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)
open import Data.List using (List ; [] ; _∷_)
import Level
import Category.Monad
import Category.Functor
open import Data.Maybe using (nothing ; just)
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.List.Any using (here ; there)
open Data.List.Any.Membership-≡ using (_∉_)
open import Data.Maybe using (just)
open import Data.Product using (∃ ; _,_)
open import Function using (flip ; _∘_)
open import Relation.Binary.PropositionalEquality using (refl ; cong ; inspect ; [_] ; sym)
open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎)

open import FinMap using (FinMapMaybe ; lookupM ; union ; fromFunc ; empty ; insert ; lemma-lookupM-empty)
import CheckInsert
open CheckInsert Carrier deq using (checkInsert ; lemma-checkInsert-new ; lemma-lookupM-checkInsert-other)
import BFF
import Bidir

open BFF.VecBFF Carrier deq using (get-type ; assoc ; enumerate ; denumerate ; bff)

{-
assoc-enough : {getlen : ℕ → ℕ} (get : get-type getlen) → {m : ℕ} → (s : Vec Carrier m) → (v : Vec Carrier (getlen m)) → ∃ (λ h → assoc (get (enumerate s)) v ≡ just h) → ∃ λ u → bff get s v ≡ just u
assoc-enough get s v (h , p) = u , cong (_<$>_ (flip map s′ ∘ flip lookup) ∘ (_<$>_ (flip union g))) p
    where s′ = enumerate s
          g  = fromFunc (denumerate s)
          u  = map (flip lookup (union h g)) s′
-}

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) ∎)