load "misc.quox"
load "bool.quox"

namespace either {

def0 Tag : ★ = {left, right}

def0 Payload : ★ → ★ → Tag → ★ =
  λ A B tag ⇒ case tag return ★ of { 'left ⇒ A; 'right ⇒ B }

def0 Either : ★ → ★ → ★ =
  λ A B ⇒ (tag : Tag) × Payload A B tag

def Left : 0.(A B : ★) → A → Either A B =
  λ A B x ⇒ ('left, x)

def Right : 0.(A B : ★) → B → Either A B =
  λ A B x ⇒ ('right, x)

def elim' :
    0.(A B : ★) → 0.(P : 0.(Either A B) → ★) →
    ω.((x : A) → P (Left  A B x)) →
    ω.((x : B) → P (Right A B x)) →
    (t : Tag) → (a : Payload A B t) → P (t, a) =
  λ A B P f g t ⇒
    case t
    return t' ⇒ (a : Payload A B t') → P (t', a)
    of { 'left ⇒ f; 'right ⇒ g }

def elim :
    0.(A B : ★) → 0.(P : 0.(Either A B) → ★) →
    ω.((x : A) → P (Left  A B x)) →
    ω.((x : B) → P (Right A B x)) →
    (x : Either A B) → P x =
  λ A B P f g e ⇒
    case e return e' ⇒ P e' of { (t, a) ⇒ elim' A B P f g t a }

def elimω' :
    0.(A B : ★) → 0.(P : 0.(Either A B) → ★) →
    ω.(ω.(x : A) → P (Left  A B x)) →
    ω.(ω.(x : B) → P (Right A B x)) →
    (t : Tag) → ω.(a : Payload A B t) → P (t, a) =
  λ A B P f g t ⇒
    case t
    return t' ⇒ ω.(a : Payload A B t') → P (t', a)
    of { 'left ⇒ f; 'right ⇒ g }

def elimω :
    0.(A B : ★) → 0.(P : 0.(Either A B) → ★) →
    ω.(ω.(x : A) → P (Left  A B x)) →
    ω.(ω.(x : B) → P (Right A B x)) →
    ω.(x : Either A B) → P x =
  λ A B P f g e ⇒ elimω' A B P f g (fst e) (snd e)

def fold :
    0.(A B C : ★) → ω.(A → C) → ω.(B → C) → Either A B → C =
  λ A B C ⇒ elim A B (λ _ ⇒ C)

def foldω :
    0.(A B C : ★) → ω.(ω.A → C) → ω.(ω.B → C) → ω.(Either A B) → C =
  λ A B C ⇒ elimω A B (λ _ ⇒ C)


}

def0 Either = either.Either
def  Left   = either.Left
def  Right  = either.Right


namespace dec {

def0 Dec : ★ → ★ = λ A ⇒ Either [0.A] [0.Not A]

def Yes : 0.(A : ★) → 0.A       → Dec A = λ A y ⇒ Left  [0.A] [0.Not A] [y]
def No  : 0.(A : ★) → 0.(Not A) → Dec A = λ A n ⇒ Right [0.A] [0.Not A] [n]

def yes-refl : 0.(A : ★) → 0.(x : A) → Dec (x ≡ x : A) =
  λ A x ⇒ Yes (x ≡ x : A) (δ 𝑖 ⇒ x)

def0 DecEq : ★ → ★ =
  λ A ⇒ ω.(x : A) → ω.(y : A) → Dec (x ≡ y : A)

def elim :
    0.(A : ★) → 0.(P : 0.(Dec A) → ★) →
    ω.(0.(y : A)     → P (Yes A y)) →
    ω.(0.(n : Not A) → P (No  A n)) →
    (x : Dec A) → P x =
  λ A P f g ⇒
    either.elim [0.A] [0.Not A] P
      (λ y ⇒ case0 y return y' ⇒ P (Left  [0.A] [0.Not A] y') of {[y'] ⇒ f y'})
      (λ n ⇒ case0 n return n' ⇒ P (Right [0.A] [0.Not A] n') of {[n'] ⇒ g n'})

def bool : 0.(A : ★) → Dec A → Bool =
  λ A ⇒ elim A (λ _ ⇒ Bool) (λ _ ⇒ 'true) (λ _ ⇒ 'false)

def drop' : 0.(A : ★) → Dec A → True =
  λ A ⇒ elim A (λ _ ⇒ True) (λ _ ⇒ 'true) (λ _ ⇒ 'true)

def drop : 0.(A B : ★) → Dec A → B → B =
  λ A B x y ⇒ true.drop B (drop' A x) y

}

def0 Dec   = dec.Dec
def0 DecEq = dec.DecEq
def  Yes   = dec.Yes
def  No    = dec.No