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module Bidir where
open import Data.Bool hiding (_≟_)
open import Data.Nat
open import Data.Fin
open import Data.Fin.Props renaming (_≟_ to _≟F_)
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.Nullary.Negation
open import Relation.Binary.Core
open import Relation.Binary.PropositionalEquality
open Relation.Binary.PropositionalEquality.≡-Reasoning
_>>=_ : {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) → lookupM f m ≡ just a → 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-lookupM-empty : {A : Set} {n : ℕ} → (i : Fin n) → lookupM {A} i empty ≡ nothing
lemma-lookupM-empty zero = refl
lemma-lookupM-empty (suc i) = lemma-lookupM-empty i
lemma-from-just : {A : Set} → {x y : A} → _≡_ {_} {Maybe A} (just x) (just y) → x ≡ y
lemma-from-just refl = refl
lemma-lookupM-insert : {A : Set} {n : ℕ} → (i : Fin n) → (a : A) → (m : FinMapMaybe n A) → lookupM i (insert i a m) ≡ just a
lemma-lookupM-insert zero _ (_ ∷ _) = refl
lemma-lookupM-insert (suc i) a (_ ∷ xs) = lemma-lookupM-insert i a xs
lemma-lookupM-insert-other : {A : Set} {n : ℕ} → (i j : Fin n) → (a : A) → (m : FinMapMaybe n A) → ¬(i ≡ j) → lookupM i m ≡ lookupM i (insert j a m)
lemma-lookupM-insert-other zero zero a m p = contradiction refl p
lemma-lookupM-insert-other zero (suc j) a (x ∷ xs) p = refl
lemma-lookupM-insert-other (suc i) zero a (x ∷ xs) p = refl
lemma-lookupM-insert-other (suc i) (suc j) a (x ∷ xs) p = lemma-lookupM-insert-other i j a xs (contraposition (cong suc) p)
lemma-lookupM-generate : {A : Set} {n : ℕ} → (i : Fin n) → (f : Fin n → A) → (is : List (Fin n)) → (a : A) → lookupM i (generate f is) ≡ just a → f i ≡ a
lemma-lookupM-generate {A} i f [] a p with begin
just a
≡⟨ sym p ⟩
lookupM i (generate f [])
≡⟨ refl ⟩
lookupM i empty
≡⟨ lemma-lookupM-empty i ⟩
nothing ∎
lemma-lookupM-generate i f [] a p | ()
lemma-lookupM-generate i f (i' ∷ is) a p with i ≟F i'
lemma-lookupM-generate i f (.i ∷ is) a p | yes refl = lemma-from-just (begin
just (f i)
≡⟨ sym (lemma-lookupM-insert i (f i) (generate f is)) ⟩
lookupM i (insert i (f i) (generate f is))
≡⟨ refl ⟩
lookupM i (generate f (i ∷ is))
≡⟨ p ⟩
just a ∎)
lemma-lookupM-generate i f (i' ∷ is) a p | no ¬p2 = lemma-lookupM-generate i f is a (begin
lookupM i (generate f is)
≡⟨ lemma-lookupM-insert-other i i' (f i') (generate f is) ¬p2 ⟩
lookupM i (insert i' (f i') (generate f is))
≡⟨ refl ⟩
lookupM i (generate f (i' ∷ is))
≡⟨ p ⟩
just a ∎)
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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