-- Martin Escardo & Paulo Oliva, Fri 24-25 Feb 2017
-- Conversion of dialogue trees to Brouwer trees.

{-# OPTIONS --without-K #-}

data  : Set where 
 zero :               
 succ :          

data _≡_ {X : Set} : X  X  Set where
 refl : {x : X}  x  x

trans : {X : Set} {x y z : X}  x  y  y  z  x  z
trans refl refl = refl

infix  1 _≡_
infixr 0 _≡⟨_⟩_

_≡⟨_⟩_ : {X : Set} (x : X) {y z : X}  x  y  y  z  x  z
_ ≡⟨ p  q = trans p q

_∎ : {X : Set } (x : X)  x  x
_∎ _ = refl

-- Dialogue trees:

data D : Set where 
 η :   D
 β : (  D)    D

-- We use φ to range over forests of dialogue trees, that is,
-- functions ℕ → D.

-- Dialogue trees represent functions (ℕ → ℕ) → ℕ:

deval : D  (  )  
deval (η k)   α = k
deval (β φ n) α = deval (φ(α n)) α

-- Brouwer trees:

data B : Set where
 η :   B
 δ : (  B)  B

-- We use γ to range over forests of Brower trees.

-- Brouwer trees represent functions (ℕ → ℕ) → ℕ too:

beval : B  (  )  
beval (η k) α = k
beval (δ γ) α = beval (γ(α zero))  i  α(succ i))

-- Conversion from dialogue to Brouwer trees, with two auxiliary
-- functions follow and β':

follow :   B  B
follow n (η k) = η k
follow n (δ γ) = γ n

-- The function β' simulates D's constructor β in B:

β' : (  B)    B
β' γ zero     = δ  i  follow i (γ i))
β' γ (succ n) = δ  i  β'  j  follow i (γ j)) n) 

-- Conversion is the unique homomorphism w.r.t. D-structure:

convert : D  B
convert (η k)   = η k
convert (β φ n) = β'  i  convert (φ i)) n

-- The correctness proof of the function convert uses two lemmas, one
-- for the function follow and the other for the function β':

-- By cases on b:

follow-lemma : (b : B) (α :   )
              beval b α  beval (follow (α zero) b)  i  α (succ i)) 
follow-lemma (η k) α = refl
follow-lemma (δ φ) α = refl

-- By induction on n, using follow-lemma both in the base case and the
-- induction step:

β'-lemma : (n : ) (φ :   B) (α :   )
          beval (φ(α n)) α  beval (β' φ n) α

β'-lemma zero φ α =
  beval (φ(α zero)) α                                    ≡⟨ follow-lemma ((φ(α zero))) α 
  beval (follow (α zero) (φ(α zero)))  i  α (succ i)) ≡⟨ refl 
  beval (δ  i  follow i (φ i))) α                     ≡⟨ refl 
  beval (β' φ zero) α 

β'-lemma (succ n) φ α = 
  beval (φ(α(succ n))) α                                       ≡⟨ follow-lemma (φ(α(succ n))) α 
  beval (follow (α zero) (φ(α(succ n))))  i  α(succ i))     ≡⟨ β'-lemma n  j  follow (α zero) (φ j))  i  α (succ i)) 
  beval (β'  j  follow (α zero) (φ j)) n)  i  α(succ i)) ≡⟨ refl 
  beval (δ  i  β'  j  follow i (φ j)) n)) α              ≡⟨ refl 
  beval (β' φ(succ n)) α 

-- By induction on d, using β'-lemma in the induction step:

convert-correctness : (d : D) (α :   )  deval d α  beval (convert d) α
convert-correctness (η k)   α = refl
convert-correctness (β φ n) α = 
  deval (φ(α n)) α                       ≡⟨ convert-correctness (φ(α n)) α 
  beval (convert (φ(α n))) α             ≡⟨ β'-lemma n  i  convert (φ i)) α 
  beval (β'  i  convert (φ i)) n) α