_ih : ∀ (f₀ : B F i₀ → CofixA F n), x (succ n) = CofixA.intro i₀ f₀ → i₀ = i₁
f₀ : B F i₀ → CofixA F (succ n)
h₀ : x (succ (succ n)) = CofixA.intro i₀ f₀
i₂ : F.A
f₂ : B F i₂ → CofixA F n
H : Agree (CofixA.intro i₂ f₂) (CofixA.intro i₀ f₀)
h₂ : x (succ n) = CofixA.intro i₂ f₂
⊢ CofixA.intro i₂ f₂ = CofixA.intro i₀ (truncate ∘ f₀)

Next tactic:
cases' H with _ _ _ _ _ _ hagree