------------------------------------------------------------------------
-- The Agda standard library
--
-- Coinductive lists where all elements satisfy a predicate
------------------------------------------------------------------------
{-# OPTIONS --cubical-compatible --guardedness #-}
module Codata.Musical.Colist.Relation.Unary.All.Properties where
open import Codata.Musical.Colist.Base using (Colist; _∷_)
open import Codata.Musical.Colist.Relation.Unary.All using (All; []; _∷_)
open import Function.Base using (_∋_)
open import Level using (Level; _⊔_)
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
open import Relation.Unary using (Pred)
private
variable
a b p : Level
A : Set a
B : Set b
P : Pred A p
∷-injectiveˡ : ∀ {x xs px qx pxs qxs} →
(All P (x ∷ xs) ∋ px ∷ pxs) ≡ qx ∷ qxs → px ≡ qx
∷-injectiveˡ refl = refl
∷-injectiveʳ : ∀ {x xs px qx pxs qxs} →
(All P (x ∷ xs) ∋ px ∷ pxs) ≡ qx ∷ qxs → pxs ≡ qxs
∷-injectiveʳ refl = refl
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