{-# OPTIONS --without-K --safe #-}
open import Relation.Binary using (Setoid; _⇒_; _Preserves_⟶_)
module Data.List.Relation.Binary.Sublist.Setoid.Properties
{c ℓ} (S : Setoid c ℓ) where
open import Data.List.Base hiding (_∷ʳ_)
import Data.List.Relation.Binary.Equality.Setoid as SetoidEquality
import Data.List.Relation.Binary.Sublist.Setoid as SetoidSublist
import Data.List.Relation.Binary.Sublist.Heterogeneous.Properties
as HeteroProperties
import Data.List.Membership.Setoid as SetoidMembership
open import Data.List.Relation.Unary.Any using (Any)
open import Data.Nat using (_≤_; _≥_; z≤n; s≤s)
import Data.Nat.Properties as ℕₚ
import Data.Maybe.Relation.Unary.All as Maybe
open import Function
open import Function.Bijection using (_⤖_)
open import Function.Equivalence using (_⇔_)
open import Relation.Binary.PropositionalEquality using (_≡_; refl)
open import Relation.Unary using (Pred; Decidable; Irrelevant)
open import Relation.Nullary using (¬_)
open import Relation.Nullary.Negation using (¬?)
open Setoid S using (_≈_) renaming (Carrier to A)
open SetoidEquality S using (_≋_; ≋-refl)
open SetoidSublist S hiding (map)
open SetoidMembership S using (_∈_)
module _ {xs ys : List A} where
∷-injectiveˡ : ∀ {x y} {px qx : x ≈ y} {pxs qxs : xs ⊆ ys} →
((x ∷ xs) ⊆ (y ∷ ys) ∋ px ∷ pxs) ≡ (qx ∷ qxs) → px ≡ qx
∷-injectiveˡ refl = refl
∷-injectiveʳ : ∀ {x y} {px qx : x ≈ y} {pxs qxs : xs ⊆ ys} →
((x ∷ xs) ⊆ (y ∷ ys) ∋ px ∷ pxs) ≡ (qx ∷ qxs) → pxs ≡ qxs
∷-injectiveʳ refl = refl
∷ʳ-injective : ∀ {y} {pxs qxs : xs ⊆ ys} → y ∷ʳ pxs ≡ y ∷ʳ qxs → pxs ≡ qxs
∷ʳ-injective refl = refl
tail-⊆ : ∀ xs → Maybe.All (_⊆ xs) (tail xs)
tail-⊆ xs = HeteroProperties.tail-Sublist ⊆-refl
take-⊆ : ∀ n xs → take n xs ⊆ xs
take-⊆ n xs = HeteroProperties.take-Sublist n ⊆-refl
drop-⊆ : ∀ n xs → drop n xs ⊆ xs
drop-⊆ n xs = HeteroProperties.drop-Sublist n ⊆-refl
module _ {p} {P : Pred A p} (P? : Decidable P) where
takeWhile-⊆ : ∀ xs → takeWhile P? xs ⊆ xs
takeWhile-⊆ xs = HeteroProperties.takeWhile-Sublist P? ⊆-refl
dropWhile-⊆ : ∀ xs → dropWhile P? xs ⊆ xs
dropWhile-⊆ xs = HeteroProperties.dropWhile-Sublist P? ⊆-refl
filter-⊆ : ∀ xs → filter P? xs ⊆ xs
filter-⊆ xs = HeteroProperties.filter-Sublist P? ⊆-refl
module _ {p} {P : Pred A p} (P? : Decidable P) where
takeWhile⊆filter : ∀ xs → takeWhile P? xs ⊆ filter P? xs
takeWhile⊆filter xs = HeteroProperties.takeWhile-filter P? {xs} ≋-refl
filter⊆dropWhile : ∀ xs → filter P? xs ⊆ dropWhile (¬? ∘ P?) xs
filter⊆dropWhile xs = HeteroProperties.filter-dropWhile P? {xs} ≋-refl
module _ {as bs : List A} where
∷ˡ⁻ : ∀ {a} → a ∷ as ⊆ bs → as ⊆ bs
∷ˡ⁻ = HeteroProperties.∷ˡ⁻
∷ʳ⁻ : ∀ {a b} → ¬ (a ≈ b) → a ∷ as ⊆ b ∷ bs → a ∷ as ⊆ bs
∷ʳ⁻ = HeteroProperties.∷ʳ⁻
∷⁻ : ∀ {a b} → a ∷ as ⊆ b ∷ bs → as ⊆ bs
∷⁻ = HeteroProperties.∷⁻
module _ {b ℓ} (R : Setoid b ℓ) where
open Setoid R using () renaming (Carrier to B; _≈_ to _≈′_)
open SetoidSublist R using () renaming (_⊆_ to _⊆′_)
map⁺ : ∀ {as bs} {f : A → B} → f Preserves _≈_ ⟶ _≈′_ →
as ⊆ bs → map f as ⊆′ map f bs
map⁺ {f = f} f-resp as⊆bs =
HeteroProperties.map⁺ f f (SetoidSublist.map S f-resp as⊆bs)
module _ {as bs : List A} where
++⁺ˡ : ∀ cs → as ⊆ bs → as ⊆ cs ++ bs
++⁺ˡ = HeteroProperties.++ˡ
++⁺ʳ : ∀ cs → as ⊆ bs → as ⊆ bs ++ cs
++⁺ʳ = HeteroProperties.++ʳ
++⁺ : ∀ {cs ds} → as ⊆ bs → cs ⊆ ds → as ++ cs ⊆ bs ++ ds
++⁺ = HeteroProperties.++⁺
++⁻ : ∀ {cs ds} → length as ≡ length bs → as ++ cs ⊆ bs ++ ds → cs ⊆ ds
++⁻ = HeteroProperties.++⁻
module _ where
take⁺ : ∀ {m n} {xs} → m ≤ n → take m xs ⊆ take n xs
take⁺ m≤n = HeteroProperties.take⁺ m≤n ≋-refl
module _ {m n} {xs ys : List A} where
drop⁺ : m ≥ n → xs ⊆ ys → drop m xs ⊆ drop n ys
drop⁺ = HeteroProperties.drop⁺
module _ {m n} {xs : List A} where
drop⁺-≥ : m ≥ n → drop m xs ⊆ drop n xs
drop⁺-≥ m≥n = drop⁺ m≥n ⊆-refl
module _ {xs ys : List A} where
drop⁺-⊆ : ∀ n → xs ⊆ ys → drop n xs ⊆ drop n ys
drop⁺-⊆ n xs⊆ys = drop⁺ {n} ℕₚ.≤-refl xs⊆ys
module _ {p q} {P : Pred A p} {Q : Pred A q}
(P? : Decidable P) (Q? : Decidable Q) where
takeWhile⁺ : ∀ {xs} → (∀ {a b} → a ≈ b → P a → Q b) →
takeWhile P? xs ⊆ takeWhile Q? xs
takeWhile⁺ {xs} P⇒Q = HeteroProperties.⊆-takeWhile-Sublist P? Q? {xs} P⇒Q ≋-refl
dropWhile⁺ : ∀ {xs} → (∀ {a b} → a ≈ b → Q b → P a) →
dropWhile P? xs ⊆ dropWhile Q? xs
dropWhile⁺ {xs} P⇒Q = HeteroProperties.⊇-dropWhile-Sublist P? Q? {xs} P⇒Q ≋-refl
module _ {p q} {P : Pred A p} {Q : Pred A q}
(P? : Decidable P) (Q? : Decidable Q) where
filter⁺ : ∀ {as bs} → (∀ {a b} → a ≈ b → P a → Q b) →
as ⊆ bs → filter P? as ⊆ filter Q? bs
filter⁺ = HeteroProperties.⊆-filter-Sublist P? Q?
module _ {as bs : List A} where
reverseAcc⁺ : ∀ {cs ds} → as ⊆ bs → cs ⊆ ds →
reverseAcc cs as ⊆ reverseAcc ds bs
reverseAcc⁺ = HeteroProperties.reverseAcc⁺
reverse⁺ : as ⊆ bs → reverse as ⊆ reverse bs
reverse⁺ = HeteroProperties.reverse⁺
reverse⁻ : reverse as ⊆ reverse bs → as ⊆ bs
reverse⁻ = HeteroProperties.reverse⁻
module _ {a b} {A : Set a} {B : Set b} {a as b bs} where
∷⁻¹ : a ≈ b → as ⊆ bs ⇔ a ∷ as ⊆ b ∷ bs
∷⁻¹ = HeteroProperties.∷⁻¹
∷ʳ⁻¹ : ¬ (a ≈ b) → a ∷ as ⊆ bs ⇔ a ∷ as ⊆ b ∷ bs
∷ʳ⁻¹ = HeteroProperties.∷ʳ⁻¹
module _ where
length-mono-≤ : ∀ {as bs} → as ⊆ bs → length as ≤ length bs
length-mono-≤ = HeteroProperties.length-mono-≤
to-≋ : ∀ {as bs} → length as ≡ length bs → as ⊆ bs → as ≋ bs
to-≋ = HeteroProperties.toPointwise
module _ {a b} {A : Set a} {B : Set b} where
[]⊆-irrelevant : Irrelevant ([] ⊆_)
[]⊆-irrelevant = HeteroProperties.Sublist-[]-irrelevant
module _ {x xs} where
to∈-injective : ∀ {p q : [ x ] ⊆ xs} → to∈ p ≡ to∈ q → p ≡ q
to∈-injective = HeteroProperties.toAny-injective
from∈-injective : ∀ {p q : x ∈ xs} → from∈ p ≡ from∈ q → p ≡ q
from∈-injective = HeteroProperties.fromAny-injective
to∈∘from∈≗id : ∀ (p : x ∈ xs) → to∈ (from∈ p) ≡ p
to∈∘from∈≗id = HeteroProperties.toAny∘fromAny≗id
[x]⊆xs⤖x∈xs : ([ x ] ⊆ xs) ⤖ (x ∈ xs)
[x]⊆xs⤖x∈xs = HeteroProperties.Sublist-[x]-bijection