3 open import Data.Nat using (â„• ; zero ; suc)
4 open import Data.Maybe using (Maybe ; just ; nothing ; maybe′)
5 open import Data.Fin using (Fin ; zero ; suc)
6 open import Data.Fin.Props using (_≟_)
7 open import Data.Vec using (Vec ; [] ; _∷_ ; _[_]≔_ ; replicate ; tabulate) renaming (lookup to lookupVec)
8 open import Data.Vec.Properties using (lookup∘tabulate)
9 open import Data.List using (List ; [] ; _∷_ ; map ; zip)
10 open import Data.Product using (_×_ ; _,_)
11 open import Function using (id ; _∘_ ; flip)
12 open import Relation.Nullary using (yes ; no)
13 open import Relation.Nullary.Negation using (contradiction)
14 open import Relation.Binary.Core using (_≡_ ; refl ; _≢_)
15 open import Relation.Binary.PropositionalEquality using (cong ; sym ; _≗_ ; trans ; cong₂)
16 open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎)
18 FinMapMaybe : ℕ → Set → Set
19 FinMapMaybe n A = Vec (Maybe A) n
21 lookupM : {A : Set} {n : ℕ} → Fin n → FinMapMaybe n A → Maybe A
24 insert : {A : Set} {n : ℕ} → Fin n → A → FinMapMaybe n A → FinMapMaybe n A
25 insert f a m = m [ f ]≔ (just a)
27 empty : {A : Set} {n : ℕ} → FinMapMaybe n A
28 empty = replicate nothing
30 fromAscList : {A : Set} {n : ℕ} → List (Fin n × A) → FinMapMaybe n A
31 fromAscList [] = empty
32 fromAscList ((f , a) ∷ xs) = insert f a (fromAscList xs)
34 FinMap : ℕ → Set → Set
37 lookup : {A : Set} {n : ℕ} → Fin n → FinMap n A → A
40 fromFunc : {A : Set} {n : ℕ} → (Fin n → A) → FinMap n A
43 union : {A : Set} {n : ℕ} → FinMapMaybe n A → FinMap n A → FinMap n A
44 union m1 m2 = fromFunc (λ f → maybe′ id (lookup f m2) (lookupM f m1))
46 restrict : {A : Set} {n : ℕ} → (Fin n → A) → List (Fin n) → FinMapMaybe n A
47 restrict f is = fromAscList (zip is (map f is))
49 lemma-just≢nothing : {A Whatever : Set} {a : A} {ma : Maybe A} → ma ≡ just a → ma ≡ nothing → Whatever
50 lemma-just≢nothing refl ()
52 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
53 lemma-insert-same [] () a p
54 lemma-insert-same (.(just a) ∷ xs) zero a refl = refl
55 lemma-insert-same (x ∷ xs) (suc i) a p = cong (_∷_ x) (lemma-insert-same xs i a p)
57 lemma-lookupM-empty : {A : Set} {n : ℕ} → (i : Fin n) → lookupM {A} i empty ≡ nothing
58 lemma-lookupM-empty zero = refl
59 lemma-lookupM-empty (suc i) = lemma-lookupM-empty i
61 lemma-lookupM-insert : {A : Set} {n : ℕ} → (i : Fin n) → (a : A) → (m : FinMapMaybe n A) → lookupM i (insert i a m) ≡ just a
62 lemma-lookupM-insert zero a (x ∷ xs) = refl
63 lemma-lookupM-insert (suc i) a (x ∷ xs) = lemma-lookupM-insert i a xs
65 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)
66 lemma-lookupM-insert-other zero zero a m p = contradiction refl p
67 lemma-lookupM-insert-other zero (suc j) a (x ∷ xs) p = refl
68 lemma-lookupM-insert-other (suc i) zero a (x ∷ xs) p = refl
69 lemma-lookupM-insert-other (suc i) (suc j) a (x ∷ xs) p = lemma-lookupM-insert-other i j a xs (p ∘ cong suc)
71 just-injective : {A : Set} → {x y : A} → _≡_ {_} {Maybe A} (just x) (just y) → x ≡ y
72 just-injective refl = refl
74 lemma-lookupM-restrict : {A : Set} {n : ℕ} → (i : Fin n) → (f : Fin n → A) → (is : List (Fin n)) → (a : A) → lookupM i (restrict f is) ≡ just a → f i ≡ a
75 lemma-lookupM-restrict i f [] a p = lemma-just≢nothing p (lemma-lookupM-empty i)
76 lemma-lookupM-restrict i f (i' ∷ is) a p with i ≟ i'
77 lemma-lookupM-restrict i f (.i ∷ is) a p | yes refl = just-injective (begin
79 ≡⟨ sym (lemma-lookupM-insert i (f i) (restrict f is)) ⟩
80 lookupM i (insert i (f i) (restrict f is))
83 lemma-lookupM-restrict i f (i' ∷ is) a p | no i≢i' = lemma-lookupM-restrict i f is a (begin
84 lookupM i (restrict f is)
85 ≡⟨ lemma-lookupM-insert-other i i' (f i') (restrict f is) i≢i' ⟩
86 lookupM i (insert i' (f i') (restrict f is))
90 lemma-tabulate-∘ : {n : ℕ} {A : Set} → {f g : Fin n → A} → f ≗ g → tabulate f ≡ tabulate g
91 lemma-tabulate-∘ {zero} {_} {f} {g} f≗g = refl
92 lemma-tabulate-∘ {suc n} {_} {f} {g} f≗g = cong₂ _∷_ (f≗g zero) (lemma-tabulate-∘ (f≗g ∘ suc))
94 lemma-union-restrict : {n : ℕ} {A : Set} → (f : Fin n → A) → (is : List (Fin n)) → union (restrict f is) (fromFunc f) ≡ fromFunc f
95 lemma-union-restrict {n} f is = begin
96 union (restrict f is) (fromFunc f)
98 tabulate (λ j → maybe′ id (lookup j (fromFunc f)) (lookupM j (restrict f is)))
99 ≡⟨ lemma-tabulate-∘ (lemma-inner is) ⟩
101 where lemma-inner : (is : List (Fin n)) → (j : Fin n) → maybe′ id (lookup j (fromFunc f)) (lookupM j (restrict f is)) ≡ f j
102 lemma-inner [] j = begin
103 maybe′ id (lookup j (fromFunc f)) (lookupM j empty)
104 ≡⟨ cong (maybe′ id (lookup j (fromFunc f))) (lemma-lookupM-empty j) ⟩
105 maybe′ id (lookup j (fromFunc f)) nothing
107 lookup j (fromFunc f)
108 ≡⟨ lookup∘tabulate f j ⟩
110 lemma-inner (i ∷ is) j with j ≟ i
111 lemma-inner (.j ∷ is) j | yes refl = cong (maybe′ id (lookup j (fromFunc f))) (lemma-lookupM-insert j (f j) (restrict f is))
112 lemma-inner (i ∷ is) j | no j≢i = begin
113 maybe′ id (lookup j (fromFunc f)) (lookupM j (insert i (f i) (restrict f is)))
114 ≡⟨ cong (maybe′ id (lookup j (fromFunc f))) (sym (lemma-lookupM-insert-other j i (f i) (restrict f is) j≢i)) ⟩
115 maybe′ id (lookup j (fromFunc f)) (lookupM j (restrict f is))
116 ≡⟨ lemma-inner is j ⟩