module Data.List.All.Properties where
open import Data.Bool
open import Data.Bool.Properties
open import Data.Empty
open import Data.List as List
import Data.List.Any as Any; open Any.Membership-≡
open import Data.List.All as All using (All; []; _∷_)
open import Data.List.Any using (Any; here; there)
open import Data.Product as Prod
open import Function
open import Function.Equality using (_⟨$⟩_)
open import Function.Equivalence using (_⇔_; module Equivalence)
open import Function.Inverse using (_↔_)
open import Function.Surjection using (_↠_)
open import Relation.Binary.PropositionalEquality as P using (_≡_)
open import Relation.Nullary
open import Relation.Unary using (Decidable) renaming (_⊆_ to _⋐_)
All-map : ∀ {a b p} {A : Set a} {B : Set b} {P : B → Set p}
{f : A → B} {xs} →
All (P ∘ f) xs → All P (List.map f xs)
All-map [] = []
All-map (p ∷ ps) = p ∷ All-map ps
map-All : ∀ {a b p} {A : Set a} {B : Set b} {P : B → Set p}
{f : A → B} {xs} →
All P (List.map f xs) → All (P ∘ f) xs
map-All {xs = []} [] = []
map-All {xs = _ ∷ _} (p ∷ ps) = p ∷ map-All ps
gmap : ∀ {a b p q}
{A : Set a} {B : Set b} {P : A → Set p} {Q : B → Set q}
{f : A → B} →
P ⋐ Q ∘ f → All P ⋐ All Q ∘ List.map f
gmap g = All-map ∘ All.map g
All-all : ∀ {a} {A : Set a} (p : A → Bool) {xs} →
All (T ∘ p) xs → T (all p xs)
All-all p [] = _
All-all p (px ∷ pxs) = Equivalence.from T-∧ ⟨$⟩ (px , All-all p pxs)
all-All : ∀ {a} {A : Set a} (p : A → Bool) xs →
T (all p xs) → All (T ∘ p) xs
all-All p [] _ = []
all-All p (x ∷ xs) px∷xs with Equivalence.to (T-∧ {p x}) ⟨$⟩ px∷xs
all-All p (x ∷ xs) px∷xs | (px , pxs) = px ∷ all-All p xs pxs
anti-mono : ∀ {a p} {A : Set a} {P : A → Set p} {xs ys} →
xs ⊆ ys → All P ys → All P xs
anti-mono xs⊆ys pys = All.tabulate (All.lookup pys ∘ xs⊆ys)
all-anti-mono : ∀ {a} {A : Set a} (p : A → Bool) {xs ys} →
xs ⊆ ys → T (all p ys) → T (all p xs)
all-anti-mono p xs⊆ys = All-all p ∘ anti-mono xs⊆ys ∘ all-All p _
private
++⁺ : ∀ {a p} {A : Set a} {P : A → Set p} {xs ys : List A} →
All P xs → All P ys → All P (xs ++ ys)
++⁺ [] pys = pys
++⁺ (px ∷ pxs) pys = px ∷ ++⁺ pxs pys
++⁻ : ∀ {a p} {A : Set a} {P : A → Set p} (xs : List A) {ys} →
All P (xs ++ ys) → All P xs × All P ys
++⁻ [] p = [] , p
++⁻ (x ∷ xs) (px ∷ pxs) = Prod.map (_∷_ px) id $ ++⁻ _ pxs
++⁺∘++⁻ : ∀ {a p} {A : Set a} {P : A → Set p} xs {ys}
(p : All P (xs ++ ys)) → uncurry′ ++⁺ (++⁻ xs p) ≡ p
++⁺∘++⁻ [] p = P.refl
++⁺∘++⁻ (x ∷ xs) (px ∷ pxs) = P.cong (_∷_ px) $ ++⁺∘++⁻ xs pxs
++⁻∘++⁺ : ∀ {a p} {A : Set a} {P : A → Set p} {xs ys}
(p : All P xs × All P ys) → ++⁻ xs (uncurry ++⁺ p) ≡ p
++⁻∘++⁺ ([] , pys) = P.refl
++⁻∘++⁺ (px ∷ pxs , pys) rewrite ++⁻∘++⁺ (pxs , pys) = P.refl
++↔ : ∀ {a p} {A : Set a} {P : A → Set p} {xs ys} →
(All P xs × All P ys) ↔ All P (xs ++ ys)
++↔ {P = P} {xs = xs} = record
{ to = P.→-to-⟶ $ uncurry ++⁺
; from = P.→-to-⟶ $ ++⁻ xs
; inverse-of = record
{ left-inverse-of = ++⁻∘++⁺
; right-inverse-of = ++⁺∘++⁻ xs
}
}
¬Any↠All¬ : ∀ {a p} {A : Set a} {P : A → Set p} {xs} →
¬ Any P xs ↠ All (¬_ ∘ P) xs
¬Any↠All¬ {P = P} = record
{ to = P.→-to-⟶ (to _)
; surjective = record
{ from = P.→-to-⟶ from
; right-inverse-of = to∘from
}
}
where
to : ∀ xs → ¬ Any P xs → All (¬_ ∘ P) xs
to [] ¬p = []
to (x ∷ xs) ¬p = ¬p ∘ here ∷ to xs (¬p ∘ there)
from : ∀ {xs} → All (¬_ ∘ P) xs → ¬ Any P xs
from [] ()
from (¬p ∷ _) (here p) = ¬p p
from (_ ∷ ¬p) (there p) = from ¬p p
to∘from : ∀ {xs} (¬p : All (¬_ ∘ P) xs) → to xs (from ¬p) ≡ ¬p
to∘from [] = P.refl
to∘from (¬p ∷ ¬ps) = P.cong₂ _∷_ P.refl (to∘from ¬ps)
from∘to : P.Extensionality _ _ →
∀ xs → (¬p : ¬ Any P xs) → from (to xs ¬p) ≡ ¬p
from∘to ext [] ¬p = ext λ ()
from∘to ext (x ∷ xs) ¬p = ext λ
{ (here p) → P.refl
; (there p) → P.cong (λ f → f p) $ from∘to ext xs (¬p ∘ there)
}
Any¬→¬All : ∀ {a p} {A : Set a} {P : A → Set p} {xs} →
Any (¬_ ∘ P) xs → ¬ All P xs
Any¬→¬All (here ¬p) = ¬p ∘ All.head
Any¬→¬All (there ¬p) = Any¬→¬All ¬p ∘ All.tail
Any¬⇔¬All : ∀ {a p} {A : Set a} {P : A → Set p} {xs} →
Decidable P → Any (¬_ ∘ P) xs ⇔ ¬ All P xs
Any¬⇔¬All {P = P} dec = record
{ to = P.→-to-⟶ Any¬→¬All
; from = P.→-to-⟶ (from _)
}
where
from : ∀ xs → ¬ All P xs → Any (¬_ ∘ P) xs
from [] ¬∀ = ⊥-elim (¬∀ [])
from (x ∷ xs) ¬∀ with dec x
... | yes p = there (from xs (¬∀ ∘ _∷_ p))
... | no ¬p = here ¬p
to∘from : P.Extensionality _ _ →
∀ {xs} (¬∀ : ¬ All P xs) → Any¬→¬All (from xs ¬∀) ≡ ¬∀
to∘from ext ¬∀ = ext (⊥-elim ∘ ¬∀)