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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)
fmap : {A B : Set} → (A → B) → Maybe A → Maybe B
fmap f = maybe′ (λ a → just (f a)) 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))
lemma-insert-same : {τ : Set} {n : ℕ} → (m : FinMapMaybe n τ) → (f : Fin n) → (a : τ) → just a ≡ lookupM f m → m ≡ insert f a m
lemma-insert-same [] () a p
lemma-insert-same (.(just a) ∷ xs) zero a refl = refl
lemma-insert-same (x ∷ xs) (suc i) a p = cong (_∷_ x) (lemma-insert-same xs i a p)
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 | _ with eq (f i) x
lemma-checkInsert-generate eq f i is | just .(f i) | Reveal_is_.[_] p | yes refl = cong just (lemma-insert-same (generate f is) i (f i) (sym p))
lemma-checkInsert-generate eq f i is | just x | _ | no ¬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′)) ∎
where open Relation.Binary.PropositionalEquality.≡-Reasoning
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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