------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties related to AllPairs
------------------------------------------------------------------------
{-# OPTIONS --cubical-compatible --safe #-}
module Data.Vec.Relation.Unary.AllPairs.Properties where
open import Data.Vec.Base using (_∷_; map; _++_; concat; take; drop; tabulate)
import Data.Vec.Properties as Vec
open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
import Data.Vec.Relation.Unary.All.Properties as All
open import Data.Vec.Relation.Unary.AllPairs as AllPairs using (AllPairs; []; _∷_)
open import Data.Bool.Base using (true; false)
open import Data.Fin.Base using (Fin)
open import Data.Fin.Properties using (suc-injective)
open import Data.Nat.Base using (zero; suc; _+_)
open import Function.Base using (_∘_)
open import Level using (Level)
open import Relation.Binary.Core using (Rel)
open import Relation.Binary.PropositionalEquality.Core using (_≢_)
private
variable
a b c p ℓ : Level
A : Set a
B : Set b
C : Set c
------------------------------------------------------------------------
-- Introduction (⁺) and elimination (⁻) rules for vector operations
------------------------------------------------------------------------
-- map
module _ {R : Rel A ℓ} {f : B → A} where
map⁺ : ∀ {n xs} → AllPairs (λ x y → R (f x) (f y)) {n} xs →
AllPairs R {n} (map f xs)
map⁺ [] = []
map⁺ (x∉xs ∷ xs!) = All.map⁺ x∉xs ∷ map⁺ xs!
------------------------------------------------------------------------
-- ++
module _ {R : Rel A ℓ} where
++⁺ : ∀ {n m xs ys} → AllPairs R {n} xs → AllPairs R {m} ys →
All (λ x → All (R x) ys) xs → AllPairs R (xs ++ ys)
++⁺ [] Rys _ = Rys
++⁺ (px ∷ Rxs) Rys (Rxys ∷ Rxsys) = All.++⁺ px Rxys ∷ ++⁺ Rxs Rys Rxsys
------------------------------------------------------------------------
-- concat
module _ {R : Rel A ℓ} where
concat⁺ : ∀ {n m xss} → All (AllPairs R {n}) {m} xss →
AllPairs (λ xs ys → All (λ x → All (R x) ys) xs) xss →
AllPairs R (concat xss)
concat⁺ [] [] = []
concat⁺ (pxs ∷ pxss) (Rxsxss ∷ Rxss) = ++⁺ pxs (concat⁺ pxss Rxss)
(All.map All.concat⁺ (All.All-swap Rxsxss))
------------------------------------------------------------------------
-- take and drop
module _ {R : Rel A ℓ} where
take⁺ : ∀ {n} m {xs} → AllPairs R {m + n} xs → AllPairs R {m} (take m xs)
take⁺ zero pxs = []
take⁺ (suc m) {x ∷ xs} (px ∷ pxs) = All.take⁺ m px ∷ take⁺ m pxs
drop⁺ : ∀ {n} m {xs} → AllPairs R {m + n} xs → AllPairs R {n} (drop m xs)
drop⁺ zero pxs = pxs
drop⁺ (suc m) (_ ∷ pxs) = drop⁺ m pxs
------------------------------------------------------------------------
-- tabulate
module _ {R : Rel A ℓ} where
tabulate⁺ : ∀ {n} {f : Fin n → A} → (∀ {i j} → i ≢ j → R (f i) (f j)) →
AllPairs R (tabulate f)
tabulate⁺ {zero} fᵢ~fⱼ = []
tabulate⁺ {suc n} fᵢ~fⱼ =
All.tabulate⁺ (λ j → fᵢ~fⱼ λ()) ∷
tabulate⁺ (fᵢ~fⱼ ∘ (_∘ suc-injective))
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