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 = {!!}