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module Bidir where
open import Data.Bool hiding (_≟_)
open import Data.Nat
open import Data.Maybe
open import Data.List hiding (replicate)
open import Data.Product hiding (zip ; map)
open import Function
open import Relation.Nullary
open import Relation.Binary.Core
module NatMap where
NatMap : Set → Set
NatMap A = List (ℕ × A)
lookup : {A : Set} → ℕ → NatMap A → Maybe A
lookup n [] = nothing
lookup n ((m , a) ∷ xs) with n ≟ m
lookup n ((.n , a) ∷ xs) | yes refl = just a
lookup n ((m , a) ∷ xs) | no ¬p = lookup n xs
notMember : {A : Set} → ℕ → NatMap A → Bool
notMember n m = not (maybeToBool (lookup n m))
-- For now we simply prepend the element. This may lead to duplicates.
insert : {A : Set} → ℕ → A → NatMap A → NatMap A
insert n a m = (n , a) ∷ m
fromAscList : {A : Set} → List (ℕ × A) → NatMap A
fromAscList [] = []
fromAscList ((n , a) ∷ xs) = insert n a (fromAscList xs)
empty : {A : Set} → NatMap A
empty = []
union : {A : Set} → NatMap A → NatMap A → NatMap A
union [] m = m
union ((n , a) ∷ xs) m = insert n a (union xs m)
open NatMap
checkInsert : {A : Set} → ((x y : A) → Dec (x ≡ y)) → ℕ → A → NatMap A → Maybe (NatMap A)
checkInsert eq i b m with lookup i m
checkInsert eq i b m | just c with eq b c
checkInsert eq i b m | just .b | yes refl = just m
checkInsert eq i b m | just c | no ¬p = nothing
checkInsert eq i b m | nothing = just (insert i b m)
assoc : {A : Set} → ((x y : A) → Dec (x ≡ y)) → List ℕ → List A → Maybe (NatMap A)
assoc _ [] [] = just empty
assoc eq (i ∷ is) (b ∷ bs) = maybe′ (checkInsert eq i b) nothing (assoc eq is bs)
assoc _ _ _ = nothing
--data Equal? where
-- same ...
-- different ...
generate : {A : Set} → (ℕ → A) → List ℕ → NatMap A
generate f [] = empty
generate f (n ∷ ns) = insert n (f n) (generate f ns)
-- this lemma is probably wrong, because two different NatMaps may represent the same semantic value.
lemma-1 : {τ : Set} → (eq : (x y : τ) → Dec (x ≡ y)) → (f : ℕ → τ) → (is : List ℕ) → assoc eq is (map f is) ≡ just (generate f is)
lemma-1 eq f [] = refl
lemma-1 eq f (i ∷ is′) = {!!}
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