Cubical.agda

{-# OPTIONS --erased-cubical --no-subtyping #-}

module Agda.Primitive.Cubical where

{-# BUILTIN INTERVAL I  #-}  -- I : Setω

{-# BUILTIN IZERO    i0 #-}
{-# BUILTIN IONE     i1 #-}

-- I is treated as the type of booleans.
{-# COMPILE JS i0 = false #-}
{-# COMPILE JS i1 = true  #-}

infix  30 primINeg
infixr 20 primIMin primIMax

primitive
    primIMin : I → I → I
    primIMax : I → I → I
    primINeg : I → I

{-# BUILTIN ISONE    IsOne    #-}  -- IsOne : I → Setω

postulate
  itIsOne : IsOne i1
  IsOne1  : ∀ i j → IsOne i → IsOne (primIMax i j)
  IsOne2  : ∀ i j → IsOne j → IsOne (primIMax i j)

{-# BUILTIN ITISONE  itIsOne  #-}
{-# BUILTIN ISONE1   IsOne1   #-}
{-# BUILTIN ISONE2   IsOne2   #-}

-- IsOne i is treated as the unit type.
{-# COMPILE JS itIsOne = { "tt" : a => a["tt"]() } #-}
{-# COMPILE JS IsOne1 =
  _ => _ => _ => { return { "tt" : a => a["tt"]() } }
  #-}
{-# COMPILE JS IsOne2 =
  _ => _ => _ => { return { "tt" : a => a["tt"]() } }
  #-}

-- Partial : ∀{ℓ} (i : I) (A : Set ℓ) → Set ℓ
-- Partial i A = IsOne i → A

{-# BUILTIN PARTIAL  Partial  #-}
{-# BUILTIN PARTIALP PartialP #-}

postulate
  isOneEmpty : ∀ {ℓ} {A : Partial i0 (Set ℓ)} → PartialP i0 A

{-# BUILTIN ISONEEMPTY isOneEmpty #-}

-- Partial i A and PartialP i A are treated as IsOne i → A.
{-# COMPILE JS isOneEmpty =
  _ => x => _ => x({ "tt" : a => a["tt"]() })
  #-}

primitive
  primPOr : ∀ {ℓ} (i j : I) {A : Partial (primIMax i j) (Set ℓ)}
            → (u : PartialP i (λ z → A (IsOne1 i j z)))
            → (v : PartialP j (λ z → A (IsOne2 i j z)))
            → PartialP (primIMax i j) A

  -- Computes in terms of primHComp and primTransp
  primComp : ∀ {ℓ} (A : (i : I) → Set (ℓ i)) {φ : I} (u : ∀ i → Partial φ (A i)) (a : A i0) → A i1

syntax primPOr p q u t = [ p ↦ u , q ↦ t ]

primitive
  primTransp : ∀ {ℓ} (A : (i : I) → Set (ℓ i)) (φ : I) (a : A i0) → A i1
  primHComp  : ∀ {ℓ} {A : Set ℓ} {φ : I} (u : ∀ i → Partial φ A) (a : A) → A