module Data.List.All.Properties where
open import Data.Bool.Base using (Bool; T)
open import Data.Bool.Properties
open import Data.Empty
open import Data.Fin using (Fin) renaming (zero to fzero; suc to fsuc)
open import Data.List.Base
open import Data.List.Any.Membership.Propositional
open import Data.List.All as All using (All; []; _∷_)
open import Data.List.Any using (Any; here; there)
open import Data.Nat using (zero; suc; z≤n; s≤s; _<_)
open import Data.Product as Prod using (_×_; _,_; uncurry; uncurry′)
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; Universal) renaming (_⊆_ to _⋐_)
module _ {a p} {A : Set a} {P : A → Set p} where
All-universal : Universal P → ∀ xs → All P xs
All-universal u [] = []
All-universal u (x ∷ xs) = u x ∷ All-universal u xs
module _ {a p} {A : Set a} {P : A → Set p} where
¬Any⇒All¬ : ∀ xs → ¬ Any P xs → All (¬_ ∘ P) xs
¬Any⇒All¬ [] ¬p = []
¬Any⇒All¬ (x ∷ xs) ¬p = ¬p ∘ here ∷ ¬Any⇒All¬ xs (¬p ∘ there)
All¬⇒¬Any : ∀ {xs} → All (¬_ ∘ P) xs → ¬ Any P xs
All¬⇒¬Any [] ()
All¬⇒¬Any (¬p ∷ _) (here p) = ¬p p
All¬⇒¬Any (_ ∷ ¬p) (there p) = All¬⇒¬Any ¬p p
¬All⇒Any¬ : Decidable P → ∀ xs → ¬ All P xs → Any (¬_ ∘ P) xs
¬All⇒Any¬ dec [] ¬∀ = ⊥-elim (¬∀ [])
¬All⇒Any¬ dec (x ∷ xs) ¬∀ with dec x
... | yes p = there (¬All⇒Any¬ dec xs (¬∀ ∘ _∷_ p))
... | no ¬p = here ¬p
Any¬→¬All : ∀ {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¬ : ∀ {xs} → (¬ Any P xs) ↠ All (¬_ ∘ P) xs
¬Any↠All¬ = record
{ to = P.→-to-⟶ (¬Any⇒All¬ _)
; surjective = record
{ from = P.→-to-⟶ All¬⇒¬Any
; right-inverse-of = to∘from
}
}
where
to∘from : ∀ {xs} (¬p : All (¬_ ∘ P) xs) → ¬Any⇒All¬ xs (All¬⇒¬Any ¬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) → All¬⇒¬Any (¬Any⇒All¬ 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 : ∀ {xs} → Decidable P → Any (¬_ ∘ P) xs ⇔ (¬ All P xs)
Any¬⇔¬All dec = record
{ to = P.→-to-⟶ Any¬→¬All
; from = P.→-to-⟶ (¬All⇒Any¬ dec _)
}
where
to∘from : P.Extensionality _ _ →
∀ {xs} (¬∀ : ¬ All P xs) → Any¬→¬All (¬All⇒Any¬ dec xs ¬∀) ≡ ¬∀
to∘from ext ¬∀ = ext (⊥-elim ∘ ¬∀)
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 _
module _{a b} {A : Set a} {B : Set b} where
All-map : ∀ {p} {P : B → Set p} {f : A → B} {xs} →
All (P ∘ f) xs → All P (map f xs)
All-map [] = []
All-map (p ∷ ps) = p ∷ All-map ps
map-All : ∀ {p} {P : B → Set p} {f : A → B} {xs} →
All P (map f xs) → All (P ∘ f) xs
map-All {xs = []} [] = []
map-All {xs = _ ∷ _} (p ∷ ps) = p ∷ map-All ps
gmap : ∀ {p q} {P : A → Set p} {Q : B → Set q} {f : A → B} →
P ⋐ Q ∘ f → All P ⋐ All Q ∘ map f
gmap g = All-map ∘ All.map g
module _ {a p} {A : Set a} {P : A → Set p} where
++⁺ : ∀ {xs ys} → All P xs → All P ys → All P (xs ++ ys)
++⁺ [] pys = pys
++⁺ (px ∷ pxs) pys = px ∷ ++⁺ pxs pys
++⁻ˡ : ∀ xs {ys} → All P (xs ++ ys) → All P xs
++⁻ˡ [] p = []
++⁻ˡ (x ∷ xs) (px ∷ pxs) = px ∷ (++⁻ˡ _ pxs)
++⁻ʳ : ∀ xs {ys} → All P (xs ++ ys) → All P ys
++⁻ʳ [] p = p
++⁻ʳ (x ∷ xs) (px ∷ pxs) = ++⁻ʳ xs pxs
++⁻ : ∀ xs {ys} → All P (xs ++ ys) → All P xs × All P ys
++⁻ [] p = [] , p
++⁻ (x ∷ xs) (px ∷ pxs) = Prod.map (px ∷_) id (++⁻ _ pxs)
++↔ : ∀ {xs ys} → (All P xs × All P ys) ↔ All P (xs ++ ys)
++↔ {xs} = record
{ to = P.→-to-⟶ $ uncurry ++⁺
; from = P.→-to-⟶ $ ++⁻ xs
; inverse-of = record
{ left-inverse-of = ++⁻∘++⁺
; right-inverse-of = ++⁺∘++⁻ xs
}
}
where
++⁺∘++⁻ : ∀ xs {ys} (p : All P (xs ++ ys)) →
uncurry′ ++⁺ (++⁻ xs p) ≡ p
++⁺∘++⁻ [] p = P.refl
++⁺∘++⁻ (x ∷ xs) (px ∷ pxs) = P.cong (_∷_ px) $ ++⁺∘++⁻ xs pxs
++⁻∘++⁺ : ∀ {xs ys} (p : All P xs × All P ys) →
++⁻ xs (uncurry ++⁺ p) ≡ p
++⁻∘++⁺ ([] , pys) = P.refl
++⁻∘++⁺ (px ∷ pxs , pys) rewrite ++⁻∘++⁺ (pxs , pys) = P.refl
module _ {a p} {A : Set a} {P : A → Set p} where
concat⁺ : ∀ {xss} → All (All P) xss → All P (concat xss)
concat⁺ [] = []
concat⁺ (pxs ∷ pxss) = ++⁺ pxs (concat⁺ pxss)
concat⁻ : ∀ {xss} → All P (concat xss) → All (All P) xss
concat⁻ {[]} [] = []
concat⁻ {xs ∷ xss} pxs = ++⁻ˡ xs pxs ∷ concat⁻ (++⁻ʳ xs pxs)
module _ {a p} {A : Set a} {P : A → Set p} where
drop⁺ : ∀ {xs} n → All P xs → All P (drop n xs)
drop⁺ zero pxs = pxs
drop⁺ (suc n) [] = []
drop⁺ (suc n) (px ∷ pxs) = drop⁺ n pxs
take⁺ : ∀ {xs} n → All P xs → All P (take n xs)
take⁺ zero pxs = []
take⁺ (suc n) [] = []
take⁺ (suc n) (px ∷ pxs) = px ∷ take⁺ n pxs
module _ {a p} {A : Set a} {P : A → Set p} where
applyUpTo⁺₁ : ∀ f n → (∀ {i} → i < n → P (f i)) → All P (applyUpTo f n)
applyUpTo⁺₁ f zero Pf = []
applyUpTo⁺₁ f (suc n) Pf = Pf (s≤s z≤n) ∷ applyUpTo⁺₁ (f ∘ suc) n (Pf ∘ s≤s)
applyUpTo⁺₂ : ∀ f n → (∀ i → P (f i)) → All P (applyUpTo f n)
applyUpTo⁺₂ f n Pf = applyUpTo⁺₁ f n (λ _ → Pf _)
applyUpTo⁻ : ∀ f n → All P (applyUpTo f n) → ∀ {i} → i < n → P (f i)
applyUpTo⁻ f zero pxs ()
applyUpTo⁻ f (suc n) (px ∷ _) (s≤s z≤n) = px
applyUpTo⁻ f (suc n) (_ ∷ pxs) (s≤s (s≤s i<n)) =
applyUpTo⁻ (f ∘ suc) n pxs (s≤s i<n)
module _ {a p} {A : Set a} {P : A → Set p} where
tabulate⁺ : ∀ {n} {f : Fin n → A} →
(∀ i → P (f i)) → All P (tabulate f)
tabulate⁺ {zero} Pf = []
tabulate⁺ {suc n} Pf = Pf fzero ∷ tabulate⁺ (Pf ∘ fsuc)
tabulate⁻ : ∀ {n} {f : Fin n → A} →
All P (tabulate f) → (∀ i → P (f i))
tabulate⁻ {zero} pf ()
tabulate⁻ {suc n} (px ∷ _) fzero = px
tabulate⁻ {suc n} (_ ∷ pf) (fsuc i) = tabulate⁻ pf i