l3lab/ntp-mathlib huggingface dataset:
theorem tail_even (s : Stream' α) : tail (even s) = even (tail (tail s)) := by
  unfold even
  rw [corec_eq]
  

Proof state:
α : Type u
β : Type v
δ : Type w
s : Stream' α
⊢ tail (head s :: corec (fun s => head s) (fun s => tail (tail s)) (tail (tail s))) =
    corec (fun s => head s) (fun s => tail (tail s)) (tail (tail s))

Next tactic:
rfl
theorem even_cons_cons (a₁ a₂ : α) (s : Stream' α) : even (a₁::a₂::s) = a₁::even s := by
  

Proof state:
α : Type u
β : Type v
δ : Type w
a₁ a₂ : α
s : Stream' α
⊢ even (a₁ :: a₂ :: s) = a₁ :: even s

Next tactic:
unfold even
theorem even_cons_cons (a₁ a₂ : α) (s : Stream' α) : even (a₁::a₂::s) = a₁::even s := by
  unfold even
  

Proof state:
α : Type u
β : Type v
δ : Type w
a₁ a₂ : α
s : Stream' α
⊢ corec (fun s => head s) (fun s => tail (tail s)) (a₁ :: a₂ :: s) =
    a₁ :: corec (fun s => head s) (fun s => tail (tail s)) s

Next tactic:
rw [corec_eq]
theorem even_cons_cons (a₁ a₂ : α) (s : Stream' α) : even (a₁::a₂::s) = a₁::even s := by
  unfold even
  rw [corec_eq]; 

Proof state:
α : Type u
β : Type v
δ : Type w
a₁ a₂ : α
s : Stream' α
⊢ head (a₁ :: a₂ :: s) :: corec (fun s => head s) (fun s => tail (tail s)) (tail (tail (a₁ :: a₂ :: s))) =
    a₁ :: corec (fun s => head s) (fun s => tail (tail s)) s

Next tactic:
rfl
theorem even_interleave (s₁ s₂ : Stream' α) : even (s₁ ⋈ s₂) = s₁ :=
  eq_of_bisim (fun s₁' s₁ => ∃ s₂, s₁' = even (s₁ ⋈ s₂))
    (fun s₁' s₁ ⟨s₂, h₁⟩ => by
      

Proof state:
α : Type u
β : Type v
δ : Type w
s₁✝ s₂✝ s₁' s₁ : Stream' α
x✝ : (fun s₁' s₁ => ∃ s₂, s₁' = even (s₁ ⋈ s₂)) s₁' s₁
s₂ : Stream' α
h₁ : s₁' = even (s₁ ⋈ s₂)
⊢ head s₁' = head s₁ ∧ (fun s₁' s₁ => ∃ s₂, s₁' = even (s₁ ⋈ s₂)) (tail s₁') (tail s₁)

Next tactic:
rw [h₁]
theorem even_interleave (s₁ s₂ : Stream' α) : even (s₁ ⋈ s₂) = s₁ :=
  eq_of_bisim (fun s₁' s₁ => ∃ s₂, s₁' = even (s₁ ⋈ s₂))
    (fun s₁' s₁ ⟨s₂, h₁⟩ => by
      rw [h₁]
      

Proof state:
α : Type u
β : Type v
δ : Type w
s₁✝ s₂✝ s₁' s₁ : Stream' α
x✝ : (fun s₁' s₁ => ∃ s₂, s₁' = even (s₁ ⋈ s₂)) s₁' s₁
s₂ : Stream' α
h₁ : s₁' = even (s₁ ⋈ s₂)
⊢ head (even (s₁ ⋈ s₂)) = head s₁ ∧ (fun s₁' s₁ => ∃ s₂, s₁' = even (s₁ ⋈ s₂)) (tail (even (s₁ ⋈ s₂))) (tail s₁)

Next tactic:
constructor
theorem even_interleave (s₁ s₂ : Stream' α) : even (s₁ ⋈ s₂) = s₁ :=
  eq_of_bisim (fun s₁' s₁ => ∃ s₂, s₁' = even (s₁ ⋈ s₂))
    (fun s₁' s₁ ⟨s₂, h₁⟩ => by
      rw [h₁]
      constructor
      · 

Proof state:
case left
α : Type u
β : Type v
δ : Type w
s₁✝ s₂✝ s₁' s₁ : Stream' α
x✝ : (fun s₁' s₁ => ∃ s₂, s₁' = even (s₁ ⋈ s₂)) s₁' s₁
s₂ : Stream' α
h₁ : s₁' = even (s₁ ⋈ s₂)
⊢ head (even (s₁ ⋈ s₂)) = head s₁

Next tactic:
rfl
theorem even_interleave (s₁ s₂ : Stream' α) : even (s₁ ⋈ s₂) = s₁ :=
  eq_of_bisim (fun s₁' s₁ => ∃ s₂, s₁' = even (s₁ ⋈ s₂))
    (fun s₁' s₁ ⟨s₂, h₁⟩ => by
      rw [h₁]
      constructor
      · rfl
      · 

Proof state:
case right
α : Type u
β : Type v
δ : Type w
s₁✝ s₂✝ s₁' s₁ : Stream' α
x✝ : (fun s₁' s₁ => ∃ s₂, s₁' = even (s₁ ⋈ s₂)) s₁' s₁
s₂ : Stream' α
h₁ : s₁' = even (s₁ ⋈ s₂)
⊢ (fun s₁' s₁ => ∃ s₂, s₁' = even (s₁ ⋈ s₂)) (tail (even (s₁ ⋈ s₂))) (tail s₁)

Next tactic:
exact ⟨tail s₂, by rw [interleave_eq, even_cons_cons, tail_cons]⟩
theorem even_interleave (s₁ s₂ : Stream' α) : even (s₁ ⋈ s₂) = s₁ :=
  eq_of_bisim (fun s₁' s₁ => ∃ s₂, s₁' = even (s₁ ⋈ s₂))
    (fun s₁' s₁ ⟨s₂, h₁⟩ => by
      rw [h₁]
      constructor
      · rfl
      · exact ⟨tail s₂, by 

Proof state:
α : Type u
β : Type v
δ : Type w
s₁✝ s₂✝ s₁' s₁ : Stream' α
x✝ : (fun s₁' s₁ => ∃ s₂, s₁' = even (s₁ ⋈ s₂)) s₁' s₁
s₂ : Stream' α
h₁ : s₁' = even (s₁ ⋈ s₂)
⊢ tail (even (s₁ ⋈ s₂)) = even (tail s₁ ⋈ tail s₂)

Next tactic:
rw [interleave_eq, even_cons_cons, tail_cons]
theorem interleave_even_odd (s₁ : Stream' α) : even s₁ ⋈ odd s₁ = s₁ :=
  eq_of_bisim (fun s' s => s' = even s ⋈ odd s)
    (fun s' s (h : s' = even s ⋈ odd s) => by
      

Proof state:
α : Type u
β : Type v
δ : Type w
s₁ s' s : Stream' α
h : s' = even s ⋈ odd s
⊢ head s' = head s ∧ (fun s' s => s' = even s ⋈ odd s) (tail s') (tail s)

Next tactic:
rw [h]
theorem interleave_even_odd (s₁ : Stream' α) : even s₁ ⋈ odd s₁ = s₁ :=
  eq_of_bisim (fun s' s => s' = even s ⋈ odd s)
    (fun s' s (h : s' = even s ⋈ odd s) => by
      rw [h]; 

Proof state:
α : Type u
β : Type v
δ : Type w
s₁ s' s : Stream' α
h : s' = even s ⋈ odd s
⊢ head (even s ⋈ odd s) = head s ∧ (fun s' s => s' = even s ⋈ odd s) (tail (even s ⋈ odd s)) (tail s)

Next tactic:
constructor
theorem interleave_even_odd (s₁ : Stream' α) : even s₁ ⋈ odd s₁ = s₁ :=
  eq_of_bisim (fun s' s => s' = even s ⋈ odd s)
    (fun s' s (h : s' = even s ⋈ odd s) => by
      rw [h]; constructor
      · 

Proof state:
case left
α : Type u
β : Type v
δ : Type w
s₁ s' s : Stream' α
h : s' = even s ⋈ odd s
⊢ head (even s ⋈ odd s) = head s

Next tactic:
rfl
theorem interleave_even_odd (s₁ : Stream' α) : even s₁ ⋈ odd s₁ = s₁ :=
  eq_of_bisim (fun s' s => s' = even s ⋈ odd s)
    (fun s' s (h : s' = even s ⋈ odd s) => by
      rw [h]; constructor
      · rfl
      · 

Proof state:
case right
α : Type u
β : Type v
δ : Type w
s₁ s' s : Stream' α
h : s' = even s ⋈ odd s
⊢ (fun s' s => s' = even s ⋈ odd s) (tail (even s ⋈ odd s)) (tail s)

Next tactic:
simp [odd_eq, odd_eq, tail_interleave, tail_even]
theorem get_even : ∀ (n : Nat) (s : Stream' α), get (even s) n = get s (2 * n)
  | 0, s => rfl
  | succ n, s => by
    

Proof state:
α : Type u
β : Type v
δ : Type w
n : ℕ
s : Stream' α
⊢ get (even s) (succ n) = get s (2 * succ n)

Next tactic:
change get (even s) (succ n) = get s (succ (succ (2 * n)))
theorem get_even : ∀ (n : Nat) (s : Stream' α), get (even s) n = get s (2 * n)
  | 0, s => rfl
  | succ n, s => by
    change get (even s) (succ n) = get s (succ (succ (2 * n)))
    

Proof state:
α : Type u
β : Type v
δ : Type w
n : ℕ
s : Stream' α
⊢ get (even s) (succ n) = get s (succ (succ (2 * n)))

Next tactic:
rw [get_succ, get_succ, tail_even, get_even n]
theorem get_even : ∀ (n : Nat) (s : Stream' α), get (even s) n = get s (2 * n)
  | 0, s => rfl
  | succ n, s => by
    change get (even s) (succ n) = get s (succ (succ (2 * n)))
    rw [get_succ, get_succ, tail_even, get_even n]; 

Proof state:
α : Type u
β : Type v
δ : Type w
n : ℕ
s : Stream' α
⊢ get (tail (tail s)) (2 * n) = get (tail s) (2 * n + 1)

Next tactic:
rfl
theorem get_odd : ∀ (n : Nat) (s : Stream' α), get (odd s) n = get s (2 * n + 1) := fun n s => by
  

Proof state:
α : Type u
β : Type v
δ : Type w
n : ℕ
s : Stream' α
⊢ get (odd s) n = get s (2 * n + 1)

Next tactic:
rw [odd_eq, get_even]
theorem get_odd : ∀ (n : Nat) (s : Stream' α), get (odd s) n = get s (2 * n + 1) := fun n s => by
  rw [odd_eq, get_even]; 

Proof state:
α : Type u
β : Type v
δ : Type w
n : ℕ
s : Stream' α
⊢ get (tail s) (2 * n) = get s (2 * n + 1)

Next tactic:
rfl
theorem mem_of_mem_even (a : α) (s : Stream' α) : a ∈ even s → a ∈ s := fun ⟨n, h⟩ =>
  Exists.intro (2 * n) (by 

Proof state:
α : Type u
β : Type v
δ : Type w
a : α
s : Stream' α
x✝ : a ∈ even s
n : ℕ
h : (fun b => a = b) (get (even s) n)
⊢ (fun b => a = b) (get s (2 * n))

Next tactic:
rw [h, get_even]
theorem mem_of_mem_odd (a : α) (s : Stream' α) : a ∈ odd s → a ∈ s := fun ⟨n, h⟩ =>
  Exists.intro (2 * n + 1) (by 

Proof state:
α : Type u
β : Type v
δ : Type w
a : α
s : Stream' α
x✝ : a ∈ odd s
n : ℕ
h : (fun b => a = b) (get (odd s) n)
⊢ (fun b => a = b) (get s (2 * n + 1))

Next tactic:
rw [h, get_odd]
theorem append_append_stream : ∀ (l₁ l₂ : List α) (s : Stream' α),
    l₁ ++ l₂ ++ₛ s = l₁ ++ₛ (l₂ ++ₛ s)
  | [], l₂, s => rfl
  | List.cons a l₁, l₂, s => by
    

Proof state:
α : Type u
β : Type v
δ : Type w
a : α
l₁ l₂ : List α
s : Stream' α
⊢ a :: l₁ ++ l₂ ++ₛ s = a :: l₁ ++ₛ (l₂ ++ₛ s)

Next tactic:
rw [List.cons_append, cons_append_stream, cons_append_stream, append_append_stream l₁]
theorem map_append_stream (f : α → β) :
    ∀ (l : List α) (s : Stream' α), map f (l ++ₛ s) = List.map f l ++ₛ map f s
  | [], s => rfl
  | List.cons a l, s => by
    

Proof state:
α : Type u
β : Type v
δ : Type w
f : α → β
a : α
l : List α
s : Stream' α
⊢ map f (a :: l ++ₛ s) = List.map f (a :: l) ++ₛ map f s

Next tactic:
rw [cons_append_stream, List.map_cons, map_cons, cons_append_stream, map_append_stream f l]
theorem drop_append_stream : ∀ (l : List α) (s : Stream' α), drop l.length (l ++ₛ s) = s
  | [], s => by 

Proof state:
α : Type u
β : Type v
δ : Type w
s : Stream' α
⊢ drop (List.length []) ([] ++ₛ s) = s

Next tactic:
rfl
theorem drop_append_stream : ∀ (l : List α) (s : Stream' α), drop l.length (l ++ₛ s) = s
  | [], s => by rfl
  | List.cons a l, s => by
    

Proof state:
α : Type u
β : Type v
δ : Type w
a : α
l : List α
s : Stream' α
⊢ drop (List.length (a :: l)) (a :: l ++ₛ s) = s

Next tactic:
rw [List.length_cons, drop_succ, cons_append_stream, tail_cons, drop_append_stream l s]
theorem append_stream_head_tail (s : Stream' α) : [head s] ++ₛ tail s = s := by
  

Proof state:
α : Type u
β : Type v
δ : Type w
s : Stream' α
⊢ [head s] ++ₛ tail s = s

Next tactic:
rw [cons_append_stream, nil_append_stream, Stream'.eta]
theorem take_succ' {s : Stream' α} : ∀ n, s.take (n+1) = s.take n ++ [s.get n]
  | 0 => rfl
  | n+1 => by 

Proof state:
α : Type u
β : Type v
δ : Type w
s : Stream' α
n : ℕ
⊢ take (n + 1 + 1) s = take (n + 1) s ++ [get s (n + 1)]

Next tactic:
rw [take_succ, take_succ' n, ← List.cons_append, ← take_succ, get_tail]
@[simp]
theorem length_take (n : ℕ) (s : Stream' α) : (take n s).length = n := by
  

Proof state:
α : Type u
β : Type v
δ : Type w
n : ℕ
s : Stream' α
⊢ List.length (take n s) = n

Next tactic:
induction n generalizing s
@[simp]
theorem length_take (n : ℕ) (s : Stream' α) : (take n s).length = n := by
  induction n generalizing s <;> 

Proof state:
case zero
α : Type u
β : Type v
δ : Type w
s : Stream' α
⊢ List.length (take zero s) = zero

Next tactic:
simp [*, take_succ]
@[simp]
theorem length_take (n : ℕ) (s : Stream' α) : (take n s).length = n := by
  induction n generalizing s <;> 

Proof state:
case succ
α : Type u
β : Type v
δ : Type w
n✝ : ℕ
n_ih✝ : ∀ (s : Stream' α), List.length (take n✝ s) = n✝
s : Stream' α
⊢ List.length (take (succ n✝) s) = succ n✝

Next tactic:
simp [*, take_succ]
@[simp]
theorem take_take {s : Stream' α} : ∀ {m n}, (s.take n).take m = s.take (min n m)
  | 0, n => by 

Proof state:
α : Type u
β : Type v
δ : Type w
s : Stream' α
n : ℕ
⊢ List.take 0 (take n s) = take (min n 0) s

Next tactic:
rw [min_zero, List.take_zero, take_zero]
@[simp]
theorem take_take {s : Stream' α} : ∀ {m n}, (s.take n).take m = s.take (min n m)
  | 0, n => by rw [min_zero, List.take_zero, take_zero]
  | m, 0 => by 

Proof state:
α : Type u
β : Type v
δ : Type w
s : Stream' α
m : ℕ
⊢ List.take m (take 0 s) = take (min 0 m) s

Next tactic:
rw [zero_min, take_zero, List.take_nil]
@[simp]
theorem take_take {s : Stream' α} : ∀ {m n}, (s.take n).take m = s.take (min n m)
  | 0, n => by rw [min_zero, List.take_zero, take_zero]
  | m, 0 => by rw [zero_min, take_zero, List.take_nil]
  | m+1, n+1 => by 

Proof state:
α : Type u
β : Type v
δ : Type w
s : Stream' α
m n : ℕ
⊢ List.take (m + 1) (take (n + 1) s) = take (min (n + 1) (m + 1)) s

Next tactic:
rw [take_succ, List.take_cons, Nat.succ_min_succ, take_succ, take_take]
theorem get?_take {s : Stream' α} : ∀ {k n}, k < n → (s.take n).get? k = s.get k
  | 0, n+1, _ => rfl
  | k+1, n+1, h => by 

Proof state:
α : Type u
β : Type v
δ : Type w
s : Stream' α
k n : ℕ
h : k + 1 < n + 1
⊢ List.get? (take (n + 1) s) (k + 1) = some (get s (k + 1))

Next tactic:
rw [take_succ, List.get?, get?_take (Nat.lt_of_succ_lt_succ h), get_succ]
@[simp] theorem dropLast_take {xs : Stream' α} :
    (Stream'.take n xs).dropLast = Stream'.take (n-1) xs := by
  

Proof state:
α : Type u
β : Type v
δ : Type w
n : ℕ
xs : Stream' α
⊢ List.dropLast (take n xs) = take (n - 1) xs

Next tactic:
cases n
@[simp] theorem dropLast_take {xs : Stream' α} :
    (Stream'.take n xs).dropLast = Stream'.take (n-1) xs := by
  cases n; 

Proof state:
case zero
α : Type u
β : Type v
δ : Type w
xs : Stream' α
⊢ List.dropLast (take zero xs) = take (zero - 1) xs
case succ
α : Type u
β : Type v
δ : Type w
xs : Stream' α
n✝ : ℕ
⊢ List.dropLast (take (succ n✝) xs) = take (succ n✝ - 1) xs

Next tactic:
case zero => simp
@[simp] theorem dropLast_take {xs : Stream' α} :
    (Stream'.take n xs).dropLast = Stream'.take (n-1) xs := by
  cases n; 

Proof state:
α : Type u
β : Type v
δ : Type w
xs : Stream' α
⊢ List.dropLast (take zero xs) = take (zero - 1) xs

Next tactic:
case zero => simp
@[simp] theorem dropLast_take {xs : Stream' α} :
    (Stream'.take n xs).dropLast = Stream'.take (n-1) xs := by
  cases n; case zero => 

Proof state:
α : Type u
β : Type v
δ : Type w
xs : Stream' α
⊢ List.dropLast (take zero xs) = take (zero - 1) xs

Next tactic:
simp
@[simp] theorem dropLast_take {xs : Stream' α} :
    (Stream'.take n xs).dropLast = Stream'.take (n-1) xs := by
  cases n; case zero => simp
  

Proof state:
case succ
α : Type u
β : Type v
δ : Type w
xs : Stream' α
n✝ : ℕ
⊢ List.dropLast (take (succ n✝) xs) = take (succ n✝ - 1) xs

Next tactic:
case succ n => rw [take_succ', List.dropLast_concat, Nat.add_one_sub_one]
@[simp] theorem dropLast_take {xs : Stream' α} :
    (Stream'.take n xs).dropLast = Stream'.take (n-1) xs := by
  cases n; case zero => simp
  

Proof state:
α : Type u
β : Type v
δ : Type w
xs : Stream' α
n : ℕ
⊢ List.dropLast (take (succ n) xs) = take (succ n - 1) xs

Next tactic:
case succ n => rw [take_succ', List.dropLast_concat, Nat.add_one_sub_one]
@[simp] theorem dropLast_take {xs : Stream' α} :
    (Stream'.take n xs).dropLast = Stream'.take (n-1) xs := by
  cases n; case zero => simp
  case succ n => 

Proof state:
α : Type u
β : Type v
δ : Type w
xs : Stream' α
n : ℕ
⊢ List.dropLast (take (succ n) xs) = take (succ n - 1) xs

Next tactic:
rw [take_succ', List.dropLast_concat, Nat.add_one_sub_one]
@[simp]
theorem append_take_drop : ∀ (n : Nat) (s : Stream' α),
    appendStream' (take n s) (drop n s) = s := by
  

Proof state:
α : Type u
β : Type v
δ : Type w
⊢ ∀ (n : ℕ) (s : Stream' α), take n s ++ₛ drop n s = s

Next tactic:
intro n
@[simp]
theorem append_take_drop : ∀ (n : Nat) (s : Stream' α),
    appendStream' (take n s) (drop n s) = s := by
  intro n
  

Proof state:
α : Type u
β : Type v
δ : Type w
n : ℕ
⊢ ∀ (s : Stream' α), take n s ++ₛ drop n s = s

Next tactic:
induction' n with n' ih
@[simp]
theorem append_take_drop : ∀ (n : Nat) (s : Stream' α),
    appendStream' (take n s) (drop n s) = s := by
  intro n
  induction' n with n' ih
  · 

Proof state:
case zero
α : Type u
β : Type v
δ : Type w
⊢ ∀ (s : Stream' α), take zero s ++ₛ drop zero s = s

Next tactic:
intro s
@[simp]
theorem append_take_drop : ∀ (n : Nat) (s : Stream' α),
    appendStream' (take n s) (drop n s) = s := by
  intro n
  induction' n with n' ih
  · intro s
    

Proof state:
case zero
α : Type u
β : Type v
δ : Type w
s : Stream' α
⊢ take zero s ++ₛ drop zero s = s

Next tactic:
rfl
@[simp]
theorem append_take_drop : ∀ (n : Nat) (s : Stream' α),
    appendStream' (take n s) (drop n s) = s := by
  intro n
  induction' n with n' ih
  · intro s
    rfl
  · 

Proof state:
case succ
α : Type u
β : Type v
δ : Type w
n' : ℕ
ih : ∀ (s : Stream' α), take n' s ++ₛ drop n' s = s
⊢ ∀ (s : Stream' α), take (succ n') s ++ₛ drop (succ n') s = s

Next tactic:
intro s
@[simp]
theorem append_take_drop : ∀ (n : Nat) (s : Stream' α),
    appendStream' (take n s) (drop n s) = s := by
  intro n
  induction' n with n' ih
  · intro s
    rfl
  · intro s
    

Proof state:
case succ
α : Type u
β : Type v
δ : Type w
n' : ℕ
ih : ∀ (s : Stream' α), take n' s ++ₛ drop n' s = s
s : Stream' α
⊢ take (succ n') s ++ₛ drop (succ n') s = s

Next tactic:
rw [take_succ, drop_succ, cons_append_stream, ih (tail s), Stream'.eta]
theorem take_theorem (s₁ s₂ : Stream' α) : (∀ n : Nat, take n s₁ = take n s₂) → s₁ = s₂ := by
  

Proof state:
α : Type u
β : Type v
δ : Type w
s₁ s₂ : Stream' α
⊢ (∀ (n : ℕ), take n s₁ = take n s₂) → s₁ = s₂

Next tactic:
intro h
theorem take_theorem (s₁ s₂ : Stream' α) : (∀ n : Nat, take n s₁ = take n s₂) → s₁ = s₂ := by
  intro h; 

Proof state:
α : Type u
β : Type v
δ : Type w
s₁ s₂ : Stream' α
h : ∀ (n : ℕ), take n s₁ = take n s₂
⊢ s₁ = s₂

Next tactic:
apply Stream'.ext
theorem take_theorem (s₁ s₂ : Stream' α) : (∀ n : Nat, take n s₁ = take n s₂) → s₁ = s₂ := by
  intro h; apply Stream'.ext; 

Proof state:
case a
α : Type u
β : Type v
δ : Type w
s₁ s₂ : Stream' α
h : ∀ (n : ℕ), take n s₁ = take n s₂
⊢ ∀ (n : ℕ), get s₁ n = get s₂ n

Next tactic:
intro n
theorem take_theorem (s₁ s₂ : Stream' α) : (∀ n : Nat, take n s₁ = take n s₂) → s₁ = s₂ := by
  intro h; apply Stream'.ext; intro n
  

Proof state:
case a
α : Type u
β : Type v
δ : Type w
s₁ s₂ : Stream' α
h : ∀ (n : ℕ), take n s₁ = take n s₂
n : ℕ
⊢ get s₁ n = get s₂ n

Next tactic:
induction' n with n _
theorem take_theorem (s₁ s₂ : Stream' α) : (∀ n : Nat, take n s₁ = take n s₂) → s₁ = s₂ := by
  intro h; apply Stream'.ext; intro n
  induction' n with n _
  · 

Proof state:
case a.zero
α : Type u
β : Type v
δ : Type w
s₁ s₂ : Stream' α
h : ∀ (n : ℕ), take n s₁ = take n s₂
⊢ get s₁ zero = get s₂ zero

Next tactic:
have aux := h 1
theorem take_theorem (s₁ s₂ : Stream' α) : (∀ n : Nat, take n s₁ = take n s₂) → s₁ = s₂ := by
  intro h; apply Stream'.ext; intro n
  induction' n with n _
  · have aux := h 1
    

Proof state:
case a.zero
α : Type u
β : Type v
δ : Type w
s₁ s₂ : Stream' α
h : ∀ (n : ℕ), take n s₁ = take n s₂
aux : take 1 s₁ = take 1 s₂
⊢ get s₁ zero = get s₂ zero

Next tactic:
simp? [take] at aux says
      simp only [take, List.cons.injEq, and_true] at aux
theorem take_theorem (s₁ s₂ : Stream' α) : (∀ n : Nat, take n s₁ = take n s₂) → s₁ = s₂ := by
  intro h; apply Stream'.ext; intro n
  induction' n with n _
  · have aux := h 1
    simp? [take] at aux says
      

Proof state:
case a.zero
α : Type u
β : Type v
δ : Type w
s₁ s₂ : Stream' α
h : ∀ (n : ℕ), take n s₁ = take n s₂
aux : take 1 s₁ = take 1 s₂
⊢ get s₁ zero = get s₂ zero

Next tactic:
simp only [take, List.cons.injEq, and_true] at aux
theorem take_theorem (s₁ s₂ : Stream' α) : (∀ n : Nat, take n s₁ = take n s₂) → s₁ = s₂ := by
  intro h; apply Stream'.ext; intro n
  induction' n with n _
  · have aux := h 1
    simp? [take] at aux says
      simp only [take, List.cons.injEq, and_true] at aux
    

Proof state:
case a.zero
α : Type u
β : Type v
δ : Type w
s₁ s₂ : Stream' α
h : ∀ (n : ℕ), take n s₁ = take n s₂
aux : head s₁ = head s₂
⊢ get s₁ zero = get s₂ zero

Next tactic:
exact aux
theorem take_theorem (s₁ s₂ : Stream' α) : (∀ n : Nat, take n s₁ = take n s₂) → s₁ = s₂ := by
  intro h; apply Stream'.ext; intro n
  induction' n with n _
  · have aux := h 1
    simp? [take] at aux says
      simp only [take, List.cons.injEq, and_true] at aux
    exact aux
  · 

Proof state:
case a.succ
α : Type u
β : Type v
δ : Type w
s₁ s₂ : Stream' α
h : ∀ (n : ℕ), take n s₁ = take n s₂
n : ℕ
n_ih✝ : get s₁ n = get s₂ n
⊢ get s₁ (succ n) = get s₂ (succ n)

Next tactic:
have h₁ : some (get s₁ (succ n)) = some (get s₂ (succ n)) := by
      rw [← get?_take_succ, ← get?_take_succ, h (succ (succ n))]
theorem take_theorem (s₁ s₂ : Stream' α) : (∀ n : Nat, take n s₁ = take n s₂) → s₁ = s₂ := by
  intro h; apply Stream'.ext; intro n
  induction' n with n _
  · have aux := h 1
    simp? [take] at aux says
      simp only [take, List.cons.injEq, and_true] at aux
    exact aux
  · have h₁ : some (get s₁ (succ n)) = some (get s₂ (succ n)) := by
      

Proof state:
α : Type u
β : Type v
δ : Type w
s₁ s₂ : Stream' α
h : ∀ (n : ℕ), take n s₁ = take n s₂
n : ℕ
n_ih✝ : get s₁ n = get s₂ n
⊢ some (get s₁ (succ n)) = some (get s₂ (succ n))

Next tactic:
rw [← get?_take_succ, ← get?_take_succ, h (succ (succ n))]
theorem take_theorem (s₁ s₂ : Stream' α) : (∀ n : Nat, take n s₁ = take n s₂) → s₁ = s₂ := by
  intro h; apply Stream'.ext; intro n
  induction' n with n _
  · have aux := h 1
    simp? [take] at aux says
      simp only [take, List.cons.injEq, and_true] at aux
    exact aux
  · have h₁ : some (get s₁ (succ n)) = some (get s₂ (succ n)) := by
      rw [← get?_take_succ, ← get?_take_succ, h (succ (succ n))]
    

Proof state:
case a.succ
α : Type u
β : Type v
δ : Type w
s₁ s₂ : Stream' α
h : ∀ (n : ℕ), take n s₁ = take n s₂
n : ℕ
n_ih✝ : get s₁ n = get s₂ n
h₁ : some (get s₁ (succ n)) = some (get s₂ (succ n))
⊢ get s₁ (succ n) = get s₂ (succ n)

Next tactic:
injection h₁
theorem cycle_eq : ∀ (l : List α) (h : l ≠ []), cycle l h = l ++ₛ cycle l h
  | [], h => absurd rfl h
  | List.cons a l, _ =>
    have gen : ∀ l' a', corec Stream'.cycleF Stream'.cycleG (a', l', a, l) =
        (a'::l') ++ₛ corec Stream'.cycleF Stream'.cycleG (a, l, a, l) := by
      

Proof state:
α : Type u
β : Type v
δ : Type w
a : α
l : List α
x✝ : a :: l ≠ []
⊢ ∀ (l' : List α) (a' : α),
    corec Stream'.cycleF Stream'.cycleG (a', l', a, l) = a' :: l' ++ₛ corec Stream'.cycleF Stream'.cycleG (a, l, a, l)

Next tactic:
intro l'
theorem cycle_eq : ∀ (l : List α) (h : l ≠ []), cycle l h = l ++ₛ cycle l h
  | [], h => absurd rfl h
  | List.cons a l, _ =>
    have gen : ∀ l' a', corec Stream'.cycleF Stream'.cycleG (a', l', a, l) =
        (a'::l') ++ₛ corec Stream'.cycleF Stream'.cycleG (a, l, a, l) := by
      intro l'
      

Proof state:
α : Type u
β : Type v
δ : Type w
a : α
l : List α
x✝ : a :: l ≠ []
l' : List α
⊢ ∀ (a' : α),
    corec Stream'.cycleF Stream'.cycleG (a', l', a, l) = a' :: l' ++ₛ corec Stream'.cycleF Stream'.cycleG (a, l, a, l)

Next tactic:
induction' l' with a₁ l₁ ih
theorem cycle_eq : ∀ (l : List α) (h : l ≠ []), cycle l h = l ++ₛ cycle l h
  | [], h => absurd rfl h
  | List.cons a l, _ =>
    have gen : ∀ l' a', corec Stream'.cycleF Stream'.cycleG (a', l', a, l) =
        (a'::l') ++ₛ corec Stream'.cycleF Stream'.cycleG (a, l, a, l) := by
      intro l'
      induction' l' with a₁ l₁ ih
      · 

Proof state:
case nil
α : Type u
β : Type v
δ : Type w
a : α
l : List α
x✝ : a :: l ≠ []
⊢ ∀ (a' : α),
    corec Stream'.cycleF Stream'.cycleG (a', [], a, l) = [a'] ++ₛ corec Stream'.cycleF Stream'.cycleG (a, l, a, l)

Next tactic:
intros
theorem cycle_eq : ∀ (l : List α) (h : l ≠ []), cycle l h = l ++ₛ cycle l h
  | [], h => absurd rfl h
  | List.cons a l, _ =>
    have gen : ∀ l' a', corec Stream'.cycleF Stream'.cycleG (a', l', a, l) =
        (a'::l') ++ₛ corec Stream'.cycleF Stream'.cycleG (a, l, a, l) := by
      intro l'
      induction' l' with a₁ l₁ ih
      · intros
        

Proof state:
case nil
α : Type u
β : Type v
δ : Type w
a : α
l : List α
x✝ : a :: l ≠ []
a'✝ : α
⊢ corec Stream'.cycleF Stream'.cycleG (a'✝, [], a, l) = [a'✝] ++ₛ corec Stream'.cycleF Stream'.cycleG (a, l, a, l)

Next tactic:
rw [corec_eq]
theorem cycle_eq : ∀ (l : List α) (h : l ≠ []), cycle l h = l ++ₛ cycle l h
  | [], h => absurd rfl h
  | List.cons a l, _ =>
    have gen : ∀ l' a', corec Stream'.cycleF Stream'.cycleG (a', l', a, l) =
        (a'::l') ++ₛ corec Stream'.cycleF Stream'.cycleG (a, l, a, l) := by
      intro l'
      induction' l' with a₁ l₁ ih
      · intros
        rw [corec_eq]
        

Proof state:
case nil
α : Type u
β : Type v
δ : Type w
a : α
l : List α
x✝ : a :: l ≠ []
a'✝ : α
⊢ Stream'.cycleF (a'✝, [], a, l) :: corec Stream'.cycleF Stream'.cycleG (Stream'.cycleG (a'✝, [], a, l)) =
    [a'✝] ++ₛ corec Stream'.cycleF Stream'.cycleG (a, l, a, l)

Next tactic:
rfl
theorem cycle_eq : ∀ (l : List α) (h : l ≠ []), cycle l h = l ++ₛ cycle l h
  | [], h => absurd rfl h
  | List.cons a l, _ =>
    have gen : ∀ l' a', corec Stream'.cycleF Stream'.cycleG (a', l', a, l) =
        (a'::l') ++ₛ corec Stream'.cycleF Stream'.cycleG (a, l, a, l) := by
      intro l'
      induction' l' with a₁ l₁ ih
      · intros
        rw [corec_eq]
        rfl
      · 

Proof state:
case cons
α : Type u
β : Type v
δ : Type w
a : α
l : List α
x✝ : a :: l ≠ []
a₁ : α
l₁ : List α
ih :
  ∀ (a' : α),
    corec Stream'.cycleF Stream'.cycleG (a', l₁, a, l) = a' :: l₁ ++ₛ corec Stream'.cycleF Stream'.cycleG (a, l, a, l)
⊢ ∀ (a' : α),
    corec Stream'.cycleF Stream'.cycleG (a', a₁ :: l₁, a, l) =
      a' :: a₁ :: l₁ ++ₛ corec Stream'.cycleF Stream'.cycleG (a, l, a, l)

Next tactic:
intros