clear this
α β α' β'✝ : Type
γ δ : β → Type
β' : Type
f : (a : β) → δ a → β'
n : ℕ
init_carry : α → Bool
next_bit : (α → Bool) → (β → Bool) → α → Bool
init_carry_x : (a : β) → γ a → Bool
next_bit_x : (a : β) → (γ a → Bool) → (δ a → Bool) → (γ a → Bool) × Bool
x : β' → ℕ → Bool
this :
  ∀ (init_carry : α → Bool) (next_bit : (α → Bool) → (β → Bool) → α → Bool) (init_carry_x : (a : β) → γ a → Bool)
    (next_bit_x : (a : β) → (γ a → Bool) → (δ a → Bool) → (γ a → Bool) × Bool) (x : β' → ℕ → Bool),
    propagateCarry init_carry next_bit (fun a => propagate (init_carry_x a) (next_bit_x a) fun d => x (f a d)) n =
      propagateCarry (fun a => Sum.elim init_carry (fun b => init_carry_x b.fst b.snd) a)
          (fun carry bits =>
            Sum.elim
              (next_bit (carry ∘ inl) fun a =>
                (next_bit_x a (fun d => carry (inr { fst := a, snd := d })) fun d => bits (f a d)).snd)
              fun x =>
              Prod.fst (next_bit_x x.fst (fun d => carry (inr { fst := x.fst, snd := d })) fun d => bits (f x.fst d))
                x.snd)
          x n ∘
        inl
⊢ (next_bit
      (propagateCarry (fun a => Sum.elim init_carry (fun b => init_carry_x b.fst b.snd) a)
          (fun carry bits =>
            Sum.elim
              (next_bit (carry ∘ inl) fun a =>
                (next_bit_x a (fun d => carry (inr { fst := a, snd := d })) fun d => bits (f a d)).snd)
              fun x =>
              Prod.fst (next_bit_x x.fst (fun d => carry (inr { fst := x.fst, snd := d })) fun d => bits (f x.fst d))
                x.snd)
          x n ∘
        inl)
      fun i =>
      (next_bit_x i (propagateCarry (init_carry_x i) (fun c b => (next_bit_x i c b).fst) (fun d => x (f i d)) n)
          fun j => x (f i j) (n + 1)).snd) =
    (Sum.elim
        (next_bit
          (propagateCarry (fun a => Sum.elim init_carry (fun b => init_carry_x b.fst b.snd) a)
              (fun carry bits =>
                Sum.elim
                  (next_bit (carry ∘ inl) fun a =>
                    (next_bit_x a (fun d => carry (inr { fst := a, snd := d })) fun d => bits (f a d)).snd)
                  fun x =>
                  Prod.fst
                    (next_bit_x x.fst (fun d => carry (inr { fst := x.fst, snd := d })) fun d => bits (f x.fst d))
                    x.snd)
              x n ∘
            inl)
          fun a =>
          (next_bit_x a
              (fun d =>
                propagateCarry (fun a => Sum.elim init_carry (fun b => init_carry_x b.fst b.snd) a)
                  (fun carry bits =>
                    Sum.elim
                      (next_bit (carry ∘ inl) fun a =>
                        (next_bit_x a (fun d => carry (inr { fst := a, snd := d })) fun d => bits (f a d)).snd)
                      fun x =>
                      Prod.fst
                        (next_bit_x x.fst (fun d => carry (inr { fst := x.fst, snd := d })) fun d => bits (f x.fst d))
                        x.snd)
                  x n (inr { fst := a, snd := d }))
              fun d => x (f a d) (n + 1)).snd)
        fun x_1 =>
        Prod.fst
          (next_bit_x x_1.fst
            (fun d =>
              propagateCarry (fun a => Sum.elim init_carry (fun b => init_carry_x b.fst b.snd) a)
                (fun carry bits =>
                  Sum.elim
                    (next_bit (carry ∘ inl) fun a =>
                      (next_bit_x a (fun d => carry (inr { fst := a, snd := d })) fun d => bits (f a d)).snd)
                    fun x =>
                    Prod.fst
                      (next_bit_x x.fst (fun d => carry (inr { fst := x.fst, snd := d })) fun d => bits (f x.fst d))
                      x.snd)
                x n (inr { fst := x_1.fst, snd := d }))
            fun d => x (f x_1.fst d) (n + 1))
          x_1.snd) ∘
      inl
α β α' β'✝ : Type
γ δ : β → Type
β' : Type
f : (a : β) → δ a → β'
n : ℕ
init_carry : α → Bool
next_bit : (α → Bool) → (β → Bool) → α → Bool
init_carry_x : (a : β) → γ a → Bool
next_bit_x : (a : β) → (γ a → Bool) → (δ a → Bool) → (γ a → Bool) × Bool
x : β' → ℕ → Bool
⊢ (next_bit
      (propagateCarry (fun a => Sum.elim init_carry (fun b => init_carry_x b.fst b.snd) a)
          (fun carry bits =>
            Sum.elim
              (next_bit (carry ∘ inl) fun a =>
                (next_bit_x a (fun d => carry (inr { fst := a, snd := d })) fun d => bits (f a d)).snd)
              fun x =>
              Prod.fst (next_bit_x x.fst (fun d => carry (inr { fst := x.fst, snd := d })) fun d => bits (f x.fst d))
                x.snd)
          x n ∘
        inl)
      fun i =>
      (next_bit_x i (propagateCarry (init_carry_x i) (fun c b => (next_bit_x i c b).fst) (fun d => x (f i d)) n)
          fun j => x (f i j) (n + 1)).snd) =
    (Sum.elim
        (next_bit
          (propagateCarry (fun a => Sum.elim init_carry (fun b => init_carry_x b.fst b.snd) a)
              (fun carry bits =>
                Sum.elim
                  (next_bit (carry ∘ inl) fun a =>
                    (next_bit_x a (fun d => carry (inr { fst := a, snd := d })) fun d => bits (f a d)).snd)
                  fun x =>
                  Prod.fst
                    (next_bit_x x.fst (fun d => carry (inr { fst := x.fst, snd := d })) fun d => bits (f x.fst d))
                    x.snd)
              x n ∘
            inl)
          fun a =>
          (next_bit_x a
              (fun d =>
                propagateCarry (fun a => Sum.elim init_carry (fun b => init_carry_x b.fst b.snd) a)
                  (fun carry bits =>
                    Sum.elim
                      (next_bit (carry ∘ inl) fun a =>
                        (next_bit_x a (fun d => carry (inr { fst := a, snd := d })) fun d => bits (f a d)).snd)
                      fun x =>
                      Prod.fst
                        (next_bit_x x.fst (fun d => carry (inr { fst := x.fst, snd := d })) fun d => bits (f x.fst d))
                        x.snd)
                  x n (inr { fst := a, snd := d }))
              fun d => x (f a d) (n + 1)).snd)
        fun x_1 =>
        Prod.fst
          (next_bit_x x_1.fst
            (fun d =>
              propagateCarry (fun a => Sum.elim init_carry (fun b => init_carry_x b.fst b.snd) a)
                (fun carry bits =>
                  Sum.elim
                    (next_bit (carry ∘ inl) fun a =>
                      (next_bit_x a (fun d => carry (inr { fst := a, snd := d })) fun d => bits (f a d)).snd)
                    fun x =>
                    Prod.fst
                      (next_bit_x x.fst (fun d => carry (inr { fst := x.fst, snd := d })) fun d => bits (f x.fst d))
                      x.snd)
                x n (inr { fst := x_1.fst, snd := d }))
            fun d => x (f x_1.fst d) (n + 1))
          x_1.snd) ∘
      inl