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 checkInsert i b m | just c with deq b c checkInsert i b m | just .b | yes refl = just m checkInsert i b m | just c | no ¬p = nothing checkInsert i b m | nothing = just (insert i b m) record checkInsertProof {n : ℕ} (i : Fin n) (x : Carrier) (m : FinMapMaybe n Carrier) (P : Set) : Set where field same : lookupM i m ≡ just x → P new : lookupM i m ≡ nothing → P wrong : (x' : Carrier) → x ≢ x' → lookupM i m ≡ just x' → P apply-checkInsertProof : {P : Set} {n : ℕ} → (i : Fin n) → (x : Carrier) → (m : FinMapMaybe n Carrier) → checkInsertProof i x m P → P apply-checkInsertProof i x m rp with lookupM i m | inspect (lookupM i) m apply-checkInsertProof i x m rp | just x' | il with deq x x' apply-checkInsertProof i x m rp | just .x | [ il ] | yes refl = checkInsertProof.same rp il apply-checkInsertProof i x m rp | just x' | [ il ] | no x≢x' = checkInsertProof.wrong rp x' x≢x' il apply-checkInsertProof i x m rp | nothing | [ il ] = checkInsertProof.new rp 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 record checkInsertEqualProof {n : ℕ} (i : Fin n) (x : Carrier) (m : FinMapMaybe n Carrier) (e : Maybe (FinMapMaybe n Carrier)) : Set where field same : lookupM i m ≡ just x → just m ≡ e new : lookupM i m ≡ nothing → just (insert i x m) ≡ e wrong : (x' : Carrier) → x ≢ x' → lookupM i m ≡ just x' → nothing ≡ e lift-checkInsertProof : {n : ℕ} {i : Fin n} {x : Carrier} {m : FinMapMaybe n Carrier} {e : Maybe (FinMapMaybe n Carrier)} → checkInsertEqualProof i x m e → checkInsertProof i x m (checkInsert i x m ≡ e) lift-checkInsertProof {_} {i} {x} {m} o = record { same = λ p → trans (lemma-checkInsert-same i x m p) (checkInsertEqualProof.same o p) ; new = λ p → trans (lemma-checkInsert-new i x m p) (checkInsertEqualProof.new o p) ; wrong = λ x' q p → trans (lemma-checkInsert-wrong i x m x' q p) (checkInsertEqualProof.wrong o x' q p) } 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 = apply-checkInsertProof i (f i) (restrict f is) (lift-checkInsertProof record { same = λ lookupM≡justx → cong just (lemma-insert-same (restrict f is) i (f i) lookupM≡justx) ; new = λ lookupM≡nothing → refl ; wrong = λ x' x≢x' lookupM≡justx' → contradiction (lemma-lookupM-restrict i f is x' lookupM≡justx') x≢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 (begin just x ≡⟨ sym pl ⟩ lookupM i h ≡⟨ pl' ⟩ (nothing ∎)) 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 ph' | just .y | pl' | yes refl = begin lookupM i h' ≡⟨ cong (lookupM i) (just-injective (sym ph')) ⟩ lookupM i h ≡⟨ pl ⟩ just x ∎ lemma-lookupM-checkInsert i j x y h h' pl () | just z | pl' | no ¬p