```------------------------------------------------------------------------
-- The Agda standard library
--
-- All (□) for containers
------------------------------------------------------------------------

{-# OPTIONS --cubical-compatible --safe #-}

module Data.Container.Relation.Unary.All where

open import Level using (_⊔_)
open import Relation.Unary using (Pred; _⊆_)
open import Data.Product.Base using (_,_; proj₁; proj₂; ∃)
open import Function.Base using (_∘′_; id)

open import Data.Container.Core hiding (map)
import Data.Container.Morphism as M

record □ {s p} (C : Container s p) {x ℓ} {X : Set x}
(P : Pred X ℓ) (cx : ⟦ C ⟧ X) : Set (p ⊔ ℓ) where
constructor all
field proof : ∀ p → P (proj₂ cx p)

module _ {s₁ p₁ s₂ p₂} {C : Container s₁ p₁} {D : Container s₂ p₂}
{x ℓ ℓ′} {X : Set x} {P : Pred X ℓ} {Q : Pred X ℓ′}
where

map : (f : C ⇒ D) → P ⊆ Q → □ C P ⊆ □ D Q ∘′ ⟪ f ⟫
map f P⊆Q (all prf) .□.proof p = P⊆Q (prf (f .position p))

module _ {s₁ p₁ s₂ p₂} {C : Container s₁ p₁} {D : Container s₂ p₂}
{x ℓ} {X : Set x} {P : Pred X ℓ}
where

map₁ : (f : C ⇒ D) → □ C P ⊆ □ D P ∘′ ⟪ f ⟫
map₁ f = map f id

module _ {s p} {C : Container s p}
{x ℓ ℓ′} {X : Set x} {P : Pred X ℓ} {Q : Pred X ℓ′}
where

map₂ : P ⊆ Q → □ C P ⊆ □ C Q
map₂ = map (M.id C)
```