module Bidir where

open import Data.Bool hiding (_≟_)
open import Data.Nat
open import Data.Fin
open import Data.Maybe
open import Data.List hiding (replicate)
open import Data.Vec hiding (map ; zip ; _>>=_) renaming (lookup to lookupVec)
open import Data.Product hiding (zip ; map)
open import Function
open import Relation.Nullary
open import Relation.Binary.Core
open import Relation.Binary.PropositionalEquality

_>>=_ : {A B : Set} → Maybe A → (A → Maybe B) → Maybe B
_>>=_ = flip (flip maybe′ nothing)

module FinMap where

  FinMapMaybe : ℕ → Set → Set
  FinMapMaybe n A = Vec (Maybe A) n

  lookupM : {A : Set} {n : ℕ} → Fin n → FinMapMaybe n A → Maybe A
  lookupM = lookupVec

  insert : {A : Set} {n : ℕ} → Fin n → A → FinMapMaybe n A → FinMapMaybe n A
  insert f a m = m [ f ]≔ (just a)

  empty : {A : Set} {n : ℕ} → FinMapMaybe n A
  empty = replicate nothing

  fromAscList : {A : Set} {n : ℕ} → List (Fin n × A) → FinMapMaybe n A
  fromAscList []             = empty
  fromAscList ((f , a) ∷ xs) = insert f a (fromAscList xs)

  FinMap : ℕ → Set → Set
  FinMap n A = Vec A n

  lookup : {A : Set} {n : ℕ} → Fin n → FinMap n A → A
  lookup = lookupVec

  fromFunc : {A : Set} {n : ℕ} → (Fin n → A) → FinMap n A
  fromFunc = tabulate

  union : {A : Set} {n : ℕ} → FinMapMaybe n A → FinMap n  A → FinMap n A
  union m1 m2 = tabulate (λ f → maybe′ id (lookup f m2) (lookupM f m1))

open FinMap

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

generate : {A : Set} {n : ℕ} → (Fin n → A) → List (Fin n) → FinMapMaybe n A
generate f is = fromAscList (zip is (map f is))

data Is-Just {A : Set} : (Maybe A) → Set where
  is-just : (x : A) → Is-Just (just x) 

the : {A : Set} {t : Maybe A} → Is-Just t → A
the (is-just x) = x

lemma-insert-same : {τ : Set} {n : ℕ} → (m : FinMapMaybe n τ) → (f : Fin n) → (a? : Is-Just (lookup f m)) → m ≡ insert f (the a?) m
lemma-insert-same [] () a?
lemma-insert-same (.(just x) ∷ xs) zero (is-just x) = refl
lemma-insert-same (x ∷ xs) (suc f′) a? = cong (_∷_ x) (lemma-insert-same xs f′ a?)

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′) with assoc eq is′ (map f is′) | generate f is′ | lemma-1 eq f is′
lemma-1 eq f (i ∷ is′) | nothing | _ | ()
lemma-1 eq f (i ∷ is′) | just m | .m | refl with lookup i m
lemma-1 eq f (i ∷ is′) | just m | .m | refl | nothing = refl
lemma-1 eq f (i ∷ is′) | just m | .m | refl | just x with eq (f i) x
lemma-1 eq f (i ∷ is′) | just m | .m | refl | just .(f i) | yes refl = cong just (lemma-insert-same m i {!!})
lemma-1 eq f (i ∷ is′) | just m | .m | refl | just x | no ¬p = {!!}

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 is v 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′ = h >>= (λ jh → just (union jh g))
                 in h′ >>= (λ jh′ → just (map (flip lookup jh′) s′))

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 = {!!}