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module Bidir where
open import Data.Nat using (ℕ)
open import Data.Fin using (Fin)
open import Data.Maybe using (Maybe ; nothing ; just ; maybe′)
open import Data.List using (List ; [] ; _∷_ ; map ; length)
open import Data.Vec using (toList ; fromList ; tabulate) renaming (lookup to lookupVec)
open import Function using (id ; _∘_ ; flip)
open import Relation.Nullary using (Dec ; yes ; no)
open import Relation.Nullary.Negation using (contradiction)
open import Relation.Binary.Core using (_≡_ ; refl)
open import Relation.Binary.PropositionalEquality using (cong ; inspect ; Reveal_is_)
open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎)
open import FinMap
_>>=_ : {A B : Set} → Maybe A → (A → Maybe B) → Maybe B
_>>=_ = flip (flip maybe′ nothing)
fmap : {A B : Set} → (A → B) → Maybe A → Maybe B
fmap f = maybe′ (λ a → just (f a)) nothing
EqInst : Set → Set
EqInst A = (x y : A) → Dec (x ≡ y)
checkInsert : {A : Set} {n : ℕ} → EqInst A → Fin n → A → FinMapMaybe n A → Maybe (FinMapMaybe n A)
checkInsert eq i b m with lookupM 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} {n : ℕ} → EqInst A → List (Fin n) → List A → Maybe (FinMapMaybe n A)
assoc _ [] [] = just empty
assoc eq (i ∷ is) (b ∷ bs) = (assoc eq is bs) >>= (checkInsert eq i b)
assoc _ _ _ = nothing
lemma-checkInsert-generate : {τ : Set} {n : ℕ} → (eq : EqInst τ) → (f : Fin n → τ) → (i : Fin n) → (is : List (Fin n)) → checkInsert eq i (f i) (generate f is) ≡ just (generate f (i ∷ is))
lemma-checkInsert-generate eq f i is with lookupM i (generate f is) | inspect (lookupM i) (generate f is)
lemma-checkInsert-generate eq f i is | nothing | _ = refl
lemma-checkInsert-generate eq f i is | just x | Reveal_is_.[_] prf with lemma-lookupM-generate i f is x prf
lemma-checkInsert-generate eq f i is | just .(f i) | Reveal_is_.[_] prf | refl with eq (f i) (f i)
lemma-checkInsert-generate eq f i is | just .(f i) | Reveal_is_.[_] prf | refl | yes refl = cong just (lemma-insert-same (generate f is) i (f i) prf)
lemma-checkInsert-generate eq f i is | just .(f i) | Reveal_is_.[_] prf | refl | no ¬p = contradiction refl ¬p
lemma-1 : {τ : Set} {n : ℕ} → (eq : EqInst τ) → (f : Fin n → τ) → (is : List (Fin n)) → assoc eq is (map f is) ≡ just (generate f is)
lemma-1 eq f [] = refl
lemma-1 eq f (i ∷ is′) = begin
(assoc eq (i ∷ is′) (map f (i ∷ is′)))
≡⟨ refl ⟩
(assoc eq is′ (map f is′) >>= checkInsert eq i (f i))
≡⟨ cong (λ m → m >>= checkInsert eq i (f i)) (lemma-1 eq f is′) ⟩
(just (generate f is′) >>= (checkInsert eq i (f i)))
≡⟨ refl ⟩
(checkInsert eq i (f i) (generate f is′))
≡⟨ lemma-checkInsert-generate eq f i is′ ⟩
just (generate f (i ∷ is′)) ∎
lemma-2 : {τ : Set} {n : ℕ} → (eq : EqInst τ) → (is : List (Fin n)) → (v : List τ) → (h : FinMapMaybe n τ) → just h ≡ assoc eq is v → map (flip lookup h) is ≡ map just v
lemma-2 eq [] [] h p = refl
lemma-2 eq [] (x ∷ xs) h ()
lemma-2 eq (x ∷ xs) [] h ()
lemma-2 eq (i ∷ is) (x ∷ xs) h p = {!!}
idrange : (n : ℕ) → List (Fin n)
idrange n = toList (tabulate id)
bff : ({A : Set} → List A → List A) → ({B : Set} → EqInst B → List B → List B → Maybe (List B))
bff get eq s v = let s′ = idrange (length s)
g = fromFunc (λ f → lookupVec f (fromList s))
h = assoc eq (get s′) v
h′ = fmap (flip union g) h
in fmap (flip map s′ ∘ flip lookup) h′
theorem-1 : (get : {α : Set} → List α → List α) → {τ : Set} → (eq : EqInst τ) → (s : List τ) → bff get eq s (get s) ≡ just s
theorem-1 get eq s = {!!}
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