------------------------------------------------------------------------ -- Pointwise equalities can be lifted ------------------------------------------------------------------------ {-# OPTIONS --universe-polymorphism #-} module Stream.Pointwise where open import Coinduction hiding (∞; unfold) open import Stream open import Stream.Equality import Stream.Programs as Prog open Prog hiding (lift; ⟦_⟧) open import Data.Nat open import Data.Fin using (Fin; zero; suc) open import Data.Vec as Vec using (Vec; _∷_) open import Relation.Binary open import Relation.Binary.PropositionalEquality private module IsEq {A : Set} = IsEquivalence (Setoid.isEquivalence (Stream.setoid A)) ------------------------------------------------------------------------ -- Definitions infix 8 _∞ infixr 7 _·_ infix 6 _⟨_⟩_ -- Expressions corresponding to pointwise definitions of streams. -- Indexed on the number of variables. -- It is possible to generalise this type, allowing variables to -- correspond to streams containing elements of arbitrary type, and -- letting the function arguments of _·_ and _⟨_⟩_ be more general. -- However, this would complicate the development, so I hesitate to do -- so without evidence that it would be genuinely useful. data Pointwise A n : Set where var : (x : Fin n) → Pointwise A n _∞ : (x : A) → Pointwise A n _·_ : (f : A → A) (xs : Pointwise A n) → Pointwise A n _⟨_⟩_ : (xs : Pointwise A n) (_∙_ : A → A → A) (ys : Pointwise A n) → Pointwise A n -- Stream semantics. ⟦_⟧ : ∀ {A n} → Pointwise A n → (Vec (Prog A) n → Prog A) ⟦ var x ⟧ ρ = Vec.lookup x ρ ⟦ x ∞ ⟧ ρ = x ∞ ⟦ f · xs ⟧ ρ = f · ⟦ xs ⟧ ρ ⟦ xs ⟨ _∙_ ⟩ ys ⟧ ρ = ⟦ xs ⟧ ρ ⟨ _∙_ ⟩ ⟦ ys ⟧ ρ -- Pointwise semantics. ⟪_⟫ : ∀ {A n} → Pointwise A n → (Vec A n → A) ⟪ var x ⟫ ρ = Vec.lookup x ρ ⟪ x ∞ ⟫ ρ = x ⟪ f · xs ⟫ ρ = f (⟪ xs ⟫ ρ) ⟪ xs ⟨ _∙_ ⟩ ys ⟫ ρ = ⟪ xs ⟫ ρ ∙ ⟪ ys ⟫ ρ ------------------------------------------------------------------------ -- Some lemmas used below private -- lookup is natural. lookup-nat : ∀ {a b n} {A : Set a} {B : Set b} (f : A → B) (x : Fin n) ρ → f (Vec.lookup x ρ) ≡ Vec.lookup x (Vec.map f ρ) lookup-nat f zero (x ∷ ρ) = refl lookup-nat f (suc i) (x ∷ ρ) = lookup-nat f i ρ ------------------------------------------------------------------------ -- The two semantics above are related via the function lift private -- Lifts a pointwise function to a function on stream programs. lift : ∀ {A B n} → (Vec A n → B) → Vec (Prog A) n → Prog B lift f xs = f (Vec.map headP xs) ≺ ♯ lift f (Vec.map tailP xs) -- lift is a congruence in its first argument. lift-cong : ∀ {A B n} {f g : Vec A n → B} → (∀ ρ → f ρ ≡ g ρ) → ∀ ρ → lift f ρ ≊ lift g ρ lift-cong hyp ρ = hyp (Vec.map headP ρ) ≺ ♯ lift-cong hyp (Vec.map tailP ρ) -- unfold xs ρ is the one-step unfolding of ⟦ xs ⟧ ρ. Note the -- similarity to lift. unfold : ∀ {A n} (xs : Pointwise A n) ρ → Prog A unfold xs ρ = ⟪ xs ⟫ (Vec.map headP ρ) ≺♯ ⟦ xs ⟧ (Vec.map tailP ρ) unfold-lemma : ∀ {A n} (xs : Pointwise A n) ρ → ⟦ xs ⟧ ρ ≊ unfold xs ρ unfold-lemma (var x) ρ = Vec.lookup x ρ ≊⟨ ≊-η (Vec.lookup x ρ) ⟩ headP (Vec.lookup x ρ) ≺♯ tailP (Vec.lookup x ρ) ≊⟨ lookup-nat headP x ρ ≺ ♯ ≈⇒≅ (IsEq.reflexive (cong Prog.⟦_⟧ (lookup-nat tailP x ρ))) ⟩ Vec.lookup x (Vec.map headP ρ) ≺♯ Vec.lookup x (Vec.map tailP ρ) ≡⟨ refl ⟩ unfold (var x) ρ ∎ unfold-lemma (x ∞) ρ = x ∞ ∎ unfold-lemma (f · xs) ρ = f · ⟦ xs ⟧ ρ ≊⟨ ·-cong f (⟦ xs ⟧ ρ) (unfold xs ρ) (unfold-lemma xs ρ) ⟩ f · unfold xs ρ ∎ unfold-lemma (xs ⟨ ∙ ⟩ ys) ρ = ⟦ xs ⟧ ρ ⟨ ∙ ⟩ ⟦ ys ⟧ ρ ≊⟨ ⟨ ∙ ⟩-cong (⟦ xs ⟧ ρ) (unfold xs ρ) (unfold-lemma xs ρ) (⟦ ys ⟧ ρ) (unfold ys ρ) (unfold-lemma ys ρ) ⟩ unfold xs ρ ⟨ ∙ ⟩ unfold ys ρ ∎ -- The two semantics are related. main-lemma : ∀ {A n} (xs : Pointwise A n) → ∀ ρ → ⟦ xs ⟧ ρ ≊ lift ⟪ xs ⟫ ρ main-lemma xs ρ = ⟦ xs ⟧ ρ ≊⟨ unfold-lemma xs ρ ⟩ unfold xs ρ ≡⟨ refl ⟩ ⟪ xs ⟫ (Vec.map headP ρ) ≺♯ ⟦ xs ⟧ (Vec.map tailP ρ) ≊⟨ refl ≺ ♯ main-lemma xs (Vec.map tailP ρ) ⟩ lift ⟪ xs ⟫ ρ ∎ ------------------------------------------------------------------------ -- To prove that two streams which are defined pointwise are equal, it -- is enough to reason about a single (arbitrary) point -- This function is a bit awkward to use, since the user has to come -- up with a suitable environment manually. The alternative function -- pointwise below may be slightly easier to use. pointwise' : ∀ {A n} (xs ys : Pointwise A n) → (∀ ρ → ⟪ xs ⟫ ρ ≡ ⟪ ys ⟫ ρ) → (∀ ρ → ⟦ xs ⟧ ρ ≊ ⟦ ys ⟧ ρ) pointwise' xs ys hyp ρ = ⟦ xs ⟧ ρ ≊⟨ main-lemma xs ρ ⟩ lift ⟪ xs ⟫ ρ ≊⟨ lift-cong hyp ρ ⟩ lift ⟪ ys ⟫ ρ ≊⟨ ≅-sym (main-lemma ys ρ) ⟩ ⟦ ys ⟧ ρ ∎ open import Data.Vec.N-ary -- Applies the function to all possible variables. app : ∀ {A} n → N-ary n (Pointwise A n) (Pointwise A n) → Pointwise A n app n f = f $ⁿ Vec.map var (Vec.allFin n) -- The type signature of this function may be a bit daunting, but once -- n, f and g are instantiated with well-behaved concrete values the -- remaining type evaluates nicely. pointwise : ∀ {A} n (f g : N-ary n (Pointwise A n) (Pointwise A n)) → Eq n _≡_ (curryⁿ ⟪ app n f ⟫) (curryⁿ ⟪ app n g ⟫) → Eq n _≊_ (curryⁿ ⟦ app n f ⟧) (curryⁿ ⟦ app n g ⟧) pointwise n f g hyp = curryⁿ-cong _≊_ ⟦ app n f ⟧ ⟦ app n g ⟧ (pointwise' (app n f) (app n g) (curryⁿ-cong⁻¹ _≡_ ⟪ app n f ⟫ ⟪ app n g ⟫ hyp)) ------------------------------------------------------------------------ -- Some examples private example₁ : suc · 0 ∞ ≊ 1 ∞ example₁ = pointwise 0 (suc · 0 ∞) (1 ∞) refl example₂ : ∀ s → suc · s ≊ 1 ∞ ⟨ _+_ ⟩ s example₂ = pointwise 1 (λ s → suc · s) (λ s → 1 ∞ ⟨ _+_ ⟩ s) (λ _ → refl) example₃ : ∀ s t u → (s ⟨ _+_ ⟩ t) ⟨ _+_ ⟩ u ≊ s ⟨ _+_ ⟩ (t ⟨ _+_ ⟩ u) example₃ = pointwise 3 (λ s t u → (s ⟨ _+_ ⟩ t) ⟨ _+_ ⟩ u) (λ s t u → s ⟨ _+_ ⟩ (t ⟨ _+_ ⟩ u)) +-assoc where open import Data.Nat.Properties