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open import Level using () renaming (zero to ℓ₀)
open import Relation.Binary using (DecSetoid)
module Bidir (A : DecSetoid ℓ₀ ℓ₀) where
open import Data.Nat using (ℕ)
open import Data.Fin using (Fin)
import Level
import Category.Monad
import Category.Functor
open import Data.Maybe using (Maybe ; nothing ; just ; maybe′) renaming (setoid to MaybeSetoid ; Eq to MaybeEq)
open Category.Monad.RawMonad {Level.zero} Data.Maybe.monad using (_>>=_)
open Category.Functor.RawFunctor {Level.zero} Data.Maybe.functor using (_<$>_)
open import Data.List using (List)
open import Data.List.All using (All)
open import Data.Vec using (Vec ; [] ; _∷_ ; toList ; map ; tabulate) renaming (lookup to lookupVec)
open import Data.Vec.Equality using () renaming (module Equality to VecEq)
open import Data.Vec.Properties using (tabulate-∘ ; lookup∘tabulate ; map-cong ; map-∘)
open import Data.Product using (∃ ; _×_ ; _,_ ; proj₁ ; proj₂)
open import Function using (id ; _∘_ ; flip)
open import Relation.Binary.Core using (refl ; _≡_)
open import Relation.Binary.PropositionalEquality using (cong ; sym ; inspect ; [_] ; trans ; cong₂ ; decSetoid ; module ≡-Reasoning) renaming (setoid to EqSetoid)
open import Relation.Binary using (Setoid ; module Setoid ; module DecSetoid)
import Relation.Binary.EqReasoning as EqR
import FreeTheorems
open FreeTheorems.VecVec using (get-type ; free-theorem)
open import Generic using (just-injective ; map-just-injective ; vecIsSetoid)
open import FinMap
import CheckInsert
open CheckInsert A
import BFF
open BFF.VecBFF A using (assoc ; enumerate ; denumerate ; bff)
open module A = DecSetoid A using (Carrier) renaming (_≟_ to deq)
module SetoidReasoning where
infix 1 begin⟨_⟩_
infixr 2 _≈⟨_⟩_ _≡⟨_⟩_
infix 2 _∎
begin⟨_⟩_ : (X : Setoid ℓ₀ ℓ₀) → {x y : Setoid.Carrier X} → EqR._IsRelatedTo_ X x y → Setoid._≈_ X x y
begin⟨_⟩_ X p = EqR.begin_ X p
_∎ : {X : Setoid ℓ₀ ℓ₀} → (x : Setoid.Carrier X) → EqR._IsRelatedTo_ X x x
_∎ {X} = EqR._∎ X
_≈⟨_⟩_ : {X : Setoid ℓ₀ ℓ₀} → (x : Setoid.Carrier X) → {y z : Setoid.Carrier X} → Setoid._≈_ X x y → EqR._IsRelatedTo_ X y z → EqR._IsRelatedTo_ X x z
_≈⟨_⟩_ {X} = EqR._≈⟨_⟩_ X
_≡⟨_⟩_ : {X : Setoid ℓ₀ ℓ₀} → (x : Setoid.Carrier X) → {y z : Setoid.Carrier X} → x ≡ y → EqR._IsRelatedTo_ X y z → EqR._IsRelatedTo_ X x z
_≡⟨_⟩_ {X} = EqR._≡⟨_⟩_ X
lemma-1 : {m n : ℕ} → (f : Fin n → Carrier) → (is : Vec (Fin n) m) → assoc is (map f is) ≡ just (restrict f (toList is))
lemma-1 f [] = refl
lemma-1 f (i ∷ is′) = begin
(assoc is′ (map f is′) >>= checkInsert i (f i))
≡⟨ cong (λ m → m >>= checkInsert i (f i)) (lemma-1 f is′) ⟩
checkInsert i (f i) (restrict f (toList is′))
≡⟨ lemma-checkInsert-restrict f i (toList is′) ⟩
just (restrict f (toList (i ∷ is′))) ∎
where open ≡-Reasoning
lemma-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (x : Carrier) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc (i ∷ is) (x ∷ xs) ≡ just h → Setoid._≈_ (MaybeSetoid A.setoid) (lookupM i h) (just x)
lemma-lookupM-assoc i is x xs h p with assoc is xs
lemma-lookupM-assoc i is x xs h () | nothing
lemma-lookupM-assoc i is x xs h p | just h' with checkInsert i x h' | insertionresult i x h'
lemma-lookupM-assoc i is x xs .h refl | just h | ._ | same x' x≈x' pl = begin
lookupM i h
≡⟨ pl ⟩
just x'
≈⟨ MaybeEq.just (Setoid.sym A.setoid x≈x') ⟩
just x ∎
where open EqR (MaybeSetoid A.setoid)
lemma-lookupM-assoc i is x xs ._ refl | just h' | ._ | new _ = Setoid.reflexive (MaybeSetoid A.setoid) (lemma-lookupM-insert i x h')
lemma-lookupM-assoc i is x xs h () | just h' | ._ | wrong _ _ _
_in-domain-of_ : {n : ℕ} {A : Set} → (is : List (Fin n)) → (FinMapMaybe n A) → Set
_in-domain-of_ is h = All (λ i → ∃ λ x → lookupM i h ≡ just x) is
lemma-assoc-domain : {m n : ℕ} → (is : Vec (Fin n) m) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc is xs ≡ just h → (toList is) in-domain-of h
lemma-assoc-domain [] [] h ph = Data.List.All.[]
lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h ph with assoc is' xs' | inspect (assoc is') xs'
lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h () | nothing | [ ph' ]
lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h ph | just h' | [ ph' ] with checkInsert i' x' h' | inspect (checkInsert i' x') h' | insertionresult i' x' h'
lemma-assoc-domain (i' ∷ is') (x' ∷ xs') .h refl | just h | [ ph' ] | ._ | _ | same x _ pl = All._∷_ (x , pl) (lemma-assoc-domain is' xs' h ph')
lemma-assoc-domain (i' ∷ is') (x' ∷ xs') ._ refl | just h' | [ ph' ] | ._ | [ cI≡ ] | new _ = All._∷_
(x' , lemma-lookupM-insert i' x' h')
(Data.List.All.map
(λ {i} p → proj₁ p , lemma-lookupM-checkInsert i i' (proj₁ p) x' h' (insert i' x' h') (proj₂ p) cI≡)
(lemma-assoc-domain is' xs' h' ph'))
lemma-assoc-domain (i' ∷ is') (x' ∷ xs') h () | just h' | [ ph' ] | ._ | _ | wrong _ _ _
lemma-map-lookupM-assoc : {m : ℕ} → (i : Fin m) → (x : Carrier) → (h : FinMapMaybe m Carrier) → (h' : FinMapMaybe m Carrier) → checkInsert i x h' ≡ just h → {n : ℕ} → (js : Vec (Fin m) n) → (toList js) in-domain-of h' → map (flip lookupM h) js ≡ map (flip lookupM h') js
lemma-map-lookupM-assoc i x h h' ph [] pj = refl
lemma-map-lookupM-assoc i x h h' ph (j ∷ js) (Data.List.All._∷_ (x' , pl) pj) = cong₂ _∷_
(trans (lemma-lookupM-checkInsert j i x' x h' h pl ph) (sym pl))
(lemma-map-lookupM-assoc i x h h' ph js pj)
lemma-2 : {m n : ℕ} → (is : Vec (Fin n) m) → (v : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc is v ≡ just h → Setoid._≈_ (vecIsSetoid (MaybeSetoid A.setoid) m) (map (flip lookupM h) is) (map just v)
lemma-2 [] [] h p = Setoid.refl (vecIsSetoid (MaybeSetoid A.setoid) _)
lemma-2 (i ∷ is) (x ∷ xs) h p with assoc is xs | inspect (assoc is) xs
lemma-2 (i ∷ is) (x ∷ xs) h () | nothing | _
lemma-2 (i ∷ is) (x ∷ xs) h p | just h' | [ ir ] = begin
lookupM i h ∷ map (flip lookupM h) is
≈⟨ lemma-lookupM-assoc i is x xs h (trans (cong (flip _>>=_ (checkInsert i x)) ir) p) VecEq.∷-cong Setoid.refl (vecIsSetoid (MaybeSetoid A.setoid) _) ⟩
just x ∷ map (flip lookupM h) is
≡⟨ cong (_∷_ (just x)) (lemma-map-lookupM-assoc i x h h' p is (lemma-assoc-domain is xs h' ir)) ⟩
just x ∷ map (flip lookupM h') is
≈⟨ Setoid.refl (MaybeSetoid A.setoid) VecEq.∷-cong (lemma-2 is xs h' ir) ⟩
just x ∷ map just xs ∎
where open EqR (vecIsSetoid (MaybeSetoid A.setoid) _)
lemma-map-denumerate-enumerate : {m : ℕ} → (as : Vec Carrier m) → map (denumerate as) (enumerate as) ≡ as
lemma-map-denumerate-enumerate [] = refl
lemma-map-denumerate-enumerate (a ∷ as) = cong (_∷_ a) (begin
map (flip lookupVec (a ∷ as)) (tabulate Fin.suc)
≡⟨ cong (map (flip lookupVec (a ∷ as))) (tabulate-∘ Fin.suc id) ⟩
map (flip lookupVec (a ∷ as)) (map Fin.suc (tabulate id))
≡⟨ refl ⟩
map (flip lookupVec (a ∷ as)) (map Fin.suc (enumerate as))
≡⟨ sym (map-∘ _ _ (enumerate as)) ⟩
map (flip lookupVec (a ∷ as) ∘ Fin.suc) (enumerate as)
≡⟨ refl ⟩
map (denumerate as) (enumerate as)
≡⟨ lemma-map-denumerate-enumerate as ⟩
as ∎)
where open ≡-Reasoning
theorem-1 : {getlen : ℕ → ℕ} → (get : get-type getlen) → {m : ℕ} → (s : Vec Carrier m) → bff get s (get s) ≡ just s
theorem-1 get s = begin
bff get s (get s)
≡⟨ cong (bff get s ∘ get) (sym (lemma-map-denumerate-enumerate s)) ⟩
bff get s (get (map (denumerate s) (enumerate s)))
≡⟨ cong (bff get s) (free-theorem get (denumerate s) (enumerate s)) ⟩
bff get s (map (denumerate s) (get (enumerate s)))
≡⟨ refl ⟩
(h′↦r ∘ h↦h′) (assoc (get (enumerate s)) (map (denumerate s) (get (enumerate s))))
≡⟨ cong (h′↦r ∘ h↦h′) (lemma-1 (denumerate s) (get (enumerate s))) ⟩
(h′↦r ∘ h↦h′ ∘ just) (restrict (denumerate s) (toList (get (enumerate s))))
≡⟨ refl ⟩
(h′↦r ∘ just) (union (restrict (denumerate s) (toList (get (enumerate s)))) (fromFunc (denumerate s)))
≡⟨ cong (h′↦r ∘ just) (lemma-union-restrict (denumerate s) (toList (get (enumerate s)))) ⟩
(h′↦r ∘ just) (fromFunc (denumerate s))
≡⟨ refl ⟩
just (map (flip lookup (fromFunc (denumerate s))) (enumerate s))
≡⟨ cong just (map-cong (lookup∘tabulate (denumerate s)) (enumerate s)) ⟩
just (map (denumerate s) (enumerate s))
≡⟨ cong just (lemma-map-denumerate-enumerate s) ⟩
just s ∎
where open ≡-Reasoning
h↦h′ = _<$>_ (flip union (fromFunc (denumerate s)))
h′↦r = _<$>_ (flip map (enumerate s) ∘ flip lookupVec)
lemma-<$>-just : {A B : Set} {f : A → B} {b : B} {ma : Maybe A} → f <$> ma ≡ just b → ∃ λ a → ma ≡ just a
lemma-<$>-just {ma = just x} f<$>ma≡just-b = x , refl
lemma-<$>-just {ma = nothing} ()
lemma-union-not-used : {m n : ℕ} {A : Set} (h : FinMapMaybe n A) → (h' : FinMap n A) → (is : Vec (Fin n) m) → (toList is) in-domain-of h → map just (map (flip lookup (union h h')) is) ≡ map (flip lookupM h) is
lemma-union-not-used h h' [] p = refl
lemma-union-not-used h h' (i ∷ is') (Data.List.All._∷_ (x , px) p') = cong₂ _∷_ (begin
just (lookup i (union h h'))
≡⟨ cong just (lookup∘tabulate (λ j → maybe′ id (lookup j h') (lookupM j h)) i) ⟩
just (maybe′ id (lookup i h') (lookupM i h))
≡⟨ cong just (cong (maybe′ id (lookup i h')) px) ⟩
just (maybe′ id (lookup i h') (just x))
≡⟨ sym px ⟩
lookupM i h ∎)
(lemma-union-not-used h h' is' p')
where open ≡-Reasoning
map-just-≈-injective : {n : ℕ} {x y : Vec Carrier n} → Setoid._≈_ (vecIsSetoid (MaybeSetoid A.setoid) n) (map just x) (map just y) → Setoid._≈_ (vecIsSetoid A.setoid n) x y
map-just-≈-injective {x = []} {y = []} VecEq.[]-cong = VecEq.[]-cong
map-just-≈-injective {x = _ ∷ _} {y = _ ∷ _} (just x≈y VecEq.∷-cong ps) = x≈y VecEq.∷-cong map-just-≈-injective ps
theorem-2 : {getlen : ℕ → ℕ} (get : get-type getlen) → {m : ℕ} → (v : Vec Carrier (getlen m)) → (s u : Vec Carrier m) → bff get s v ≡ just u → Setoid._≈_ (vecIsSetoid A.setoid (getlen m)) (get u) v
theorem-2 get v s u p with lemma-<$>-just (proj₂ (lemma-<$>-just p))
theorem-2 get v s u p | h , ph = begin⟨ vecIsSetoid A.setoid _ ⟩
get u
≡⟨ just-injective (begin⟨ EqSetoid _ ⟩
get <$> (just u)
≡⟨ cong (_<$>_ get) (sym p) ⟩
get <$> (bff get s v)
≡⟨ cong (_<$>_ get ∘ _<$>_ h′↦r ∘ _<$>_ h↦h′) ph ⟩
get <$> (h′↦r <$> (h↦h′ <$> just h)) ∎) ⟩
get (map (flip lookup (h↦h′ h)) s′)
≡⟨ free-theorem get (flip lookup (h↦h′ h)) s′ ⟩
map (flip lookup (h↦h′ h)) (get s′)
≈⟨ map-just-≈-injective (begin⟨ vecIsSetoid (MaybeSetoid A.setoid) _ ⟩
map just (map (flip lookup (union h g)) (get s′))
≡⟨ lemma-union-not-used h g (get s′) (lemma-assoc-domain (get s′) v h ph) ⟩
map (flip lookupM h) (get s′)
≈⟨ lemma-2 (get s′) v h ph ⟩
map just v
∎) ⟩
v ∎
where open SetoidReasoning
s′ = enumerate s
g = fromFunc (denumerate s)
h↦h′ = flip union g
h′↦r = flip map s′ ∘ flip lookupVec
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