open import Relation.Binary.Core using (Decidable ; _≡_) module CheckInsert (Carrier : Set) (deq : Decidable {A = Carrier} _≡_) where open import Data.Nat using (ℕ) open import Data.Fin using (Fin) open import Data.Fin.Props using (_≟_) open import Data.Maybe using (Maybe ; nothing ; just) open import Data.List using (List ; [] ; _∷_) open import Relation.Nullary using (Dec ; yes ; no) open import Relation.Nullary.Negation using (contradiction) open import Relation.Binary.Core using (refl ; _≢_) open import Relation.Binary.PropositionalEquality using (cong ; sym ; inspect ; [_] ; trans) open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎) open import FinMap checkInsert : {n : ℕ} → Fin n → Carrier → FinMapMaybe n Carrier → Maybe (FinMapMaybe n Carrier) checkInsert i b m with lookupM i m ... | nothing = just (insert i b m) ... | just c with deq b c ... | yes b≡c = just m ... | no b≢c = nothing data InsertionResult {n : ℕ} (i : Fin n) (x : Carrier) (h : FinMapMaybe n Carrier) : Maybe (FinMapMaybe n Carrier) → Set where insert-same : lookupM i h ≡ just x → InsertionResult i x h (just h) insert-new : lookupM i h ≡ nothing → InsertionResult i x h (just (insert i x h)) insert-wrong : (x' : Carrier) → x ≢ x' → lookupM i h ≡ just x' → InsertionResult i x h nothing insertionresult : {n : ℕ} → (i : Fin n) → (x : Carrier) → (h : FinMapMaybe n Carrier) → InsertionResult i x h (checkInsert i x h) insertionresult i x h with lookupM i h | inspect (lookupM i) h insertionresult i x h | just x' | _ with deq x x' insertionresult i x h | just .x | [ il ] | yes refl = insert-same il insertionresult i x h | just x' | [ il ] | no x≢x' = insert-wrong x' x≢x' il insertionresult i x h | nothing | [ il ] = insert-new il lemma-checkInsert-same : {n : ℕ} → (i : Fin n) → (x : Carrier) → (m : FinMapMaybe n Carrier) → lookupM i m ≡ just x → checkInsert i x m ≡ just m lemma-checkInsert-same i x m p with lookupM i m lemma-checkInsert-same i x m refl | .(just x) with deq x x lemma-checkInsert-same i x m refl | .(just x) | yes refl = refl lemma-checkInsert-same i x m refl | .(just x) | no x≢x = contradiction refl x≢x lemma-checkInsert-new : {n : ℕ} → (i : Fin n) → (x : Carrier) → (m : FinMapMaybe n Carrier) → lookupM i m ≡ nothing → checkInsert i x m ≡ just (insert i x m) lemma-checkInsert-new i x m p with lookupM i m lemma-checkInsert-new i x m refl | .nothing = refl lemma-checkInsert-wrong : {n : ℕ} → (i : Fin n) → (x : Carrier) → (m : FinMapMaybe n Carrier) → (x' : Carrier) → x ≢ x' → lookupM i m ≡ just x' → checkInsert i x m ≡ nothing lemma-checkInsert-wrong i x m x' d p with lookupM i m lemma-checkInsert-wrong i x m x' d refl | .(just x') with deq x x' lemma-checkInsert-wrong i x m x' d refl | .(just x') | yes q = contradiction q d lemma-checkInsert-wrong i x m x' d refl | .(just x') | no ¬q = refl lemma-checkInsert-restrict : {n : ℕ} → (f : Fin n → Carrier) → (i : Fin n) → (is : List (Fin n)) → checkInsert i (f i) (restrict f is) ≡ just (restrict f (i ∷ is)) lemma-checkInsert-restrict f i is with checkInsert i (f i) (restrict f is) | insertionresult i (f i) (restrict f is) lemma-checkInsert-restrict f i is | ._ | insert-same p = cong just (lemma-insert-same _ i (f i) p) lemma-checkInsert-restrict f i is | ._ | insert-new _ = refl lemma-checkInsert-restrict f i is | ._ | insert-wrong x fi≢x p = contradiction (lemma-lookupM-restrict i f is x p) fi≢x lemma-lookupM-checkInsert : {n : ℕ} → (i j : Fin n) → (x y : Carrier) → (h h' : FinMapMaybe n Carrier) → lookupM i h ≡ just x → checkInsert j y h ≡ just h' → lookupM i h' ≡ just x lemma-lookupM-checkInsert i j x y h h' pl ph' with lookupM j h | inspect (lookupM j) h lemma-lookupM-checkInsert i j x y h .(insert j y h) pl refl | nothing | pl' with i ≟ j lemma-lookupM-checkInsert i .i x y h .(insert i y h) pl refl | nothing | [ pl' ] | yes refl = lemma-just≢nothing (trans (sym pl) pl') lemma-lookupM-checkInsert i j x y h .(insert j y h) pl refl | nothing | pl' | no ¬p = begin lookupM i (insert j y h) ≡⟨ sym (lemma-lookupM-insert-other i j y h ¬p) ⟩ lookupM i h ≡⟨ pl ⟩ just x ∎ lemma-lookupM-checkInsert i j x y h h' pl ph' | just z | pl' with deq y z lemma-lookupM-checkInsert i j x y h .h pl refl | just .y | pl' | yes refl = pl lemma-lookupM-checkInsert i j x y h h' pl () | just z | pl' | no ¬p lemma-lookupM-checkInsert-other : {n : ℕ} → (i j : Fin n) → i ≢ j → (x : Carrier) → (h h' : FinMapMaybe n Carrier) → checkInsert j x h ≡ just h' → lookupM i h ≡ lookupM i h' lemma-lookupM-checkInsert-other i j i≢j x h h' ph' with lookupM j h lemma-lookupM-checkInsert-other i j i≢j x h h' ph' | just y with deq x y lemma-lookupM-checkInsert-other i j i≢j x h .h refl | just .x | yes refl = refl lemma-lookupM-checkInsert-other i j i≢j x h h' () | just y | no x≢y lemma-lookupM-checkInsert-other i j i≢j x h .(insert j x h) refl | nothing = lemma-lookupM-insert-other i j x h i≢j