port theorem-{1,2} to PartialShapeVec

2 files changed, 127 insertions, 50 deletions

diff --git a/Bidir.agda b/Bidir.agda
index ae99c5a..1a661de 100644
--- a/Bidir.agda
+++ b/Bidir.agda
@@ -13,7 +13,7 @@ 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) renaming (lookup to lookupVec)
+open import Data.Vec using (Vec ; [] ; _∷_ ; toList ; map ; allFin) renaming (lookup to lookupVec)
 open import Data.Vec.Equality using () renaming (module Equality to VecEq)
 open import Data.Vec.Properties using (lookup∘tabulate ; map-cong ; map-∘ ; map-lookup-allFin)
 open import Data.Product using (∃ ; _×_ ; _,_ ; proj₁ ; proj₂)
@@ -24,14 +24,16 @@ open import Relation.Binary.PropositionalEquality using (cong ; sym ; inspect ;
 open import Relation.Binary using (Setoid ; module Setoid ; module DecSetoid)
 import Relation.Binary.EqReasoning as EqR
 
+open import Structures using (Functor ; IsFunctor ; Shaped ; module Shaped)
+open import Instances using (MaybeFunctor)
 import GetTypes
-open GetTypes.PartialVecVec using (Get ; module Get)
+open GetTypes.PartialShapeVec using (Get ; module Get)
 open import Generic using (mapMV ; mapMV-cong ; mapMV-purity ; sequenceV ; VecISetoid ; just-injective)
 open import FinMap
 import CheckInsert
 open CheckInsert A
 import BFF
-open BFF.PartialVecBFF A using (assoc ; enumerate ; enumeratel ; denumerate ; bff)
+open BFF.PartialShapeBFF A using (assoc ; enumerate ; denumerate ; bff)
 open Setoid using () renaming (_≈_ to _∋_≈_)
 open module A = DecSetoid A using (Carrier) renaming (_≟_ to deq)
 
@@ -107,11 +109,24 @@ lemma-2 (i ∷ is) (x ∷ xs) h p | just h' | [ ir ] = begin
   just x ∷ map just xs ∎
   where open EqR (VecISetoid (MaybeSetoid A.setoid) at _)
 
-theorem-1 : (G : Get) → {i : Get.|I| G} → (s : Vec Carrier (Get.|gl₁| G i)) → bff G i s (Get.get G s) ≡ just (map just s)
+lemma-fmap-denumerate-enumerate : {S : Set} {C : Set → S → Set} → (ShapeT : Shaped S C) → {α : Set} {s : S} → (c : C α s) → Shaped.fmap ShapeT (denumerate ShapeT c) (enumerate ShapeT s) ≡ c
+lemma-fmap-denumerate-enumerate {S} {C} ShapeT {s = s} c = begin
+  fmap (denumerate ShapeT c) (fill s (allFin (arity s)))
+    ≡⟨ fill-fmap (denumerate ShapeT c) s (allFin (arity s)) ⟩
+  fill s (map (flip lookupVec (content c)) (allFin (arity s)))
+    ≡⟨ cong (fill s) (map-lookup-allFin (content c)) ⟩
+  fill s (content c)
+    ≡⟨ content-fill c ⟩
+  c ∎
+  where open ≡-Reasoning
+        open Shaped ShapeT
+
+
+theorem-1 : (G : Get) → {i : Get.|I| G} → (s : Get.Container G Carrier (Get.|gl₁| G i)) → bff G i s (Get.get G s) ≡ just (Get.fmap G just s)
 theorem-1 G {i} s = begin
   bff G i s (get s)
-    ≡⟨ cong (bff G i s ∘ get) (sym (map-lookup-allFin s)) ⟩
-  bff G i s (get (map f t))
+    ≡⟨ cong (bff G i s ∘ get) (sym (lemma-fmap-denumerate-enumerate ShapeT s)) ⟩
+  bff G i s (get (fmap f t))
     ≡⟨ cong (bff G i s) (free-theorem f t) ⟩
   bff G i s (map f (get t))
     ≡⟨ refl ⟩
@@ -119,26 +134,26 @@ theorem-1 G {i} s = begin
     ≡⟨ cong (_<$>_ h′↦r ∘ _<$>_ h↦h′) (lemma-1 f (get t)) ⟩
   (Maybe.just ∘ h′↦r ∘ h↦h′) (restrict f (toList (get t)))
     ≡⟨ cong just (begin
-      h′↦r (union (restrict f (toList (get t))) (reshape g′ (|gl₁| i)))
+      h′↦r (union (restrict f (toList (get t))) (reshape g′ (arity (|gl₁| i))))
         ≡⟨ cong (h′↦r ∘ union (restrict f (toList (get t)))) (lemma-reshape-id g′) ⟩
       h′↦r (union (restrict f (toList (get t))) g′)
         ≡⟨ cong h′↦r (lemma-disjoint-union f (get t)) ⟩
       h′↦r (fromFunc f)
         ≡⟨ refl ⟩
-      map (flip lookupM (fromFunc f)) t
-        ≡⟨ map-cong (lemma-lookupM-fromFunc f) t ⟩
-      map (Maybe.just ∘ f) t
-        ≡⟨ map-∘ just f t ⟩
-      map just (map f t)
-        ≡⟨ cong (map just) (map-lookup-allFin s) ⟩
-      map just s ∎) ⟩ _ ∎
+      fmap (flip lookupM (fromFunc f)) t
+        ≡⟨ IsFunctor.cong (isFunctor (|gl₁| i)) (lemma-lookupM-fromFunc f) t ⟩
+      fmap (Maybe.just ∘ f) t
+        ≡⟨ IsFunctor.composition (isFunctor (|gl₁| i)) just f t ⟩
+      fmap just (fmap f t)
+        ≡⟨ cong (fmap just) (lemma-fmap-denumerate-enumerate ShapeT s) ⟩
+      fmap just s ∎) ⟩ _ ∎
     where open ≡-Reasoning
           open Get G
-          t    = enumeratel (|gl₁| i)
-          f    = denumerate s
+          t    = enumerate ShapeT (|gl₁| i)
+          f    = denumerate ShapeT s
           g′   = delete-many (get t) (fromFunc f)
-          h↦h′ = flip union (reshape g′ (|gl₁| i))
-          h′↦r = flip map t ∘ flip lookupM
+          h↦h′ = flip union (reshape g′ (arity (|gl₁| i)))
+          h′↦r = (λ f′ → fmap f′ t) ∘ flip lookupM
 
 
 lemma-<$>-just : {A B : Set} {f : A → B} {b : B} (ma : Maybe A) → f <$> ma ≡ just b → ∃ λ a → ma ≡ just a
@@ -162,9 +177,21 @@ lemma->>=-just : {A B : Set} (ma : Maybe A) {f : A → Maybe B} {b : B} → (ma
 lemma->>=-just (just a) p = a , refl
 lemma->>=-just nothing  ()
 
-lemma-just-sequence : {A : Set} {n : ℕ} → (v : Vec A n) → sequenceV (map just v) ≡ just v
-lemma-just-sequence []       = refl
-lemma-just-sequence (x ∷ xs) = cong (_<$>_ (_∷_ x)) (lemma-just-sequence xs)
+lemma-just-sequenceV : {A : Set} {n : ℕ} → (v : Vec A n) → sequenceV (map just v) ≡ just v
+lemma-just-sequenceV []       = refl
+lemma-just-sequenceV (x ∷ xs) = cong (_<$>_ (_∷_ x)) (lemma-just-sequenceV xs)
+
+lemma-just-sequence : (G : Get) → {A : Set} {i : Get.|I| G} → (c : Get.Container G A (Get.|gl₁| G i)) → Get.sequence G (Get.fmap G just c) ≡ just c
+lemma-just-sequence G {i = i} c = begin
+  fill (|gl₁| i) <$> sequenceV (content (fmap just c))
+    ≡⟨ cong (_<$>_ (fill (|gl₁| i)) ∘ sequenceV) (fmap-content just c) ⟩
+  fill (|gl₁| i) <$> sequenceV (map just (content c))
+    ≡⟨ cong (_<$>_ (fill (|gl₁| i))) (lemma-just-sequenceV (content c)) ⟩
+  fill (|gl₁| i) <$> just (content c)
+    ≡⟨ cong just (content-fill c) ⟩
+  just c ∎
+  where open ≡-Reasoning
+        open Get G
 
 lemma-sequenceV-successful : {A : Set} {n : ℕ} → (v : Vec (Maybe A) n) → {r : Vec A n} → sequenceV v ≡ just r → v ≡ map just r
 lemma-sequenceV-successful []             {r = []}       p = refl
@@ -173,18 +200,44 @@ lemma-sequenceV-successful (just x ∷ xs)                 () | nothing | _
 lemma-sequenceV-successful (just x ∷ xs)  {r = .x ∷ .ys} refl  | just ys | [ p′ ] = cong (_∷_ (just x)) (lemma-sequenceV-successful xs p′)
 lemma-sequenceV-successful (nothing ∷ xs)                ()
 
-lemma-get-sequenceV : {A : Set} → (G : Get) → {i : Get.|I| G} {v : Vec (Maybe A) (Get.|gl₁| G i)} {r : Vec A (Get.|gl₁| G i)} → sequenceV v ≡ just r → Get.get G <$> sequenceV v ≡ sequenceV (Get.get G v)
-lemma-get-sequenceV G {v = v} {r = r} p = begin
-  get <$> sequenceV v
-    ≡⟨ cong (_<$>_ get ∘ sequenceV) (lemma-sequenceV-successful v p) ⟩
-  get <$> sequenceV (map just r)
-    ≡⟨ cong (_<$>_ get) (lemma-just-sequence r) ⟩
+lemma-sequence-successful : (G : Get) → {A : Set} {i : Get.|I| G} → (c : Get.Container G (Maybe A) (Get.|gl₁| G i)) → {r : Get.Container G A (Get.|gl₁| G i)} → Get.sequence G c ≡ just r → c ≡ Get.fmap G just r
+lemma-sequence-successful G {i = i} c {r} p = just-injective (sym (begin
+  fill (|gl₁| i) <$> (map just <$> (content <$> just r))
+    ≡⟨ cong (_<$>_ (fill (|gl₁| i)) ∘ _<$>_ (map just)) (begin
+      content <$> just r
+        ≡⟨ cong (_<$>_ content) (sym p) ⟩
+      content <$> (fill (|gl₁| i) <$> sequenceV (content c))
+        ≡⟨ sym (Functor.composition MaybeFunctor content (fill (|gl₁| i)) (sequenceV (content c))) ⟩
+      content ∘ fill (|gl₁| i) <$> sequenceV (content c)
+        ≡⟨ Functor.cong MaybeFunctor (fill-content (|gl₁| i)) (sequenceV (content c)) ⟩
+      id <$> sequenceV (content c)
+        ≡⟨ Functor.identity MaybeFunctor (sequenceV (content c)) ⟩
+      sequenceV (content c)
+        ≡⟨ cong sequenceV (lemma-sequenceV-successful (content c) (proj₂ wp)) ⟩
+      sequenceV (map just (proj₁ wp))
+        ≡⟨ lemma-just-sequenceV (proj₁ wp) ⟩
+      just (proj₁ (lemma-<$>-just (sequenceV (content c)) p)) ∎) ⟩
+  fill (|gl₁| i) <$> (map just <$> just (proj₁ (lemma-<$>-just (sequenceV (content c)) p)))
+    ≡⟨ cong (_<$>_ (fill (|gl₁| i)) ∘ just) (sym (lemma-sequenceV-successful (content c) (proj₂ wp))) ⟩
+  fill (|gl₁| i) <$> just (content c)
+    ≡⟨ cong just (content-fill c) ⟩
+  just c ∎))
+  where open ≡-Reasoning
+        open Get G
+        wp = lemma-<$>-just (sequenceV (content c)) p
+
+lemma-get-sequence : {A : Set} → (G : Get) → {i : Get.|I| G} {v : Get.Container G (Maybe A) (Get.|gl₁| G i)} {r : Get.Container G A (Get.|gl₁| G i)} → Get.sequence G v ≡ just r → Get.get G <$> Get.sequence G v ≡ sequenceV (Get.get G v)
+lemma-get-sequence G {v = v} {r = r} p = begin
+  get <$> sequence v
+    ≡⟨ cong (_<$>_ get ∘ sequence) (lemma-sequence-successful G v p) ⟩
+  get <$> sequence (fmap just r)
+    ≡⟨ cong (_<$>_ get) (lemma-just-sequence G r) ⟩
   get <$> just r
-    ≡⟨ sym (lemma-just-sequence (get r)) ⟩
+    ≡⟨ sym (lemma-just-sequenceV (get r)) ⟩
   sequenceV (map just (get r))
     ≡⟨ cong sequenceV (sym (free-theorem just r)) ⟩
-  sequenceV (get (map just r))
-    ≡⟨ cong (sequenceV ∘ get) (sym (lemma-sequenceV-successful v p)) ⟩
+  sequenceV (get (fmap just r))
+    ≡⟨ cong (sequenceV ∘ get) (sym (lemma-sequence-successful G v p)) ⟩
   sequenceV (get v) ∎
   where open ≡-Reasoning
         open Get G
@@ -198,16 +251,16 @@ sequence-cong {S} {m₁ = just x ∷ xs} {m₂ = just y ∷ ys} (VecEq._∷-cong
 sequence-cong {S} {m₁ = just x ∷ xs} {m₂ = just y ∷ ys} (VecEq._∷-cong_ (just x≈y) xs≈ys) | nothing | nothing | nothing = Setoid.refl (MaybeSetoid (VecISetoid S at _))
 sequence-cong {S}                                       (VecEq._∷-cong_ nothing xs≈ys) = Setoid.refl (MaybeSetoid (VecISetoid S at _))
 
-theorem-2 : (G : Get) → {i : Get.|I| G} → (j : Get.|I| G) → (s : Vec Carrier (Get.|gl₁| G i)) → (v : Vec Carrier (Get.|gl₂| G j)) → (u : Vec (Maybe Carrier) (Get.|gl₁| G j)) → bff G j s v ≡ just u → VecISetoid (MaybeSetoid A.setoid) at _ ∋ Get.get G u ≈ map just v
-theorem-2 G j s v u p with (lemma-<$>-just ((flip union (reshape (delete-many (Get.get G (enumerate s)) (fromFunc (denumerate s))) (Get.|gl₁| G j))) <$> (assoc (Get.get G (enumeratel (Get.|gl₁| G j))) v)) p)
-theorem-2 G j s v u p | h′ , ph′ with (lemma-<$>-just (assoc (Get.get G (enumeratel (Get.|gl₁| G j))) v) ph′)
-theorem-2 G j s v u p | h′ , ph′ | h , ph = begin⟨ VecISetoid (MaybeSetoid A.setoid) at _ ⟩
+theorem-2 : (G : Get) → {i : Get.|I| G} → (j : Get.|I| G) → (s : Get.Container G Carrier (Get.|gl₁| G i)) → (v : Vec Carrier (Get.|gl₂| G j)) → (u : Get.Container G (Maybe Carrier) (Get.|gl₁| G j)) → bff G j s v ≡ just u → VecISetoid (MaybeSetoid A.setoid) at _ ∋ Get.get G u ≈ map just v
+theorem-2 G {i} j s v u p with (lemma-<$>-just ((flip union (reshape (delete-many (Get.get G (enumerate (Get.ShapeT G) (Get.|gl₁| G i))) (fromFunc (denumerate (Get.ShapeT G) s))) (Get.arity G (Get.|gl₁| G j)))) <$> (assoc (Get.get G (enumerate (Get.ShapeT G) (Get.|gl₁| G j))) v)) p)
+theorem-2 G {i} j s v u p | h′ , ph′ with (lemma-<$>-just (assoc (Get.get G (enumerate (Get.ShapeT G) (Get.|gl₁| G j))) v) ph′)
+theorem-2 G {i} j s v u p | h′ , ph′ | h , ph = begin⟨ VecISetoid (MaybeSetoid A.setoid) at _ ⟩
   get u
     ≡⟨ just-injective (trans (cong (_<$>_ get) (sym p))
                              (cong (_<$>_ get ∘ _<$>_ h′↦r ∘ _<$>_ h↦h′) ph)) ⟩
   get (h′↦r (h↦h′ h))
     ≡⟨ refl ⟩
-  get (map (flip lookupM (h↦h′ h)) t)
+  get (fmap (flip lookupM (h↦h′ h)) t)
     ≡⟨ free-theorem (flip lookupM (h↦h′ h)) t ⟩
   map (flip lookupM (h↦h′ h)) (get t)
     ≡⟨ lemma-union-not-used h g′ (get t) (lemma-assoc-domain (get t) v h ph) ⟩
@@ -216,23 +269,23 @@ theorem-2 G j s v u p | h′ , ph′ | h , ph = begin⟨ VecISetoid (MaybeSetoid
   map just v ∎
     where open SetoidReasoning
           open Get G
-          s′   = enumerate s
-          g    = fromFunc (denumerate s)
+          s′   = enumerate ShapeT (|gl₁| i)
+          g    = fromFunc (denumerate ShapeT s)
           g′   = delete-many (get s′) g
-          t    = enumeratel (Get.|gl₁| G j)
-          h↦h′ = flip union (reshape g′ (Get.|gl₁| G j))
-          h′↦r = flip map t ∘ flip lookupM
+          t    = enumerate ShapeT (|gl₁| j)
+          h↦h′ = flip union (reshape g′ (arity (|gl₁| j)))
+          h′↦r = (λ f → fmap f t) ∘ flip lookupM
 
-theorem-2′ : (G : Get) → {i : Get.|I| G} → (j : Get.|I| G) → (s : Vec Carrier (Get.|gl₁| G i)) → (v : Vec Carrier (Get.|gl₂| G j)) → (u : Vec Carrier (Get.|gl₁| G j)) → bff G j s v ≡ just (map just u) → VecISetoid A.setoid at _ ∋ Get.get G u ≈ v
+theorem-2′ : (G : Get) → {i : Get.|I| G} → (j : Get.|I| G) → (s : Get.Container G Carrier (Get.|gl₁| G i)) → (v : Vec Carrier (Get.|gl₂| G j)) → (u : Get.Container G Carrier (Get.|gl₁| G j)) → bff G j s v ≡ just (Get.fmap G just u) → VecISetoid A.setoid at _ ∋ Get.get G u ≈ v
 theorem-2′ G j s v u p = drop-just (begin
   get <$> just u
-    ≡⟨ cong (_<$>_ get) (sym (lemma-just-sequence u)) ⟩
-  get <$> sequenceV (map just u)
-    ≡⟨ lemma-get-sequenceV G (lemma-just-sequence u) ⟩
-  sequenceV (get (map just u))
-    ≈⟨ sequence-cong (theorem-2 G j s v (map just u) p) ⟩
+    ≡⟨ cong (_<$>_ get) (sym (lemma-just-sequence G u)) ⟩
+  get <$> sequence (fmap just u)
+    ≡⟨ lemma-get-sequence G (lemma-just-sequence G u) ⟩
+  sequenceV (get (fmap just u))
+    ≈⟨ sequence-cong (theorem-2 G j s v (fmap just u) p) ⟩
   sequenceV (map just v)
-    ≡⟨ lemma-just-sequence v ⟩
+    ≡⟨ lemma-just-sequenceV v ⟩
   just v ∎)
   where open EqR (MaybeSetoid (VecISetoid A.setoid at _))
         open Get G
diff --git a/Instances.agda b/Instances.agda
index faff6f8..b41b0a2 100644
--- a/Instances.agda
+++ b/Instances.agda
@@ -1,11 +1,35 @@
 module Instances where
 
+open import Category.Functor using (RawFunctor)
+open import Data.Maybe as M using (Maybe)
 open import Data.Nat using (ℕ)
 open import Data.Vec using (Vec)
-open import Function using (id)
-open import Relation.Binary.PropositionalEquality using (refl)
+open import Function using (_∘_ ; id)
+open import Relation.Binary.PropositionalEquality as P using (_≗_ ; refl)
 
-open import Structures using (Shaped)
+open import Structures using (Functor ; Shaped)
+
+MaybeFunctor : Functor Maybe
+MaybeFunctor = record
+  { rawfunctor = M.functor
+  ; isFunctor = record
+    { cong = cong
+    ; identity = identity
+    ; composition = composition
+    } }
+  where _<$>_ = RawFunctor._<$>_ M.functor
+
+        cong : {α β : Set} {g h : α → β} → g ≗ h → _<$>_ g ≗ _<$>_ h
+        cong g≗h (M.just x) = P.cong M.just (g≗h x)
+        cong g≗h M.nothing  = refl
+
+        identity : {α : Set} → _<$>_ {α} id ≗ id
+        identity (M.just x) = refl
+        identity M.nothing  = refl
+
+        composition : {α β γ : Set} → (g : β → γ) → (h : α → β) → _<$>_ (g ∘ h) ≗ _<$>_ g ∘ _<$>_ h
+        composition g h (M.just x) = refl
+        composition g h M.nothing = refl
 
 VecShaped : Shaped ℕ Vec
 VecShaped = record