2 open import Data.Bool hiding (_≟_)
5 open import Data.Fin.Props renaming (_≟_ to _≟F_)
7 open import Data.List hiding (replicate)
8 open import Data.Vec hiding (map ; zip ; _>>=_) renaming (lookup to lookupVec)
9 open import Data.Product hiding (zip ; map)
11 open import Relation.Nullary
12 open import Relation.Nullary.Negation
13 open import Relation.Binary.Core
14 open import Relation.Binary.PropositionalEquality
15 open Relation.Binary.PropositionalEquality.≡-Reasoning
19 _>>=_ : {A B : Set} → Maybe A → (A → Maybe B) → Maybe B
20 _>>=_ = flip (flip maybe′ nothing)
22 fmap : {A B : Set} → (A → B) → Maybe A → Maybe B
23 fmap f = maybe′ (λ a → just (f a)) nothing
26 EqInst A = (x y : A) → Dec (x ≡ y)
28 checkInsert : {A : Set} {n : ℕ} → EqInst A → Fin n → A → FinMapMaybe n A → Maybe (FinMapMaybe n A)
29 checkInsert eq i b m with lookupM i m
30 checkInsert eq i b m | just c with eq b c
31 checkInsert eq i b m | just .b | yes refl = just m
32 checkInsert eq i b m | just c | no ¬p = nothing
33 checkInsert eq i b m | nothing = just (insert i b m)
34 assoc : {A : Set} {n : ℕ} → EqInst A → List (Fin n) → List A → Maybe (FinMapMaybe n A)
35 assoc _ [] [] = just empty
36 assoc eq (i ∷ is) (b ∷ bs) = (assoc eq is bs) >>= (checkInsert eq i b)
39 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))
40 lemma-checkInsert-generate eq f i is with lookupM i (generate f is) | inspect (lookupM i) (generate f is)
41 lemma-checkInsert-generate eq f i is | nothing | _ = refl
42 lemma-checkInsert-generate eq f i is | just x | Reveal_is_.[_] prf with lemma-lookupM-generate i f is x prf
43 lemma-checkInsert-generate eq f i is | just .(f i) | Reveal_is_.[_] prf | refl with eq (f i) (f i)
44 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)
45 lemma-checkInsert-generate eq f i is | just .(f i) | Reveal_is_.[_] prf | refl | no ¬p = contradiction refl ¬p
47 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)
48 lemma-1 eq f [] = refl
49 lemma-1 eq f (i ∷ is′) = begin
50 (assoc eq (i ∷ is′) (map f (i ∷ is′)))
52 (assoc eq is′ (map f is′) >>= checkInsert eq i (f i))
53 ≡⟨ cong (λ m → m >>= checkInsert eq i (f i)) (lemma-1 eq f is′) ⟩
54 (just (generate f is′) >>= (checkInsert eq i (f i)))
56 (checkInsert eq i (f i) (generate f is′))
57 ≡⟨ lemma-checkInsert-generate eq f i is′ ⟩
58 just (generate f (i ∷ is′)) ∎
60 lemma-2 : {τ : Set} {n : ℕ} → (eq : EqInst τ) → (is : List (Fin n)) → (v : List τ) → (h : FinMapMaybe n τ) → just h ≡ assoc eq is v → map (flip lookup h) is ≡ map just v
61 lemma-2 eq [] [] h p = refl
62 lemma-2 eq [] (x ∷ xs) h ()
63 lemma-2 eq (x ∷ xs) [] h ()
64 lemma-2 eq (i ∷ is) (x ∷ xs) h p = {!!}
66 idrange : (n : ℕ) → List (Fin n)
67 idrange n = toList (tabulate id)
69 bff : ({A : Set} → List A → List A) → ({B : Set} → EqInst B → List B → List B → Maybe (List B))
70 bff get eq s v = let s′ = idrange (length s)
71 g = fromFunc (λ f → lookupVec f (fromList s))
72 h = assoc eq (get s′) v
73 h′ = fmap (flip union g) h
74 in fmap (flip map s′ ∘ flip lookup) h′
76 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
77 theorem-1 get eq s = {!!}