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An implementation of infinite queues fashioned after the von Neuman ordinals
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open import Data.Nat | |
open import Data.Maybe | |
open import Data.Product | |
-- An implementation of infinite queues fashioned after the von Neuman ordinals | |
module Queue where | |
infixl 5 _⊕_ | |
data queue (A : Set) : Set where | |
∙ : queue A -- the empty queue | |
[_] : A → queue A -- a singleton queue | |
_⊕_ : queue A → queue A → queue A -- join two queues, one after the other | |
⟪_⟫ : (ℕ → A × queue A) → queue A -- a queue of infinitely many non-empty queues | |
-- push an element to the end of the queue | |
push : ∀ {A} → queue A → A → queue A | |
push q x = q ⊕ [ x ] | |
-- pop the first element off the queue | |
pop : ∀ {A} → queue A → Maybe (A × queue A) | |
pop ∙ = nothing | |
pop [ x ] = just (x , ∙) | |
pop (q₁ ⊕ q₂) with pop q₁ | pop q₂ | |
... | nothing | nothing = nothing | |
... | nothing | just (y , r) = just (y , r) | |
... | just (x , r) | _ = just (x , r ⊕ q₂) | |
pop {A} ⟪ qs ⟫ with (x , q ) ← (qs 0) = just (x , r) | |
where | |
r : queue A | |
r with pop q | |
... | nothing = ⟪ (λ n → qs (suc n)) ⟫ | |
... | just (y , s) = ⟪ (λ {0 → y , s ; (suc n) → qs (suc n)}) ⟫ | |
-- the queue 0, 1, 2, 3 | |
example₁ = [ 0 ] ⊕ [ 1 ] ⊕ [ 2 ] ⊕ [ 3 ] | |
-- the queue 0, 1, 4, 9, 16, ... | |
example₂ = ⟪ (λ n → (n * n , ∙)) ⟫ | |
-- the queue 0, 2, 4, 6, ..., 3 | |
example₃ = push ⟪ (λ n → (2 * n , ∙)) ⟫ 3 | |
-- the queue 0, 2, 4, ..., 1, 3, 5, ... | |
example₄ = ⟪ (λ n → (2 * n , ∙)) ⟫ ⊕ ⟪ (λ n → (2 * n + 1 , ∙)) ⟫ |
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