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- module Lib.List where
- open import Lib.Prelude
- open import Lib.Id
- infix 30 _∈_
- infixr 40 _::_ _++_
- infix 45 _!_
- data List (A : Set) : Set where
- [] : List A
- _::_ : A -> List A -> List A
- {-# COMPILE GHC List = data [] ([] | (:)) #-}
- {-# BUILTIN LIST List #-}
- {-# BUILTIN NIL [] #-}
- {-# BUILTIN CONS _::_ #-}
- _++_ : {A : Set} -> List A -> List A -> List A
- [] ++ ys = ys
- (x :: xs) ++ ys = x :: (xs ++ ys)
- foldr : {A B : Set} -> (A -> B -> B) -> B -> List A -> B
- foldr f z [] = z
- foldr f z (x :: xs) = f x (foldr f z xs)
- map : {A B : Set} -> (A -> B) -> List A -> List B
- map f [] = []
- map f (x :: xs) = f x :: map f xs
- map-id : forall {A}(xs : List A) -> map id xs ≡ xs
- map-id [] = refl
- map-id (x :: xs) with map id xs | map-id xs
- ... | .xs | refl = refl
- data _∈_ {A : Set} : A -> List A -> Set where
- hd : forall {x xs} -> x ∈ x :: xs
- tl : forall {x y xs} -> x ∈ xs -> x ∈ y :: xs
- data All {A : Set}(P : A -> Set) : List A -> Set where
- [] : All P []
- _::_ : forall {x xs} -> P x -> All P xs -> All P (x :: xs)
- head : forall {A}{P : A -> Set}{x xs} -> All P (x :: xs) -> P x
- head (x :: xs) = x
- tail : forall {A}{P : A -> Set}{x xs} -> All P (x :: xs) -> All P xs
- tail (x :: xs) = xs
- _!_ : forall {A}{P : A -> Set}{x xs} -> All P xs -> x ∈ xs -> P x
- xs ! hd = head xs
- xs ! tl n = tail xs ! n
- tabulate : forall {A}{P : A -> Set}{xs} -> ({x : A} -> x ∈ xs -> P x) -> All P xs
- tabulate {xs = []} f = []
- tabulate {xs = x :: xs} f = f hd :: tabulate (f ∘ tl)
- mapAll : forall {A B}{P : A -> Set}{Q : B -> Set}{xs}(f : A -> B) ->
- ({x : A} -> P x -> Q (f x)) -> All P xs -> All Q (map f xs)
- mapAll f h [] = []
- mapAll f h (x :: xs) = h x :: mapAll f h xs
- mapAll- : forall {A}{P Q : A -> Set}{xs} ->
- ({x : A} -> P x -> Q x) -> All P xs -> All Q xs
- mapAll- {xs = xs} f zs with map id xs | map-id xs | mapAll id f zs
- ... | .xs | refl | r = r
- infix 20 _⊆_
- data _⊆_ {A : Set} : List A -> List A -> Set where
- stop : [] ⊆ []
- drop : forall {xs y ys} -> xs ⊆ ys -> xs ⊆ y :: ys
- keep : forall {x xs ys} -> xs ⊆ ys -> x :: xs ⊆ x :: ys
- ⊆-refl : forall {A}{xs : List A} -> xs ⊆ xs
- ⊆-refl {xs = []} = stop
- ⊆-refl {xs = x :: xs} = keep ⊆-refl
- _∣_ : forall {A}{P : A -> Set}{xs ys} -> All P ys -> xs ⊆ ys -> All P xs
- [] ∣ stop = []
- (x :: xs) ∣ drop p = xs ∣ p
- (x :: xs) ∣ keep p = x :: (xs ∣ p)
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