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open import Relation.Binary.Core using (Decidable ; _≡_)
module Precond (Carrier : Set) (deq : Decidable {A = Carrier} _≡_) where
open import Data.Nat using (ℕ)
open import Data.Fin using (Fin ; zero ; suc)
open import Data.Fin.Props using (_≟_)
open import Data.List using (List ; [] ; _∷_)
open import Level using () renaming (zero to ℓ₀)
import Category.Monad
import Category.Functor
open import Data.Maybe using (Maybe ; nothing ; just ; maybe′)
open Category.Monad.RawMonad {Level.zero} Data.Maybe.monad using (_>>=_)
open Category.Functor.RawFunctor {Level.zero} Data.Maybe.functor using (_<$>_)
open import Data.Vec using (Vec ; [] ; _∷_ ; map ; lookup ; toList)
open import Data.Vec.Properties using (map-cong ; map-∘ ; tabulate-∘)
import Data.List.All
open import Data.List.Any using (here ; there)
open Data.List.Any.Membership-≡ using (_∉_)
open import Data.Maybe using (just)
open import Data.Product using (∃ ; _,_ ; proj₁ ; proj₂)
open import Function using (flip ; _∘_ ; id)
open import Relation.Binary using (Setoid)
open import Relation.Binary.PropositionalEquality using (refl ; cong ; inspect ; [_] ; sym ; decSetoid)
open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎)
open import Relation.Nullary using (yes ; no)
open import FinMap using (FinMapMaybe ; lookupM ; union ; fromFunc ; empty ; insert ; lemma-lookupM-empty ; delete-many ; lemma-tabulate-∘ ; delete ; lemma-lookupM-delete ; lemma-lookupM-fromFunc ; reshape ; lemma-reshape-id)
import CheckInsert
open CheckInsert (decSetoid deq) using (checkInsert ; lemma-checkInsert-new ; lemma-lookupM-checkInsert-other)
import BFF
import Bidir
open Bidir (decSetoid deq) using (_in-domain-of_ ; lemma-assoc-domain)
import GetTypes
open GetTypes.PartialVecVec using (Get ; module Get)
open BFF.PartialVecBFF (decSetoid deq) using (assoc ; enumerate ; denumerate ; bff ; enumeratel)
lemma-maybe-just : {A : Set} → (a : A) → (ma : Maybe A) → maybe′ Maybe.just (just a) ma ≡ Maybe.just (maybe′ id a ma)
lemma-maybe-just a (just x) = refl
lemma-maybe-just a nothing = refl
lemma-union-delete-fromFunc : {m n : ℕ} {A : Set} {is : Vec (Fin n) m} {h : FinMapMaybe n A} {g : Fin n → A} → (toList is) in-domain-of h → ∃ λ v → union h (delete-many is (fromFunc g)) ≡ fromFunc v
lemma-union-delete-fromFunc {is = []} {h = h} {g = g} p = _ , (lemma-tabulate-∘ (λ f → begin
maybe′ just (lookupM f (fromFunc g)) (lookupM f h)
≡⟨ cong (flip (maybe′ just) (lookupM f h)) (lemma-lookupM-fromFunc g f) ⟩
maybe′ just (just (g f)) (lookupM f h)
≡⟨ lemma-maybe-just (g f) (lookupM f h) ⟩
just (maybe′ id (g f) (lookupM f h)) ∎))
lemma-union-delete-fromFunc {n = n} {is = i ∷ is} {h = h} {g = g} (Data.List.All._∷_ (x , px) ps) = _ , (begin
union h (delete i (delete-many is (fromFunc g)))
≡⟨ lemma-tabulate-∘ inner ⟩
union h (delete-many is (fromFunc g))
≡⟨ proj₂ (lemma-union-delete-fromFunc ps) ⟩
_ ∎)
where inner : (f : Fin n) → maybe′ just (lookupM f (delete i (delete-many is (fromFunc g)))) (lookup f h) ≡ maybe′ just (lookupM f (delete-many is (fromFunc g))) (lookup f h)
inner f with f ≟ i
inner .i | yes refl = begin
maybe′ just (lookupM i (delete i (delete-many is (fromFunc g)))) (lookup i h)
≡⟨ cong (maybe′ just _) px ⟩
just x
≡⟨ cong (maybe′ just _) (sym px) ⟩
maybe′ just (lookupM i (delete-many is (fromFunc g))) (lookup i h) ∎
inner f | no f≢i = cong (flip (maybe′ just) (lookup f h)) (lemma-lookupM-delete (delete-many is (fromFunc g)) f≢i)
assoc-enough : (G : Get) → {i : Get.|I| G} → (j : Get.|I| G) → (s : Vec Carrier (Get.|gl₁| G i)) → (v : Vec Carrier (Get.|gl₂| G j)) → ∃ (λ h → assoc (Get.get G (enumeratel (Get.|gl₁| G j))) v ≡ just h) → ∃ λ u → bff G j s v ≡ just u
assoc-enough G {i} j s v (h , p) = _ , cong (_<$>_ (flip map t ∘ flip lookupM) ∘ _<$>_ (flip union (reshape g′ (|gl₁| j)))) p
where open Get G
g′ = delete-many (get (enumerate s)) (fromFunc (denumerate s))
t = enumeratel (|gl₁| j)
assoc-enough′ : (G : Get) → {i : Get.|I| G} → (s : Vec Carrier (Get.|gl₁| G i)) → (v : Vec Carrier (Get.|gl₂| G i)) → ∃ (λ h → assoc (Get.get G (enumeratel (Get.|gl₁| G i))) v ≡ just h) → ∃ λ u → bff G i s v ≡ just (map just u)
assoc-enough′ G {i} s v (h , p) = _ , (begin
bff G i s v
≡⟨ proj₂ (assoc-enough G i s v (h , p)) ⟩
just (map (flip lookupM (union h (reshape g′ (|gl₁| i)))) t)
≡⟨ cong just (begin _
≡⟨ cong (flip map t ∘ flip lookupM ∘ union h) (lemma-reshape-id g′) ⟩
map (flip lookupM (union h g′)) t
≡⟨ cong (flip map t ∘ flip lookupM) (proj₂ wp) ⟩
map (flip lookupM (fromFunc (proj₁ wp))) t
≡⟨ map-cong (lemma-lookupM-fromFunc (proj₁ wp)) t ⟩
map (Maybe.just ∘ proj₁ wp) t
≡⟨ map-∘ just (proj₁ wp) t ⟩
map Maybe.just (map (proj₁ wp) t) ∎) ⟩ _ ∎)
where open Get G
s′ = enumerate s
g = fromFunc (denumerate s)
g′ = delete-many (get s′) g
t = enumeratel (Get.|gl₁| G i)
wp = lemma-union-delete-fromFunc (lemma-assoc-domain (get t) v h p)
data All-different {A : Set} : List A → Set where
different-[] : All-different []
different-∷ : {x : A} {xs : List A} → x ∉ xs → All-different xs → All-different (x ∷ xs)
lemma-∉-lookupM-assoc : {m n : ℕ} → (i : Fin n) → (is : Vec (Fin n) m) → (xs : Vec Carrier m) → (h : FinMapMaybe n Carrier) → assoc is xs ≡ just h → (i ∉ toList is) → lookupM i h ≡ nothing
lemma-∉-lookupM-assoc i [] [] .empty refl i∉is = lemma-lookupM-empty i
lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is with assoc is' xs' | inspect (assoc is') xs'
lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h () i∉is | nothing | [ ph' ]
lemma-∉-lookupM-assoc i (i' ∷ is') (x' ∷ xs') h ph i∉is | just h' | [ ph' ] = begin
lookupM i h
≡⟨ sym (lemma-lookupM-checkInsert-other i i' (i∉is ∘ here) x' h' h ph) ⟩
lookupM i h'
≡⟨ lemma-∉-lookupM-assoc i is' xs' h' ph' (i∉is ∘ there) ⟩
nothing ∎
different-assoc : {m n : ℕ} → (u : Vec (Fin n) m) → (v : Vec Carrier m) → All-different (toList u) → ∃ λ h → assoc u v ≡ just h
different-assoc [] [] p = empty , refl
different-assoc (u ∷ us) (v ∷ vs) (different-∷ u∉us diff-us) with different-assoc us vs diff-us
different-assoc (u ∷ us) (v ∷ vs) (different-∷ u∉us diff-us) | h , p' = insert u v h , (begin
assoc (u ∷ us) (v ∷ vs)
≡⟨ refl ⟩
(assoc us vs >>= checkInsert u v)
≡⟨ cong (flip _>>=_ (checkInsert u v)) p' ⟩
checkInsert u v h
≡⟨ lemma-checkInsert-new u v h (lemma-∉-lookupM-assoc u us vs h p' u∉us) ⟩
just (insert u v h) ∎)
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