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module FinMap where
open import Data.Nat using (ℕ)
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) renaming (lookup to lookupVec)
open import Data.List using (List ; [] ; _∷_ ; map ; zip)
open import Data.Product using (_×_ ; _,_)
open import Function using (id)
open import Relation.Nullary using (¬_ ; yes ; no)
open import Relation.Nullary.Negation using (contradiction ; contraposition)
open import Relation.Binary.Core using (_≡_ ; refl)
open import Relation.Binary.PropositionalEquality using (cong ; sym)
open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎)
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))
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-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-from-just : {A : Set} → {x y : A} → _≡_ {_} {Maybe A} (just x) (just y) → x ≡ y
lemma-from-just refl = refl
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 ≟ 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 ∎)
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