{-# OPTIONS --cubical-compatible --sized-types #-}
module Codata.Sized.Stream.Properties where
open import Level using (Level)
open import Size
open import Codata.Sized.Thunk as Thunk using (Thunk; force)
open import Codata.Sized.Stream
open import Codata.Sized.Stream.Bisimilarity
open import Data.Nat.Base
open import Data.Nat.GeneralisedArithmetic using (fold; fold-pull)
open import Data.List.Base as List using ([]; _∷_)
open import Data.List.NonEmpty as List⁺ using (List⁺; _∷_)
import Data.List.Relation.Binary.Equality.Propositional as ≋
open import Data.Product.Base as Product using (_,_)
open import Data.Vec.Base as Vec using (_∷_)
open import Function.Base using (id; _$_; _∘′_; const)
open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_; _≢_)
open import Relation.Binary.PropositionalEquality.Properties
using (module ≡-Reasoning)
private
variable
a b c : Level
A : Set a
B : Set b
C : Set c
i : Size
lookup-repeat-identity : (n : ℕ) (a : A) → lookup (repeat a) n ≡ a
lookup-repeat-identity zero a = ≡.refl
lookup-repeat-identity (suc n) a = lookup-repeat-identity n a
take-repeat-identity : (n : ℕ) (a : A) → take n (repeat a) ≡ Vec.replicate n a
take-repeat-identity zero a = ≡.refl
take-repeat-identity (suc n) a = ≡.cong (a Vec.∷_) (take-repeat-identity n a)
splitAt-repeat-identity : (n : ℕ) (a : A) → splitAt n (repeat a) ≡ (Vec.replicate n a , repeat a)
splitAt-repeat-identity zero a = ≡.refl
splitAt-repeat-identity (suc n) a = ≡.cong (Product.map₁ (a ∷_)) (splitAt-repeat-identity n a)
replicate-repeat : ∀ {i} (n : ℕ) (a : A) → i ⊢ List.replicate n a ++ repeat a ≈ repeat a
replicate-repeat zero a = refl
replicate-repeat (suc n) a = ≡.refl ∷ λ where .force → replicate-repeat n a
cycle-replicate : ∀ {i} (n : ℕ) (n≢0 : n ≢ 0) (a : A) → i ⊢ cycle (List⁺.replicate n n≢0 a) ≈ repeat a
cycle-replicate {i} n n≢0 a = let as = List⁺.replicate n n≢0 a in begin
cycle as ≡⟨⟩
as ⁺++ _ ≈⟨ ⁺++⁺ ≋.≋-refl (λ where .force → cycle-replicate n n≢0 a) ⟩
as ⁺++ (λ where .force → repeat a) ≈⟨ ≡.refl ∷ (λ where .force → replicate-repeat (pred n) a) ⟩
repeat a ∎ where open ≈-Reasoning
module _ {a b} {A : Set a} {B : Set b} where
map-repeat : ∀ (f : A → B) a {i} → i ⊢ map f (repeat a) ≈ repeat (f a)
map-repeat f a = ≡.refl ∷ λ where .force → map-repeat f a
ap-repeat : ∀ (f : A → B) a {i} → i ⊢ ap (repeat f) (repeat a) ≈ repeat (f a)
ap-repeat f a = ≡.refl ∷ λ where .force → ap-repeat f a
ap-repeatˡ : ∀ (f : A → B) as {i} → i ⊢ ap (repeat f) as ≈ map f as
ap-repeatˡ f (a ∷ as) = ≡.refl ∷ λ where .force → ap-repeatˡ f (as .force)
ap-repeatʳ : ∀ (fs : Stream (A → B) ∞) (a : A) {i} → i ⊢ ap fs (repeat a) ≈ map (_$ a) fs
ap-repeatʳ (f ∷ fs) a = ≡.refl ∷ λ where .force → ap-repeatʳ (fs .force) a
map-++ : ∀ {i} (f : A → B) as xs → i ⊢ map f (as ++ xs) ≈ List.map f as ++ map f xs
map-++ f [] xs = refl
map-++ f (a ∷ as) xs = ≡.refl ∷ λ where .force → map-++ f as xs
map-⁺++ : ∀ {i} (f : A → B) as xs → i ⊢ map f (as ⁺++ xs) ≈ List⁺.map f as ⁺++ Thunk.map (map f) xs
map-⁺++ f (a ∷ as) xs = ≡.refl ∷ (λ where .force → map-++ f as (xs .force))
map-cycle : ∀ {i} (f : A → B) as → i ⊢ map f (cycle as) ≈ cycle (List⁺.map f as)
map-cycle f as = begin
map f (cycle as) ≈⟨ map-⁺++ f as _ ⟩
List⁺.map f as ⁺++ _ ≈⟨ ⁺++⁺ ≋.≋-refl (λ where .force → map-cycle f as) ⟩
cycle (List⁺.map f as) ∎ where open ≈-Reasoning
map-id : ∀ (as : Stream A ∞) → i ⊢ map id as ≈ as
map-id (a ∷ as) = ≡.refl ∷ λ where .force → map-id (as .force)
map-∘ : ∀ (f : A → B) (g : B → C) as → i ⊢ map g (map f as) ≈ map (g ∘′ f) as
map-∘ f g (a ∷ as) = ≡.refl ∷ λ where .force → map-∘ f g (as .force)
splitAt-map : ∀ n (f : A → B) xs →
splitAt n (map f xs) ≡ Product.map (Vec.map f) (map f) (splitAt n xs)
splitAt-map zero f xs = ≡.refl
splitAt-map (suc n) f (x ∷ xs) =
≡.cong (Product.map₁ (f x Vec.∷_)) (splitAt-map n f (xs .force))
lookup-iterate-identity : ∀ n f (a : A) → lookup (iterate f a) n ≡ fold a f n
lookup-iterate-identity zero f a = ≡.refl
lookup-iterate-identity (suc n) f a = begin
lookup (iterate f a) (suc n) ≡⟨⟩
lookup (iterate f (f a)) n ≡⟨ lookup-iterate-identity n f (f a) ⟩
fold (f a) f n ≡⟨ fold-pull a f (const ∘′ f) (f a) ≡.refl (λ _ → ≡.refl) n ⟩
f (fold a f n) ≡⟨⟩
fold a f (suc n) ∎ where open ≡-Reasoning
map-identity = map-id
{-# WARNING_ON_USAGE map-identity
"Warning: map-identity was deprecated in v2.0.
Please use map-id instead."
#-}
map-map-fusion = map-∘
{-# WARNING_ON_USAGE map-map-fusion
"Warning: map-map-fusion was deprecated in v2.0.
Please use map-∘ instead."
#-}