3 open import Data.Nat using (â„•)
4 open import Data.Fin using (Fin)
5 open import Data.Maybe using (Maybe ; nothing ; just ; maybe′)
6 open import Data.List using (List ; [] ; _∷_ ; map ; length)
7 open import Data.Vec using (toList ; fromList ; tabulate) renaming (lookup to lookupVec)
8 open import Function using (id ; _∘_ ; flip)
9 open import Relation.Nullary using (Dec ; yes ; no)
10 open import Relation.Nullary.Negation using (contradiction)
11 open import Relation.Binary.Core using (_≡_ ; refl)
12 open import Relation.Binary.PropositionalEquality using (cong ; sym ; inspect ; Reveal_is_)
13 open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎)
17 _>>=_ : {A B : Set} → Maybe A → (A → Maybe B) → Maybe B
18 _>>=_ = flip (flip maybe′ nothing)
20 fmap : {A B : Set} → (A → B) → Maybe A → Maybe B
21 fmap f = maybe′ (λ a → just (f a)) nothing
24 EqInst A = (x y : A) → Dec (x ≡ y)
26 checkInsert : {A : Set} {n : ℕ} → EqInst A → Fin n → A → FinMapMaybe n A → Maybe (FinMapMaybe n A)
27 checkInsert eq i b m with lookupM i m
28 checkInsert eq i b m | just c with eq b c
29 checkInsert eq i b m | just .b | yes refl = just m
30 checkInsert eq i b m | just c | no ¬p = nothing
31 checkInsert eq i b m | nothing = just (insert i b m)
32 assoc : {A : Set} {n : ℕ} → EqInst A → List (Fin n) → List A → Maybe (FinMapMaybe n A)
33 assoc _ [] [] = just empty
34 assoc eq (i ∷ is) (b ∷ bs) = (assoc eq is bs) >>= (checkInsert eq i b)
37 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))
38 lemma-checkInsert-generate eq f i is with lookupM i (generate f is) | inspect (lookupM i) (generate f is)
39 lemma-checkInsert-generate eq f i is | nothing | _ = refl
40 lemma-checkInsert-generate eq f i is | just x | Reveal_is_.[_] prf with lemma-lookupM-generate i f is x prf
41 lemma-checkInsert-generate eq f i is | just .(f i) | Reveal_is_.[_] prf | refl with eq (f i) (f i)
42 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)
43 lemma-checkInsert-generate eq f i is | just .(f i) | Reveal_is_.[_] prf | refl | no ¬p = contradiction refl ¬p
45 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)
46 lemma-1 eq f [] = refl
47 lemma-1 eq f (i ∷ is′) = begin
48 (assoc eq (i ∷ is′) (map f (i ∷ is′)))
50 (assoc eq is′ (map f is′) >>= checkInsert eq i (f i))
51 ≡⟨ cong (λ m → m >>= checkInsert eq i (f i)) (lemma-1 eq f is′) ⟩
52 (just (generate f is′) >>= (checkInsert eq i (f i)))
54 (checkInsert eq i (f i) (generate f is′))
55 ≡⟨ lemma-checkInsert-generate eq f i is′ ⟩
56 just (generate f (i ∷ is′)) ∎
58 lemma-lookupM-assoc : {A : Set} {n : ℕ} → (eq : EqInst A) → (i : Fin n) → (is : List (Fin n)) → (x : A) → (xs : List A) → (h : FinMapMaybe n A) → assoc eq (i ∷ is) (x ∷ xs) ≡ just h → lookupM i h ≡ just x
59 lemma-lookupM-assoc eq i is x xs h p with assoc eq is xs
60 lemma-lookupM-assoc eq i is x xs h () | nothing
61 lemma-lookupM-assoc eq i is x xs h p | just h' with lookupM i h' | inspect (lookupM i) h'
62 lemma-lookupM-assoc eq i is x xs .(insert i x h') refl | just h' | nothing | _ = lemma-lookupM-insert i x h'
63 lemma-lookupM-assoc eq i is x xs h p | just h' | just y | _ with eq x y
64 lemma-lookupM-assoc eq i is x xs h () | just h' | just y | _ | no ¬prf
65 lemma-lookupM-assoc eq i is x xs h p | just h' | just .x | Reveal_is_.[_] p' | yes refl = begin
67 ≡⟨ cong (lookupM i) (lemma-from-just (sym p)) ⟩
72 lemma-2 : {τ : Set} {n : ℕ} → (eq : EqInst τ) → (is : List (Fin n)) → (v : List τ) → (h : FinMapMaybe n τ) → assoc eq is v ≡ just h → map (flip lookupM h) is ≡ map just v
73 lemma-2 eq [] [] h p = refl
74 lemma-2 eq [] (x ∷ xs) h ()
75 lemma-2 eq (x ∷ xs) [] h ()
76 lemma-2 eq (i ∷ is) (x ∷ xs) h p = begin
77 map (flip lookupM h) (i ∷ is)
79 lookup i h ∷ map (flip lookupM h) is
80 ≡⟨ cong (flip _∷_ (map (flip lookup h) is)) (lemma-lookupM-assoc eq i is x xs h p) ⟩
81 just x ∷ map (flip lookupM h) is
82 ≡⟨ cong (_∷_ (just x)) (lemma-2 eq is xs h {!!}) ⟩
83 just x ∷ map just xs
85 map just (x ∷ xs) ∎
87 idrange : (n : ℕ) → List (Fin n)
88 idrange n = toList (tabulate id)
90 bff : ({A : Set} → List A → List A) → ({B : Set} → EqInst B → List B → List B → Maybe (List B))
91 bff get eq s v = let s′ = idrange (length s)
92 g = fromFunc (λ f → lookupVec f (fromList s))
93 h = assoc eq (get s′) v
94 h′ = fmap (flip union g) h
95 in fmap (flip map s′ ∘ flip lookup) h′
97 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
98 theorem-1 get eq s = {!!}