Created
November 9, 2020 09:20
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Guarded Cubical with clocks
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module Prims where | |
primitive | |
primLockUniv : Set₁ | |
open Prims renaming (primLockUniv to LockU) public | |
postulate | |
Cl : Set | |
k0 : Cl | |
Tick : Cl → LockU | |
▹ : ∀ {l} → Cl → Set l → Set l | |
▹ k A = (@tick x : Tick k) -> A | |
▸ : ∀ {l} k → ▹ k (Set l) → Set l | |
▸ k A = (@tick x : Tick k) → A x | |
postulate | |
tick-irr : ∀ {A : Set}{k : Cl} (x : ▹ k A) → ▸ k \ α → ▸ k \ β → x α ≡ x β | |
postulate | |
dfix : ∀ {k} {l} {A : Set l} → (▹ k A → A) → ▹ k A | |
pfix : ∀ {k} {l} {A : Set l} (f : ▹ k A → A) → dfix f ≡ (\ _ → f (dfix f)) | |
force : ∀ {l} {A : Cl → Set l} → (∀ k → ▹ k (A k)) → ∀ k → A k | |
force-delay : ∀ {l} {A : Cl → Set l} → (f : ∀ k → ▹ k (A k)) → ∀ k → ▸ k \ α → force f k ≡ f k α | |
delay-force : ∀ {l} {A : Cl → Set l} → (f : ∀ k → A k) → ∀ k → force (\ k α → f k) k ≡ f k | |
force^ : ∀ {l} {A : ∀ k → ▹ k (Set l)} → (∀ k → ▸ k (A k)) → ∀ k → force A k | |
-- No more postulates after this line. | |
private | |
variable | |
l : Level | |
A B : Set l | |
k : Cl | |
next : A → ▹ k A | |
next x _ = x | |
_⊛_ : ▹ k (A → B) → ▹ k A → ▹ k B | |
_⊛_ f x a = f a (x a) | |
map▹ : (f : A → B) → ▹ k A → ▹ k B | |
map▹ f x α = f (x α) | |
later-ext : ∀ {l} {A : Set l} → {f g : ▹ k A} → (▸ k \ α → f α ≡ g α) → f ≡ g | |
later-ext eq = \ i α → eq α i | |
pfix' : ∀ {l} {A : Set l} (f : ▹ k A → A) → ▸ k \ α → dfix f α ≡ f (dfix f) | |
pfix' f α i = pfix f i α | |
fix : ∀ {l} {A : Set l} → (▹ k A → A) → A | |
fix f = f (dfix f) | |
fix-eq : ∀ {l} {A : Set l} → (f : ▹ k A → A) → fix f ≡ f (\ _ → fix f) | |
fix-eq f = \ i → f (pfix f i) | |
delay : ∀ {A : Cl → Set} → (∀ k → A k) → ∀ k → ▹ k (A k) | |
delay a k _ = a k | |
Later-Alg[_]_ : ∀ {l} → Cl → Set l → Set l | |
Later-Alg[ k ] A = ▹ k A → A |
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But the current implementation only accepts lock variables,
so the following postulate
is not helpful to express the judgemental equalities about the diamond tick.
Maybe I will just wait for it to be implemented. :-)