Merge branch feature-decsetoid

1 files changed, 58 insertions, 29 deletions

diff --git a/Bidir.agda b/Bidir.agda
index 9cc0ca6..03daf9d 100644
--- a/Bidir.agda
+++ b/Bidir.agda
@@ -1,33 +1,51 @@
-open import Relation.Binary.Core using (Decidable ; _≡_)
+open import Level using () renaming (zero to ℓ₀)
+open import Relation.Binary using (DecSetoid)
 
-module Bidir (Carrier : Set) (deq : Decidable {A = Carrier} _≡_) where
+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′)
+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)
-open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎)
+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)
+open import Generic using (just-injective ; map-just-injective ; vecIsSetoid)
 open import FinMap
 import CheckInsert
-open CheckInsert Carrier deq
+open CheckInsert A
 import BFF
-open BFF.VecBFF Carrier deq using (assoc ; enumerate ; denumerate ; 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
@@ -37,13 +55,20 @@ lemma-1 f (i ∷ is′) = begin
   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 → lookupM i h ≡ just x
+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 pl = pl
-lemma-lookupM-assoc i is x xs ._ refl | just h' | ._ | new _ = lemma-lookupM-insert 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
@@ -54,7 +79,7 @@ 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 pl = All._∷_ (x' , pl) (lemma-assoc-domain is' xs' h ph')
+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
@@ -68,23 +93,19 @@ lemma-map-lookupM-assoc i x h h' ph (j ∷ js) (Data.List.All._∷_ (x' , pl) pj
   (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 → map (flip lookupM h) is ≡ map just v
-lemma-2 []       []       h p = refl
+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
-    ≡⟨ cong (flip _∷_ (map (flip lookupM h) is)) (lemma-lookupM-assoc i is x xs h (begin
-      assoc (i ∷ is) (x ∷ xs)
-        ≡⟨ cong (flip _>>=_ (checkInsert i x)) ir ⟩
-      checkInsert i x h'
-        ≡⟨ p ⟩
-      just h ∎) ) ⟩
+    ≈⟨ 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
-    ≡⟨ cong (_∷_ (just x)) (lemma-2 is xs h' ir) ⟩
+    ≈⟨ 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
@@ -100,6 +121,7 @@ lemma-map-denumerate-enumerate (a ∷ as) = cong (_∷_ a) (begin
   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
@@ -122,7 +144,8 @@ theorem-1 get s = begin
   just (map (denumerate s) (enumerate s))
     ≡⟨ cong just (lemma-map-denumerate-enumerate s) ⟩
   just s ∎
-    where h↦h′ = _<$>_ (flip union (fromFunc (denumerate s)))
+    where open ≡-Reasoning
+          h↦h′ = _<$>_ (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
@@ -140,12 +163,17 @@ lemma-union-not-used h h' (i ∷ is') (Data.List.All._∷_ (x , px) p') = cong�
         ≡⟨ 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 → get u ≡ v
+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
+theorem-2 get v s u p | h , ph = begin⟨ vecIsSetoid A.setoid _ ⟩
   get u
-    ≡⟨ just-injective (begin
+    ≡⟨ just-injective (begin⟨ EqSetoid _ ⟩
       get <$> (just u)
         ≡⟨ cong (_<$>_ get) (sym p) ⟩
       get <$> (bff get s v)
@@ -154,15 +182,16 @@ theorem-2 get v s u p | h , ph = begin
   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
+     ≈⟨ 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 ⟩
+         ≈⟨ lemma-2 (get s′) v h ph ⟩
        map just v
          ∎) ⟩
   v ∎
-    where s′   = enumerate s
+    where open SetoidReasoning
+          s′   = enumerate s
           g    = fromFunc (denumerate s)
           h↦h′ = flip union g
           h′↦r = flip map s′ ∘ flip lookupVec