module FinMap where open import Data.Nat using (ℕ ; zero ; suc) open import Data.Maybe using (Maybe ; just ; nothing ; maybe′) open import Data.Fin using (Fin ; zero ; suc) open import Data.Fin.Props using (_≟_) open import Data.Vec using (Vec ; [] ; _∷_ ; _[_]≔_ ; replicate ; tabulate ; foldr ; toList) renaming (lookup to lookupVec ; map to mapV) open import Data.Vec.Properties using (lookup∘tabulate) open import Data.List using (List ; [] ; _∷_ ; map ; zip) open import Data.Product using (_×_ ; _,_) open import Function using (id ; _∘_ ; flip ; const) open import Relation.Nullary using (yes ; no) open import Relation.Nullary.Negation using (contradiction) open import Relation.Binary.Core using (_≡_ ; refl ; _≢_) open import Relation.Binary.PropositionalEquality using (cong ; sym ; _≗_ ; trans ; cong₂) open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎) open import Generic using (just-injective) 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) fromFunc : {A : Set} {n : ℕ} → (Fin n → A) → FinMapMaybe n A fromFunc = mapV just ∘ tabulate union : {A : Set} {n : ℕ} → FinMapMaybe n A → FinMapMaybe n A → FinMapMaybe n A union m1 m2 = tabulate (λ f → maybe′ just (lookupM f m2) (lookupM f m1)) restrict : {A : Set} {n : ℕ} → (Fin n → A) → List (Fin n) → FinMapMaybe n A restrict f is = fromAscList (zip is (map f is)) delete : {A : Set} {n : ℕ} → Fin n → FinMapMaybe n A → FinMapMaybe n A delete i m = m [ i ]≔ nothing delete-many : {A : Set} {n m : ℕ} → Vec (Fin n) m → FinMapMaybe n A → FinMapMaybe n A delete-many = flip (foldr (const _) delete) lemma-just≢nothing : {A Whatever : Set} {a : A} {ma : Maybe A} → ma ≡ just a → ma ≡ nothing → Whatever lemma-just≢nothing refl () 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-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 a (x ∷ xs) = refl lemma-lookupM-insert (suc i) a (x ∷ 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 (p ∘ cong suc) lemma-lookupM-restrict : {A : Set} {n : ℕ} → (i : Fin n) → (f : Fin n → A) → (is : List (Fin n)) → (a : A) → lookupM i (restrict f is) ≡ just a → f i ≡ a lemma-lookupM-restrict i f [] a p = lemma-just≢nothing p (lemma-lookupM-empty i) lemma-lookupM-restrict i f (i' ∷ is) a p with i ≟ i' lemma-lookupM-restrict i f (.i ∷ is) a p | yes refl = just-injective (begin just (f i) ≡⟨ sym (lemma-lookupM-insert i (f i) (restrict f is)) ⟩ lookupM i (insert i (f i) (restrict f is)) ≡⟨ p ⟩ just a ∎) lemma-lookupM-restrict i f (i' ∷ is) a p | no i≢i' = lemma-lookupM-restrict i f is a (begin lookupM i (restrict f is) ≡⟨ lemma-lookupM-insert-other i i' (f i') (restrict f is) i≢i' ⟩ lookupM i (insert i' (f i') (restrict f is)) ≡⟨ p ⟩ just a ∎) lemma-tabulate-∘ : {n : ℕ} {A : Set} → {f g : Fin n → A} → f ≗ g → tabulate f ≡ tabulate g lemma-tabulate-∘ {zero} {_} {f} {g} f≗g = refl lemma-tabulate-∘ {suc n} {_} {f} {g} f≗g = cong₂ _∷_ (f≗g zero) (lemma-tabulate-∘ (f≗g ∘ suc)) lemma-fromFunc-tabulate : {n : ℕ} {A : Set} → (f : Fin n → A) → fromFunc f ≡ tabulate (just ∘ f) lemma-fromFunc-tabulate {zero} f = refl lemma-fromFunc-tabulate {suc _} f = cong (_∷_ (just (f zero))) (lemma-fromFunc-tabulate (f ∘ suc)) lemma-lookupM-delete : {n : ℕ} {A : Set} {i j : Fin n} → (f : FinMapMaybe n A) → i ≢ j → lookupM i (delete j f) ≡ lookupM i f lemma-lookupM-delete {i = zero} {j = zero} (_ ∷ _) p with p refl ... | () lemma-lookupM-delete {i = zero} {j = suc j} (_ ∷ _) p = refl lemma-lookupM-delete {i = suc i} {j = zero} (x ∷ xs) p = refl lemma-lookupM-delete {i = suc i} {j = suc j} (x ∷ xs) p = lemma-lookupM-delete xs (p ∘ cong suc) lemma-disjoint-union : {n m : ℕ} {A : Set} → (f : Fin n → A) → (t : Vec (Fin n) m) → union (restrict f (toList t)) (delete-many t (fromFunc f)) ≡ fromFunc f lemma-disjoint-union {n} {m} f t = trans (lemma-tabulate-∘ (lemma-inner t)) (sym (lemma-fromFunc-tabulate f)) where lemma-inner : {m : ℕ} → (t : Vec (Fin n) m) → (x : Fin n) → maybe′ just (lookupM x (delete-many t (fromFunc f))) (lookupM x (restrict f (toList t))) ≡ just (f x) lemma-inner [] x = begin maybe′ just (lookupM x (fromFunc f)) (lookupM x empty) ≡⟨ cong (maybe′ just (lookupM x (fromFunc f))) (lemma-lookupM-empty x) ⟩ lookupM x (fromFunc f) ≡⟨ cong (lookupM x) (lemma-fromFunc-tabulate f) ⟩ lookupM x (tabulate (just ∘ f)) ≡⟨ lookup∘tabulate (just ∘ f) x ⟩ just (f x) ∎ lemma-inner (t ∷ ts) x with x ≟ t lemma-inner (.x ∷ ts) x | yes refl = cong (maybe′ just (lookupM x (delete-many (x ∷ ts) (fromFunc f)))) (lemma-lookupM-insert x (f x) (restrict f (toList ts))) lemma-inner (t ∷ ts) x | no ¬p = begin maybe′ just (lookupM x (delete-many (t ∷ ts) (fromFunc f))) (lookupM x (restrict f (toList (t ∷ ts)))) ≡⟨ cong (maybe′ just (lookupM x (delete-many (t ∷ ts) (fromFunc f)))) (sym (lemma-lookupM-insert-other x t (f t) (restrict f (toList ts)) ¬p)) ⟩ maybe′ just (lookupM x (delete-many (t ∷ ts) (fromFunc f))) (lookupM x (restrict f (toList ts))) ≡⟨ cong (flip (maybe′ just) (lookupM x (restrict f (toList ts)))) (lemma-lookupM-delete (delete-many ts (fromFunc f)) ¬p) ⟩ maybe′ just (lookupM x (delete-many ts (fromFunc f))) (lookupM x (restrict f (toList ts))) ≡⟨ lemma-inner ts x ⟩ just (f x) ∎