{-# OPTIONS --without-K --safe #-}
module Data.List.Relation.Unary.All.Properties where
open import Axiom.Extensionality.Propositional using (Extensionality)
open import Data.Bool.Base using (Bool; T; true; false)
open import Data.Bool.Properties using (T-∧)
open import Data.Empty
open import Data.Fin.Base using (Fin) renaming (zero to fzero; suc to fsuc)
open import Data.List.Base as List hiding (lookup)
open import Data.List.Membership.Propositional
open import Data.List.Membership.Propositional.Properties
import Data.List.Membership.Setoid as SetoidMembership
open import Data.List.Relation.Unary.All as All using
( All; []; _∷_; lookup; updateAt
; _[_]=_; here; there
; Null
)
open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
import Data.List.Relation.Binary.Equality.Setoid as ListEq using (_≋_; []; _∷_)
open import Data.List.Relation.Binary.Pointwise using (Pointwise; []; _∷_)
open import Data.List.Relation.Binary.Subset.Propositional using (_⊆_)
open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
open import Data.Maybe.Relation.Unary.All as MAll using (just; nothing)
open import Data.Nat.Base using (zero; suc; z≤n; s≤s; _<_)
open import Data.Nat.Properties using (≤-refl; ≤-step)
open import Data.Product as Prod using (_×_; _,_; uncurry; uncurry′)
open import Function.Base
open import Function.Equality using (_⟨$⟩_)
open import Function.Equivalence using (_⇔_; equivalence; Equivalence)
open import Function.Inverse using (_↔_; inverse)
open import Function.Surjection using (_↠_; surjection)
open import Level using (Level)
open import Relation.Binary as B using (REL; Setoid; _Respects_)
open import Relation.Binary.PropositionalEquality
using (_≡_; refl; cong; cong₂; _≗_)
open import Relation.Nullary.Reflects using (invert)
open import Relation.Nullary
open import Relation.Nullary.Negation using (¬?; contradiction; decidable-stable)
open import Relation.Unary
using (Decidable; Pred; Universal) renaming (_⊆_ to _⋐_)
private
variable
a b c p q r ℓ ℓ₁ ℓ₂ : Level
A : Set a
B : Set b
C : Set c
Null⇒null : ∀ {xs : List A} → Null xs → T (null xs)
Null⇒null [] = _
null⇒Null : ∀ {xs : List A} → T (null xs) → Null xs
null⇒Null {xs = [] } _ = []
null⇒Null {xs = _ ∷ _} ()
module _ {p} {P : Pred A p} where
[]=-injective : ∀ {x xs} {px qx : P x} {pxs : All P xs} {i : x ∈ xs} →
pxs [ i ]= px →
pxs [ i ]= qx →
px ≡ qx
[]=-injective here here = refl
[]=-injective (there x↦px) (there x↦qx) = []=-injective x↦px x↦qx
module _ {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 (¬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
... | true because [p] = there (¬All⇒Any¬ dec xs (¬∀ ∘ _∷_ (invert [p])))
... | false because [¬p] = here (invert [¬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¬ = surjection (¬Any⇒All¬ _) All¬⇒¬Any to∘from
where
to∘from : ∀ {xs} (¬p : All (¬_ ∘ P) xs) → ¬Any⇒All¬ xs (All¬⇒¬Any ¬p) ≡ ¬p
to∘from [] = refl
to∘from (¬p ∷ ¬ps) = cong₂ _∷_ refl (to∘from ¬ps)
from∘to : 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) → refl
; (there 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 = equivalence Any¬⇒¬All (¬All⇒Any¬ dec _)
where
to∘from : Extensionality _ _ →
∀ {xs} (¬∀ : ¬ All P xs) → Any¬⇒¬All (¬All⇒Any¬ dec xs ¬∀) ≡ ¬∀
to∘from ext ¬∀ = ext (⊥-elim ∘ ¬∀)
module _ {_~_ : REL A B ℓ} where
All-swap : ∀ {xs ys} →
All (λ x → All (x ~_) ys) xs →
All (λ y → All (_~ y) xs) ys
All-swap {ys = []} _ = []
All-swap {ys = y ∷ ys} [] = All.universal (λ _ → []) (y ∷ ys)
All-swap {ys = y ∷ ys} ((x~y ∷ x~ys) ∷ pxs) =
(x~y ∷ (All.map All.head pxs)) ∷
All-swap (x~ys ∷ (All.map All.tail pxs))
module _ {P : A → Set p} where
[]=lookup : ∀ {x xs} (pxs : All P xs) (i : x ∈ xs) →
pxs [ i ]= lookup pxs i
[]=lookup (px ∷ pxs) (here refl) = here
[]=lookup (px ∷ pxs) (there i) = there ([]=lookup pxs i)
[]=⇒lookup : ∀ {x xs} {px : P x} {pxs : All P xs} {i : x ∈ xs} →
pxs [ i ]= px →
lookup pxs i ≡ px
[]=⇒lookup x↦px = []=-injective ([]=lookup _ _) x↦px
lookup⇒[]= : ∀{x xs} {px : P x} (pxs : All P xs) (i : x ∈ xs) →
lookup pxs i ≡ px →
pxs [ i ]= px
lookup⇒[]= pxs i refl = []=lookup pxs i
module _ {P : Pred A p} {Q : Pred A q} {f : P ⋐ Q} where
map-cong : ∀ {xs} {g : P ⋐ Q} (pxs : All P xs) →
(∀ {x} → f {x} ≗ g) → All.map f pxs ≡ All.map g pxs
map-cong [] _ = refl
map-cong (px ∷ pxs) feq = cong₂ _∷_ (feq px) (map-cong pxs feq)
map-id : ∀ {xs} (pxs : All P xs) → All.map id pxs ≡ pxs
map-id [] = refl
map-id (px ∷ pxs) = cong (px ∷_) (map-id pxs)
map-compose : ∀ {r} {R : Pred A r} {xs} {g : Q ⋐ R} (pxs : All P xs) →
All.map g (All.map f pxs) ≡ All.map (g ∘ f) pxs
map-compose [] = refl
map-compose (px ∷ pxs) = cong (_ ∷_) (map-compose pxs)
lookup-map : ∀ {xs x} (pxs : All P xs) (i : x ∈ xs) →
lookup (All.map f pxs) i ≡ f (lookup pxs i)
lookup-map (px ∷ pxs) (here refl) = refl
lookup-map (px ∷ pxs) (there i) = lookup-map pxs i
module _ {P : Pred A p} where
updateAt-updates : ∀ {x xs} (i : x ∈ xs) {f : P x → P x} {px : P x} (pxs : All P xs) →
pxs [ i ]= px →
updateAt i f pxs [ i ]= f px
updateAt-updates (here refl) (px ∷ pxs) here = here
updateAt-updates (there i) (px ∷ pxs) (there x↦px) =
there (updateAt-updates i pxs x↦px)
updateAt-minimal : ∀ {x y xs} (i : x ∈ xs) (j : y ∈ xs) →
∀ {f : P y → P y} {px : P x} (pxs : All P xs) →
i ≢∈ j →
pxs [ i ]= px →
updateAt j f pxs [ i ]= px
updateAt-minimal (here .refl) (here refl) (px ∷ pxs) i≢j here =
⊥-elim (i≢j refl refl)
updateAt-minimal (here .refl) (there j) (px ∷ pxs) i≢j here = here
updateAt-minimal (there i) (here refl) (px ∷ pxs) i≢j (there val) = there val
updateAt-minimal (there i) (there j) (px ∷ pxs) i≢j (there val) =
there (updateAt-minimal i j pxs (there-injective-≢∈ i≢j) val)
lookup∘updateAt : ∀ {x xs} (pxs : All P xs) (i : x ∈ xs) {f : P x → P x} →
lookup (updateAt i f pxs) i ≡ f (lookup pxs i)
lookup∘updateAt pxs i =
[]=⇒lookup (updateAt-updates i pxs (lookup⇒[]= pxs i refl))
lookup∘updateAt′ : ∀ {x y xs} (i : x ∈ xs) (j : y ∈ xs) →
∀ {f : P y → P y} {px : P x} (pxs : All P xs) →
i ≢∈ j →
lookup (updateAt j f pxs) i ≡ lookup pxs i
lookup∘updateAt′ i j pxs i≢j =
[]=⇒lookup (updateAt-minimal i j pxs i≢j (lookup⇒[]= pxs i refl))
updateAt-id-relative : ∀ {x xs} (i : x ∈ xs) {f : P x → P x} (pxs : All P xs) →
f (lookup pxs i) ≡ lookup pxs i →
updateAt i f pxs ≡ pxs
updateAt-id-relative (here refl)(px ∷ pxs) eq = cong (_∷ pxs) eq
updateAt-id-relative (there i) (px ∷ pxs) eq = cong (px ∷_) (updateAt-id-relative i pxs eq)
updateAt-id : ∀ {x xs} (i : x ∈ xs) (pxs : All P xs) →
updateAt i id pxs ≡ pxs
updateAt-id i pxs = updateAt-id-relative i pxs refl
updateAt-compose-relative : ∀ {x xs} (i : x ∈ xs) {f g h : P x → P x} (pxs : All P xs) →
f (g (lookup pxs i)) ≡ h (lookup pxs i) →
updateAt i f (updateAt i g pxs) ≡ updateAt i h pxs
updateAt-compose-relative (here refl) (px ∷ pxs) fg=h = cong (_∷ pxs) fg=h
updateAt-compose-relative (there i) (px ∷ pxs) fg=h =
cong (px ∷_) (updateAt-compose-relative i pxs fg=h)
updateAt-compose : ∀ {x xs} (i : x ∈ xs) {f g : P x → P x} →
updateAt {P = P} i f ∘ updateAt i g ≗ updateAt i (f ∘ g)
updateAt-compose (here refl) (px ∷ pxs) = refl
updateAt-compose (there i) (px ∷ pxs) = cong (px ∷_) (updateAt-compose i pxs)
updateAt-cong-relative : ∀ {x xs} (i : x ∈ xs) {f g : P x → P x} (pxs : All P xs) →
f (lookup pxs i) ≡ g (lookup pxs i) →
updateAt i f pxs ≡ updateAt i g pxs
updateAt-cong-relative (here refl) (px ∷ pxs) f=g = cong (_∷ pxs) f=g
updateAt-cong-relative (there i) (px ∷ pxs) f=g = cong (px ∷_) (updateAt-cong-relative i pxs f=g)
updateAt-cong : ∀ {x xs} (i : x ∈ xs) {f g : P x → P x} →
f ≗ g →
updateAt i f ≗ updateAt i g
updateAt-cong i f≗g pxs = updateAt-cong-relative i pxs (f≗g (lookup pxs i))
updateAt-commutes : ∀ {x y xs} (i : x ∈ xs) (j : y ∈ xs) →
∀ {f : P x → P x} {g : P y → P y} →
i ≢∈ j →
updateAt i f ∘ updateAt j g ≗ updateAt j g ∘ updateAt i f
updateAt-commutes (here refl) (here refl) i≢j (px ∷ pxs) =
⊥-elim (i≢j refl refl)
updateAt-commutes (here refl) (there j) i≢j (px ∷ pxs) = refl
updateAt-commutes (there i) (here refl) i≢j (px ∷ pxs) = refl
updateAt-commutes (there i) (there j) i≢j (px ∷ pxs) =
cong (px ∷_) (updateAt-commutes i j (there-injective-≢∈ i≢j) pxs)
module _ {P : Pred A p} {Q : Pred A q} where
map-updateAt : ∀ {x xs} {f : P ⋐ Q} {g : P x → P x} {h : Q x → Q x}
(pxs : All P xs) (i : x ∈ xs) →
f (g (lookup pxs i)) ≡ h (f (lookup pxs i)) →
All.map f (pxs All.[ i ]%= g) ≡ (All.map f pxs) All.[ i ]%= h
map-updateAt (px ∷ pxs) (here refl) = cong (_∷ _)
map-updateAt (px ∷ pxs) (there i) feq = cong (_ ∷_) (map-updateAt pxs i feq)
module _ {P : B → Set p} {f : A → B} where
map⁺ : ∀ {xs} → All (P ∘ f) xs → All P (map f xs)
map⁺ [] = []
map⁺ (p ∷ ps) = p ∷ map⁺ ps
map⁻ : ∀ {xs} → All P (map f xs) → All (P ∘ f) xs
map⁻ {xs = []} [] = []
map⁻ {xs = _ ∷ _} (p ∷ ps) = p ∷ map⁻ ps
module _ {P : A → Set p} {Q : B → Set q} {f : A → B} where
gmap : P ⋐ Q ∘ f → All P ⋐ All Q ∘ map f
gmap g = map⁺ ∘ All.map g
module _ (P : B → Set p) {f : A → Maybe B} where
mapMaybe⁺ : ∀ {xs} → All (MAll.All P) (map f xs) → All P (mapMaybe f xs)
mapMaybe⁺ {[]} [] = []
mapMaybe⁺ {x ∷ xs} (px ∷ pxs) with f x
... | nothing = mapMaybe⁺ pxs
... | just v with px
... | just pv = pv ∷ mapMaybe⁺ pxs
module _ {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} = inverse (uncurry ++⁺) (++⁻ xs) ++⁻∘++⁺ (++⁺∘++⁻ xs)
where
++⁺∘++⁻ : ∀ xs {ys} (p : All P (xs ++ ys)) →
uncurry′ ++⁺ (++⁻ xs p) ≡ p
++⁺∘++⁻ [] p = refl
++⁺∘++⁻ (x ∷ xs) (px ∷ pxs) = cong (_∷_ px) $ ++⁺∘++⁻ xs pxs
++⁻∘++⁺ : ∀ {xs ys} (p : All P xs × All P ys) →
++⁻ xs (uncurry ++⁺ p) ≡ p
++⁻∘++⁺ ([] , pys) = refl
++⁻∘++⁺ (px ∷ pxs , pys) rewrite ++⁻∘++⁺ (pxs , pys) = refl
module _ {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 _ (S₁ : Setoid a ℓ₁) (S₂ : Setoid b ℓ₂) where
open SetoidMembership S₁ using () renaming (_∈_ to _∈₁_)
open SetoidMembership S₂ using () renaming (_∈_ to _∈₂_)
cartesianProductWith⁺ : ∀ {P : Pred C p} f xs ys →
(∀ {x y} → x ∈₁ xs → y ∈₂ ys → P (f x y)) →
All P (cartesianProductWith f xs ys)
cartesianProductWith⁺ f [] ys pres = []
cartesianProductWith⁺ f (x ∷ xs) ys pres = ++⁺
(map⁺ (All.tabulateₛ S₂ (pres (here (Setoid.refl S₁)))))
(cartesianProductWith⁺ f xs ys (pres ∘ there))
cartesianProduct⁺ : ∀ {P : Pred _ p} xs ys →
(∀ {x y} → x ∈₁ xs → y ∈₂ ys → P (x , y)) →
All P (cartesianProduct xs ys)
cartesianProduct⁺ = cartesianProductWith⁺ _,_
module _ {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 _ {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 (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 _ {P : A → Set p} where
applyDownFrom⁺₁ : ∀ f n → (∀ {i} → i < n → P (f i)) → All P (applyDownFrom f n)
applyDownFrom⁺₁ f zero Pf = []
applyDownFrom⁺₁ f (suc n) Pf = Pf ≤-refl ∷ applyDownFrom⁺₁ f n (Pf ∘ ≤-step)
applyDownFrom⁺₂ : ∀ f n → (∀ i → P (f i)) → All P (applyDownFrom f n)
applyDownFrom⁺₂ f n Pf = applyDownFrom⁺₁ f n (λ _ → Pf _)
module _ {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⁻ {suc n} (px ∷ _) fzero = px
tabulate⁻ {suc n} (_ ∷ pf) (fsuc i) = tabulate⁻ pf i
module _ {P : A → Set p} {Q : A → Set q} where
─⁺ : ∀ {xs} (p : Any P xs) → All Q xs → All Q (xs Any.─ p)
─⁺ (here px) (_ ∷ qs) = qs
─⁺ (there p) (q ∷ qs) = q ∷ ─⁺ p qs
─⁻ : ∀ {xs} (p : Any P xs) → Q (Any.lookup p) → All Q (xs Any.─ p) → All Q xs
─⁻ (here px) q qs = q ∷ qs
─⁻ (there p) q (q′ ∷ qs) = q′ ∷ ─⁻ p q qs
module _ {P : A → Set p} (P? : Decidable P) where
all-filter : ∀ xs → All P (filter P? xs)
all-filter [] = []
all-filter (x ∷ xs) with P? x
... | true because [Px] = invert [Px] ∷ all-filter xs
... | false because _ = all-filter xs
module _ {P : A → Set p} {Q : A → Set q} (P? : Decidable P) where
filter⁺ : ∀ {xs} → All Q xs → All Q (filter P? xs)
filter⁺ {xs = _} [] = []
filter⁺ {xs = x ∷ _} (Qx ∷ Qxs) with does (P? x)
... | false = filter⁺ Qxs
... | true = Qx ∷ filter⁺ Qxs
filter⁻ : ∀ {xs} → All Q (filter P? xs) → All Q (filter (¬? ∘ P?) xs) → All Q xs
filter⁻ {[]} [] [] = []
filter⁻ {x ∷ xs} all⁺ all⁻ with P? x | ¬? (P? x)
filter⁻ {x ∷ xs} all⁺ all⁻ | yes Px | yes ¬Px = contradiction Px ¬Px
filter⁻ {x ∷ xs} (qx ∷ all⁺) all⁻ | yes Px | no ¬¬Px = qx ∷ filter⁻ all⁺ all⁻
filter⁻ {x ∷ xs} all⁺ (qx ∷ all⁻) | no _ | yes ¬Px = qx ∷ filter⁻ all⁺ all⁻
filter⁻ {x ∷ xs} all⁺ all⁻ | no ¬Px | no ¬¬Px = contradiction ¬Px ¬¬Px
module _ {P : A → Set p} {R : A → A → Set q} (R? : B.Decidable R) where
derun⁺ : ∀ {xs} → All P xs → All P (derun R? xs)
derun⁺ {[]} [] = []
derun⁺ {x ∷ []} (px ∷ []) = px ∷ []
derun⁺ {x ∷ y ∷ xs} (px ∷ all[P,y∷xs]) with does (R? x y)
... | false = px ∷ derun⁺ all[P,y∷xs]
... | true = derun⁺ all[P,y∷xs]
deduplicate⁺ : ∀ {xs} → All P xs → All P (deduplicate R? xs)
deduplicate⁺ [] = []
deduplicate⁺ {x ∷ _} (px ∷ all[P,xs]) = px ∷ filter⁺ (¬? ∘ R? x) (deduplicate⁺ all[P,xs])
derun⁻ : P B.Respects (flip R) → ∀ xs → All P (derun R? xs) → All P xs
derun⁻ P-resp-R [] [] = []
derun⁻ P-resp-R (x ∷ xs) all[P,x∷xs] = aux x xs all[P,x∷xs] where
aux : ∀ x xs → All P (derun R? (x ∷ xs)) → All P (x ∷ xs)
aux x [] (px ∷ []) = px ∷ []
aux x (y ∷ xs) all[P,x∷y∷xs] with R? x y
aux x (y ∷ xs) all[P,y∷xs] | yes Rxy with aux y xs all[P,y∷xs]
aux x (y ∷ xs) all[P,y∷xs] | yes Rxy | r@(py ∷ _) = P-resp-R Rxy py ∷ r
aux x (y ∷ xs) (px ∷ all[P,y∷xs]) | no _ = px ∷ aux y xs all[P,y∷xs]
module _ (P-resp-R : P B.Respects R) where
deduplicate⁻ : ∀ xs → All P (deduplicate R? xs) → All P xs
deduplicate⁻ [] [] = []
deduplicate⁻ (x ∷ xs) (px ∷ app[P,dedup[xs]]) = px ∷ deduplicate⁻ xs (filter⁻ (¬? ∘ R? x) app[P,dedup[xs]] (All.tabulate aux)) where
aux : ∀ {z} → z ∈ filter (¬? ∘ ¬? ∘ R? x) (deduplicate R? xs) → P z
aux {z} z∈filter = P-resp-R (decidable-stable (R? x z) (Prod.proj₂ (∈-filter⁻ (¬? ∘ ¬? ∘ R? x) {z} {deduplicate R? xs} z∈filter))) px
module _ (P : C → Set p) (f : A → B → C) where
zipWith⁺ : ∀ {xs ys} → Pointwise (λ x y → P (f x y)) xs ys →
All P (zipWith f xs ys)
zipWith⁺ [] = []
zipWith⁺ (Pfxy ∷ Pfxsys) = Pfxy ∷ zipWith⁺ Pfxsys
module _ {P : A → Set p} where
singleton⁻ : ∀ {x} → All P [ x ] → P x
singleton⁻ (px ∷ []) = px
module _ {P : A → Set p} {x xs} where
∷ʳ⁺ : All P xs → P x → All P (xs ∷ʳ x)
∷ʳ⁺ pxs px = ++⁺ pxs (px ∷ [])
∷ʳ⁻ : All P (xs ∷ʳ x) → All P xs × P x
∷ʳ⁻ pxs = Prod.map₂ singleton⁻ $ ++⁻ xs pxs
module _ {P : A → Set p} where
fromMaybe⁺ : ∀ {mx} → MAll.All P mx → All P (fromMaybe mx)
fromMaybe⁺ (just px) = px ∷ []
fromMaybe⁺ nothing = []
fromMaybe⁻ : ∀ mx → All P (fromMaybe mx) → MAll.All P mx
fromMaybe⁻ (just x) (px ∷ []) = just px
fromMaybe⁻ nothing p = nothing
module _ {P : A → Set p} where
replicate⁺ : ∀ n {x} → P x → All P (replicate n x)
replicate⁺ zero px = []
replicate⁺ (suc n) px = px ∷ replicate⁺ n px
replicate⁻ : ∀ {n x} → All P (replicate (suc n) x) → P x
replicate⁻ (px ∷ _) = px
module _ {P : A → Set p} where
inits⁺ : ∀ {xs} → All P xs → All (All P) (inits xs)
inits⁺ [] = [] ∷ []
inits⁺ (px ∷ pxs) = [] ∷ gmap (px ∷_) (inits⁺ pxs)
inits⁻ : ∀ xs → All (All P) (inits xs) → All P xs
inits⁻ [] pxs = []
inits⁻ (x ∷ []) ([] ∷ p[x] ∷ []) = p[x]
inits⁻ (x ∷ xs@(_ ∷ _)) ([] ∷ pxs@(p[x] ∷ _)) =
singleton⁻ p[x] ∷ inits⁻ xs (All.map (drop⁺ 1) (map⁻ pxs))
module _ {P : A → Set p} where
tails⁺ : ∀ {xs} → All P xs → All (All P) (tails xs)
tails⁺ [] = [] ∷ []
tails⁺ pxxs@(_ ∷ pxs) = pxxs ∷ tails⁺ pxs
tails⁻ : ∀ xs → All (All P) (tails xs) → All P xs
tails⁻ [] pxs = []
tails⁻ (x ∷ xs) (pxxs ∷ _) = pxxs
module _ (p : A → Bool) where
all⁺ : ∀ xs → T (all p xs) → All (T ∘ p) xs
all⁺ [] _ = []
all⁺ (x ∷ xs) px∷xs with Equivalence.to (T-∧ {p x}) ⟨$⟩ px∷xs
... | (px , pxs) = px ∷ all⁺ xs pxs
all⁻ : ∀ {xs} → All (T ∘ p) xs → T (all p xs)
all⁻ [] = _
all⁻ (px ∷ pxs) = Equivalence.from T-∧ ⟨$⟩ (px , all⁻ pxs)
anti-mono : ∀ {P : A → Set p} {xs ys} → xs ⊆ ys → All P ys → All P xs
anti-mono xs⊆ys pys = All.tabulate (lookup pys ∘ xs⊆ys)
all-anti-mono : ∀ (p : A → Bool) {xs ys} →
xs ⊆ ys → T (all p ys) → T (all p xs)
all-anti-mono p xs⊆ys = all⁻ p ∘ anti-mono xs⊆ys ∘ all⁺ p _
module _ {c ℓ} (S : Setoid c ℓ) where
open Setoid S
open ListEq S
respects : {P : Pred Carrier p} → P Respects _≈_ → (All P) Respects _≋_
respects p≈ [] [] = []
respects p≈ (x≈y ∷ xs≈ys) (px ∷ pxs) = p≈ x≈y px ∷ respects p≈ xs≈ys pxs
All-all = all⁻
{-# WARNING_ON_USAGE All-all
"Warning: All-all was deprecated in v0.16.
Please use all⁻ instead."
#-}
all-All = all⁺
{-# WARNING_ON_USAGE all-All
"Warning: all-All was deprecated in v0.16.
Please use all⁺ instead."
#-}
All-map = map⁺
{-# WARNING_ON_USAGE All-map
"Warning: All-map was deprecated in v0.16.
Please use map⁺ instead."
#-}
map-All = map⁻
{-# WARNING_ON_USAGE map-All
"Warning: map-All was deprecated in v0.16.
Please use map⁻ instead."
#-}
filter⁺₁ = all-filter
{-# WARNING_ON_USAGE filter⁺₁
"Warning: filter⁺₁ was deprecated in v1.0.
Please use all-filter instead."
#-}
filter⁺₂ = filter⁺
{-# WARNING_ON_USAGE filter⁺₂
"Warning: filter⁺₂ was deprecated in v1.0.
Please use filter⁺ instead."
#-}
Any¬→¬All = Any¬⇒¬All
{-# WARNING_ON_USAGE Any¬→¬All
"Warning: Any¬→¬All was deprecated in v1.3.
Please use Any¬⇒¬All instead."
#-}