Merge branch feature-shaped into master

Generalizing both Vecs in the type of get to instances of Shaped. Thus allowing
trees and other data structures to be used.

8 files changed, 452 insertions, 119 deletions

diff --git a/BFF.agda b/BFF.agda
index 2ce24db..26d1d5e 100644
--- a/BFF.agda
+++ b/BFF.agda
@@ -15,11 +15,13 @@ open import Relation.Binary using (Setoid ; DecSetoid ; module DecSetoid)
 
 open import FinMap
 open import Generic using (sequenceV ; ≡-to-Π)
+open import Structures using (Shaped ; module Shaped)
+open import Instances using (VecShaped)
 import CheckInsert
-open import GetTypes using (VecVec-to-PartialVecVec)
+open import GetTypes using (VecVec-to-PartialVecVec ; PartialVecVec-to-PartialShapeShape)
 
-module PartialVecBFF (A : DecSetoid ℓ₀ ℓ₀) where
-  open GetTypes.PartialVecVec public using (Get)
+module PartialShapeBFF (A : DecSetoid ℓ₀ ℓ₀) where
+  open GetTypes.PartialShapeShape public using (Get ; module Get)
   open module A = DecSetoid A using (Carrier) renaming (_≟_ to deq)
   open CheckInsert A
 
@@ -27,27 +29,48 @@ module PartialVecBFF (A : DecSetoid ℓ₀ ℓ₀) where
   assoc []V       []V       = just empty
   assoc (i ∷V is) (b ∷V bs) = (assoc is bs) >>= (checkInsert i b)
 
+  enumerate : {S : Set} {C : Set → S → Set} → (ShapeT : Shaped S C) → (s : S) → C (Fin (Shaped.arity ShapeT s)) s
+  enumerate ShapeT s = fill s (allFin (arity s))
+    where open Shaped ShapeT
+
+  denumerate : {S : Set} {C : Set → S → Set} → (ShapeT : Shaped S C) → {α : Set} {s : S} → (c : C α s) → Fin (Shaped.arity ShapeT s) → α
+  denumerate ShapeT c = flip lookupV (Shaped.content ShapeT c)
+
+  bff : (G : Get) → {i : Get.|I| G} → (j : Get.|I| G) → Get.SourceContainer G Carrier (Get.|gl₁| G i) → Get.ViewContainer G Carrier (Get.|gl₂| G j) → Maybe (Get.SourceContainer G (Maybe Carrier) (Get.|gl₁| G j))
+  bff G {i} j s v = let s′ = enumerate SourceShapeT (|gl₁| i)
+                        t′ = get s′
+                        g  = fromFunc (denumerate SourceShapeT s)
+                        g′ = delete-many (Shaped.content ViewShapeT t′) g
+                        t  = enumerate SourceShapeT (|gl₁| j)
+                        h  = assoc (Shaped.content ViewShapeT (get t)) (Shaped.content ViewShapeT v)
+                        h′ = (flip union (reshape g′ (Shaped.arity SourceShapeT (|gl₁| j)))) <$> h
+                    in ((λ f → fmapS f t) ∘ flip lookupM) <$> h′
+    where open Get G
+
+  sbff : (G : Get) → {i : Get.|I| G} → (j : Get.|I| G) → Get.SourceContainer G Carrier (Get.|gl₁| G i) → Get.ViewContainer G Carrier (Get.|gl₂| G j) → Maybe (Get.SourceContainer G Carrier (Get.|gl₁| G j))
+  sbff G j s v = bff G j s v >>= Shaped.sequence (Get.SourceShapeT G)
+
+module PartialVecBFF (A : DecSetoid ℓ₀ ℓ₀) where
+  open GetTypes.PartialVecVec public using (Get)
+  open module A = DecSetoid A using (Carrier) renaming (_≟_ to deq)
+  open CheckInsert A
+
+  open PartialShapeBFF A public using (assoc)
+
   enumerate : {n : ℕ} → Vec Carrier n → Vec (Fin n) n
-  enumerate {n} _ = allFin n
+  enumerate {n} _ = PartialShapeBFF.enumerate A VecShaped n
 
   enumeratel : (n : ℕ) → Vec (Fin n) n
-  enumeratel = allFin
+  enumeratel = PartialShapeBFF.enumerate A VecShaped
 
   denumerate : {n : ℕ} → Vec Carrier n → Fin n → Carrier
-  denumerate = flip lookupV
+  denumerate = PartialShapeBFF.denumerate A VecShaped
 
   bff : (G : Get) → {i : Get.|I| G} → (j : Get.|I| G) → Vec Carrier (Get.|gl₁| G i) → Vec Carrier (Get.|gl₂| G j) → Maybe (Vec (Maybe Carrier) (Get.|gl₁| G j))
-  bff G i s v = let s′ = enumerate s
-                    t′ = Get.get G s′
-                    g  = fromFunc (denumerate s)
-                    g′ = delete-many t′ g
-                    t  = enumeratel (Get.|gl₁| G i)
-                    h  = assoc (Get.get G t) v
-                    h′ = (flip union (reshape g′ (Get.|gl₁| G i))) <$> h
-                in (flip mapV t ∘ flip lookupM) <$> h′
+  bff G j s v = PartialShapeBFF.bff A (PartialVecVec-to-PartialShapeShape G) j s v
 
   sbff : (G : Get) → {i : Get.|I| G} → (j : Get.|I| G) → Vec Carrier (Get.|gl₁| G i) → Vec Carrier (Get.|gl₂| G j) → Maybe (Vec Carrier (Get.|gl₁| G j))
-  sbff G j s v = bff G j s v >>= sequenceV
+  sbff  G j s v = PartialShapeBFF.sbff A (PartialVecVec-to-PartialShapeShape G) j s v
 
 module VecBFF (A : DecSetoid ℓ₀ ℓ₀) where
   open GetTypes.VecVec public using (Get)
diff --git a/Bidir.agda b/Bidir.agda
index ae99c5a..1513a17 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 ; ShapedISetoid)
 import GetTypes
-open GetTypes.PartialVecVec using (Get ; module Get)
+open GetTypes.PartialShapeShape 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)
 
@@ -107,38 +109,53 @@ 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.SourceContainer G Carrier (Get.|gl₁| G i)) → bff G i s (Get.get G s) ≡ just (Get.fmapS 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 SourceShapeT s)) ⟩
+  bff G i s (get (fmapS f t))
     ≡⟨ cong (bff G i s) (free-theorem f t) ⟩
-  bff G i s (map f (get t))
+  bff G i s (fmapV f (get t))
     ≡⟨ refl ⟩
-  h′↦r <$> (h↦h′ <$> (assoc (get t) (map f (get t))))
-    ≡⟨ cong (_<$>_ h′↦r ∘ _<$>_ h↦h′) (lemma-1 f (get t)) ⟩
-  (Maybe.just ∘ h′↦r ∘ h↦h′) (restrict f (toList (get t)))
+  h′↦r <$> (h↦h′ <$> (assoc (Shaped.content ViewShapeT (get t)) (Shaped.content ViewShapeT (fmapV f (get t)))))
+    ≡⟨ cong (_<$>_ h′↦r ∘ _<$>_ h↦h′ ∘ assoc (Shaped.content ViewShapeT (get t))) (Shaped.fmap-content ViewShapeT f (get t)) ⟩
+  h′↦r <$> (h↦h′ <$> (assoc (Shaped.content ViewShapeT (get t)) (map f (Shaped.content ViewShapeT (get t)))))
+    ≡⟨ cong (_<$>_ h′↦r ∘ _<$>_ h↦h′) (lemma-1 f (Shaped.content ViewShapeT (get t))) ⟩
+  (Maybe.just ∘ h′↦r ∘ h↦h′) (restrict f (toList (Shaped.content ViewShapeT (get t))))
     ≡⟨ cong just (begin
-      h′↦r (union (restrict f (toList (get t))) (reshape g′ (|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 (union (restrict f (toList (Shaped.content ViewShapeT (get t)))) (reshape g′ (Shaped.arity SourceShapeT (|gl₁| i))))
+        ≡⟨ cong (h′↦r ∘ union (restrict f (toList (Shaped.content ViewShapeT (get t))))) (lemma-reshape-id g′) ⟩
+      h′↦r (union (restrict f (toList (Shaped.content ViewShapeT (get t)))) g′)
+        ≡⟨ cong h′↦r (lemma-disjoint-union f (Shaped.content ViewShapeT (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 ∎) ⟩ _ ∎
+      fmapS (flip lookupM (fromFunc f)) t
+        ≡⟨ IsFunctor.cong (Shaped.isFunctor SourceShapeT (|gl₁| i)) (lemma-lookupM-fromFunc f) t ⟩
+      fmapS (Maybe.just ∘ f) t
+        ≡⟨ IsFunctor.composition (Shaped.isFunctor SourceShapeT (|gl₁| i)) just f t ⟩
+      fmapS just (fmapS f t)
+        ≡⟨ cong (fmapS just) (lemma-fmap-denumerate-enumerate SourceShapeT s) ⟩
+      fmapS just s ∎) ⟩ _ ∎
     where open ≡-Reasoning
           open Get G
-          t    = enumeratel (|gl₁| i)
-          f    = denumerate s
-          g′   = delete-many (get t) (fromFunc f)
-          h↦h′ = flip union (reshape g′ (|gl₁| i))
-          h′↦r = flip map t ∘ flip lookupM
+          t    = enumerate SourceShapeT (|gl₁| i)
+          f    = denumerate SourceShapeT s
+          g′   = delete-many (Shaped.content ViewShapeT (get t)) (fromFunc f)
+          h↦h′ = flip union (reshape g′ (Shaped.arity SourceShapeT (|gl₁| i)))
+          h′↦r = (λ f′ → fmapS 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 +179,21 @@ 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 : {S : Set} {C : Set → S → Set} → (ShapeT : Shaped S C) → {A : Set} {s : S} → (c : C A s) → Shaped.sequence ShapeT (Shaped.fmap ShapeT just c) ≡ just c
+lemma-just-sequence ShapeT {s = s} c = begin
+  fill s <$> sequenceV (content (fmap just c))
+    ≡⟨ cong (_<$>_ (fill s) ∘ sequenceV) (fmap-content just c) ⟩
+  fill s <$> sequenceV (map just (content c))
+    ≡⟨ cong (_<$>_ (fill s)) (lemma-just-sequenceV (content c)) ⟩
+  fill s <$> just (content c)
+    ≡⟨ cong just (content-fill c) ⟩
+  just c ∎
+  where open ≡-Reasoning
+        open Shaped ShapeT
 
 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
@@ -173,66 +202,111 @@ 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 (_<$>_ get) (lemma-just-sequence r) ⟩
+lemma-sequence-successful : {S : Set} {C : Set → S → Set} → (ShapeT : Shaped S C) → {A : Set} {s : S} → (c : C (Maybe A) s) → {r : C A s} → Shaped.sequence ShapeT c ≡ just r → c ≡ Shaped.fmap ShapeT just r
+lemma-sequence-successful ShapeT {s = s} c {r} p = just-injective (sym (begin
+  fill s <$> (map just <$> (content <$> just r))
+    ≡⟨ cong (_<$>_ (fill s) ∘ _<$>_ (map just)) (begin
+      content <$> just r
+        ≡⟨ cong (_<$>_ content) (sym p) ⟩
+      content <$> (fill s <$> sequenceV (content c))
+        ≡⟨ sym (Functor.composition MaybeFunctor content (fill s) (sequenceV (content c))) ⟩
+      content ∘ fill s <$> sequenceV (content c)
+        ≡⟨ Functor.cong MaybeFunctor (fill-content s) (sequenceV (content c)) ⟩
+      id <$> 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-<$>-just (sequenceV (content c)) p)) ∎) ⟩
+  fill s <$> (map just <$> just (proj₁ (lemma-<$>-just (sequenceV (content c)) p)))
+    ≡⟨ cong (_<$>_ (fill s) ∘ just) (sym (lemma-sequenceV-successful (content c) (proj₂ wp))) ⟩
+  fill s <$> just (content c)
+    ≡⟨ cong just (content-fill c) ⟩
+  just c ∎))
+  where open ≡-Reasoning
+        open Shaped ShapeT
+        wp = lemma-<$>-just (sequenceV (content c)) p
+
+lemma-get-sequence : {A : Set} → (G : Get) → {i : Get.|I| G} {v : Get.SourceContainer G (Maybe A) (Get.|gl₁| G i)} {r : Get.SourceContainer G A (Get.|gl₁| G i)} → Shaped.sequence (Get.SourceShapeT G) v ≡ just r → Get.get G <$> Shaped.sequence (Get.SourceShapeT G) v ≡ Shaped.sequence (Get.ViewShapeT G) (Get.get G v)
+lemma-get-sequence G {v = v} {r = r} p = begin
+  get <$> Shaped.sequence SourceShapeT v
+    ≡⟨ cong (_<$>_ get ∘ Shaped.sequence SourceShapeT) (lemma-sequence-successful SourceShapeT v p) ⟩
+  get <$> Shaped.sequence SourceShapeT (fmapS just r)
+    ≡⟨ cong (_<$>_ get) (lemma-just-sequence SourceShapeT r) ⟩
   get <$> just r
-    ≡⟨ sym (lemma-just-sequence (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 v) ∎
+    ≡⟨ sym (lemma-just-sequence ViewShapeT (get r)) ⟩
+  Shaped.sequence ViewShapeT (fmapV just (get r))
+    ≡⟨ cong (Shaped.sequence ViewShapeT) (sym (free-theorem just r)) ⟩
+  Shaped.sequence ViewShapeT (get (fmapS just r))
+    ≡⟨ cong (Shaped.sequence ViewShapeT ∘ get) (sym (lemma-sequence-successful SourceShapeT v p)) ⟩
+  Shaped.sequence ViewShapeT (get v) ∎
   where open ≡-Reasoning
         open Get G
 
-sequence-cong : {S : Setoid ℓ₀ ℓ₀} {n : ℕ} {m₁ m₂ : Setoid.Carrier (VecISetoid (MaybeSetoid S) at n)} → VecISetoid (MaybeSetoid S) at _ ∋ m₁ ≈ m₂ → MaybeSetoid (VecISetoid S at n) ∋ sequenceV m₁ ≈ sequenceV m₂
-sequence-cong {S}                                       VecEq.[]-cong = Setoid.refl (MaybeSetoid (VecISetoid S at _))
-sequence-cong {S} {m₁ = just x ∷ xs} {m₂ = just y ∷ ys} (VecEq._∷-cong_ (just x≈y) xs≈ys) with sequenceV xs | sequenceV ys | sequence-cong xs≈ys
-sequence-cong {S} {m₁ = just x ∷ xs} {m₂ = just y ∷ ys} (VecEq._∷-cong_ (just x≈y) xs≈ys) | just sxs | just sys | just p = MaybeEq.just (VecEq._∷-cong_ x≈y p)
-sequence-cong {S} {m₁ = just x ∷ xs} {m₂ = just y ∷ ys} (VecEq._∷-cong_ (just x≈y) xs≈ys) | nothing | just sys | ()
-sequence-cong {S} {m₁ = just x ∷ xs} {m₂ = just y ∷ ys} (VecEq._∷-cong_ (just x≈y) xs≈ys) | just sxs | nothing | ()
-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-<$>-just (assoc (Get.get G (enumeratel (Get.|gl₁| G j))) v) ph′)
-theorem-2 G j s v u p | h′ , ph′ | h , ph = begin⟨ VecISetoid (MaybeSetoid A.setoid) at _ ⟩
-  get u
-    ≡⟨ just-injective (trans (cong (_<$>_ get) (sym p))
-                             (cong (_<$>_ get ∘ _<$>_ h′↦r ∘ _<$>_ h↦h′) ph)) ⟩
-  get (h′↦r (h↦h′ h))
+sequenceV-cong : {S : Setoid ℓ₀ ℓ₀} {n : ℕ} {m₁ m₂ : Setoid.Carrier (VecISetoid (MaybeSetoid S) at n)} → VecISetoid (MaybeSetoid S) at _ ∋ m₁ ≈ m₂ → MaybeSetoid (VecISetoid S at n) ∋ sequenceV m₁ ≈ sequenceV m₂
+sequenceV-cong {S}                                       VecEq.[]-cong = Setoid.refl (MaybeSetoid (VecISetoid S at _))
+sequenceV-cong {S} {m₁ = just x ∷ xs} {m₂ = just y ∷ ys} (VecEq._∷-cong_ (just x≈y) xs≈ys) with sequenceV xs | sequenceV ys | sequenceV-cong xs≈ys
+sequenceV-cong {S} {m₁ = just x ∷ xs} {m₂ = just y ∷ ys} (VecEq._∷-cong_ (just x≈y) xs≈ys) | just sxs | just sys | just p = MaybeEq.just (VecEq._∷-cong_ x≈y p)
+sequenceV-cong {S} {m₁ = just x ∷ xs} {m₂ = just y ∷ ys} (VecEq._∷-cong_ (just x≈y) xs≈ys) | nothing | just sys | ()
+sequenceV-cong {S} {m₁ = just x ∷ xs} {m₂ = just y ∷ ys} (VecEq._∷-cong_ (just x≈y) xs≈ys) | just sxs | nothing | ()
+sequenceV-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 _))
+sequenceV-cong {S}                                       (VecEq._∷-cong_ nothing xs≈ys) = Setoid.refl (MaybeSetoid (VecISetoid S at _))
+
+sequence-cong : {S : Set} {C : Set → S → Set} → (ShapeT : Shaped S C) → (α : Setoid ℓ₀ ℓ₀) → {s : S} {x y : C (Maybe (Setoid.Carrier α)) s} → ShapedISetoid (EqSetoid S) ShapeT (MaybeSetoid α) at _ ∋ x ≈ y → MaybeSetoid (ShapedISetoid (EqSetoid S) ShapeT α at _) ∋ Shaped.sequence ShapeT x ≈ Shaped.sequence ShapeT y
+sequence-cong ShapeT α {x = x} {y = y} (shape≈ , content≈) with sequenceV (Shaped.content ShapeT x) | sequenceV (Shaped.content ShapeT y) | sequenceV-cong content≈
+sequence-cong ShapeT α {s} (shape≈ , content≈) | .(just x) | .(just y) | just {x} {y} x≈y = just (refl , (begin
+  content (fill s x)
+    ≡⟨ fill-content s x ⟩
+  x
+    ≈⟨ x≈y ⟩
+  y
+    ≡⟨ sym (fill-content s y) ⟩
+  content (fill s y) ∎))
+  where open EqR (VecISetoid α at _)
+        open Shaped ShapeT
+sequence-cong ShapeT α (shape≈ , content≈) | .nothing  | .nothing  | nothing = nothing
+
+theorem-2 : (G : Get) → {i : Get.|I| G} → (j : Get.|I| G) → (s : Get.SourceContainer G Carrier (Get.|gl₁| G i)) → (v : Get.ViewContainer G Carrier (Get.|gl₂| G j)) → (u : Get.SourceContainer G (Maybe Carrier) (Get.|gl₁| G j)) → bff G j s v ≡ just u → ShapedISetoid (EqSetoid _) (Get.ViewShapeT G) (MaybeSetoid A.setoid) at _ ∋ Get.get G u ≈ Get.fmapV G just v
+theorem-2 G {i} j s v u p with (lemma-<$>-just ((flip union (reshape (delete-many (Shaped.content (Get.ViewShapeT G) (Get.get G (enumerate (Get.SourceShapeT G) (Get.|gl₁| G i)))) (fromFunc (denumerate (Get.SourceShapeT G) s))) (Shaped.arity (Get.SourceShapeT G) (Get.|gl₁| G j)))) <$> (assoc (Shaped.content (Get.ViewShapeT G) (Get.get G (enumerate (Get.SourceShapeT G) (Get.|gl₁| G j)))) (Shaped.content (Get.ViewShapeT G) v))) p)
+theorem-2 G {i} j s v u p | h′ , ph′ with (lemma-<$>-just (assoc (Shaped.content (Get.ViewShapeT G) (Get.get G (enumerate (Get.SourceShapeT G) (Get.|gl₁| G j)))) (Shaped.content (Get.ViewShapeT G) v)) ph′)
+theorem-2 G {i} j s v u p | h′ , ph′ | h , ph = refl , (begin⟨ VecISetoid (MaybeSetoid A.setoid) at _ ⟩
+  content (get u)
+    ≡⟨ cong content (just-injective (trans (cong (_<$>_ get) (sym p))
+                                           (cong (_<$>_ get ∘ _<$>_ h′↦r ∘ _<$>_ h↦h′) ph))) ⟩
+  content (get (h′↦r (h↦h′ h)))
     ≡⟨ refl ⟩
-  get (map (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) ⟩
-  map (flip lookupM h) (get t)
-    ≈⟨ lemma-2 (get t) v h ph ⟩
-  map just v ∎
+  content (get (fmapS (flip lookupM (h↦h′ h)) t))
+    ≡⟨ cong content (free-theorem (flip lookupM (h↦h′ h)) t) ⟩
+  content (fmapV (flip lookupM (h↦h′ h)) (get t))
+    ≡⟨ Shaped.fmap-content ViewShapeT (flip lookupM (h↦h′ h)) (get t) ⟩
+  map (flip lookupM (h↦h′ h)) (content (get t))
+    ≡⟨ lemma-union-not-used h g′ (content (get t)) (lemma-assoc-domain (content (get t)) (content v) h ph) ⟩
+  map (flip lookupM h) (content (get t))
+    ≈⟨ lemma-2 (content (get t)) (content v) h ph ⟩
+  map just (content v)
+    ≡⟨ sym (Shaped.fmap-content ViewShapeT just v) ⟩
+  content (fmapV just v) ∎)
     where open SetoidReasoning
           open Get G
-          s′   = enumerate s
-          g    = fromFunc (denumerate 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
-
-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
+          open Shaped ViewShapeT using (content)
+          s′   = enumerate SourceShapeT (|gl₁| i)
+          g    = fromFunc (denumerate SourceShapeT s)
+          g′   = delete-many (Shaped.content ViewShapeT (get s′)) g
+          t    = enumerate SourceShapeT (|gl₁| j)
+          h↦h′ = flip union (reshape g′ (Shaped.arity SourceShapeT (|gl₁| j)))
+          h′↦r = (λ f → fmapS f t) ∘ flip lookupM
+
+theorem-2′ : (G : Get) → {i : Get.|I| G} → (j : Get.|I| G) → (s : Get.SourceContainer G Carrier (Get.|gl₁| G i)) → (v : Get.ViewContainer G Carrier (Get.|gl₂| G j)) → (u : Get.SourceContainer G Carrier (Get.|gl₁| G j)) → bff G j s v ≡ just (Get.fmapS G just u) → ShapedISetoid (EqSetoid _) (Get.ViewShapeT G) 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 <$> sequenceV (map just u)
-    ≡⟨ lemma-get-sequenceV G (lemma-just-sequence u) ⟩
-  sequenceV (get (map just u))
-    ≈⟨ sequence-cong (theorem-2 G j s v (map just u) p) ⟩
-  sequenceV (map just v)
-    ≡⟨ lemma-just-sequence v ⟩
+    ≡⟨ cong (_<$>_ get) (sym (lemma-just-sequence SourceShapeT u)) ⟩
+  get <$> Shaped.sequence SourceShapeT (fmapS just u)
+    ≡⟨ lemma-get-sequence G (lemma-just-sequence SourceShapeT u) ⟩
+  Shaped.sequence ViewShapeT (get (fmapS just u))
+    ≈⟨ sequence-cong ViewShapeT A.setoid (theorem-2 G j s v (fmapS just u) p) ⟩
+  Shaped.sequence ViewShapeT (fmapV just v)
+    ≡⟨ lemma-just-sequence ViewShapeT v ⟩
   just v ∎)
-  where open EqR (MaybeSetoid (VecISetoid A.setoid at _))
-        open Get G
+  where open Get G
+        open EqR (MaybeSetoid (ShapedISetoid (EqSetoid _) ViewShapeT A.setoid at _))
diff --git a/Everything.agda b/Everything.agda
index e37c76e..5ca5550 100644
--- a/Everything.agda
+++ b/Everything.agda
@@ -2,6 +2,8 @@
 module Everything where
 
 import Generic
+import Structures
+import Instances
 import FinMap
 import CheckInsert
 import GetTypes
diff --git a/Examples.agda b/Examples.agda
index c82bcf4..eca3c90 100644
--- a/Examples.agda
+++ b/Examples.agda
@@ -5,12 +5,14 @@ open import Data.Nat.Properties using (cancel-+-left)
 import Algebra.Structures
 open Algebra.Structures.IsCommutativeSemiring Data.Nat.Properties.isCommutativeSemiring using (+-isCommutativeMonoid)
 open Algebra.Structures.IsCommutativeMonoid +-isCommutativeMonoid using () renaming (comm to +-comm)
+open import Data.List using (List ; length) renaming ([] to []L ; _∷_ to _∷L_)
 open import Data.Vec using (Vec ; [] ; _∷_ ; reverse ; _++_ ; tail ; take ; drop)
 open import Function using (id)
 open import Function.Injection using () renaming (Injection to _↪_ ; id to id↪)
-open import Relation.Binary.PropositionalEquality using (_≡_ ; refl) renaming (setoid to EqSetoid)
+open import Relation.Binary.PropositionalEquality using (_≡_ ; refl ; cong) renaming (setoid to EqSetoid)
 
 open import Generic using (≡-to-Π)
+open import Structures using (Shaped)
 import GetTypes
 import FreeTheorems
 
@@ -71,3 +73,32 @@ intersperse' : Get
 intersperse' = assume-get suc-injection (≡-to-Π intersperse-len) f
   where f : {A : Set} {n : ℕ} → Vec A (suc n) → Vec A (intersperse-len n)
         f (s ∷ v)        = intersperse s v
+
+data PairVec (α : Set) (β : Set) : List α → Set where
+  []P : PairVec α β []L
+  _,_∷P_ : (x : α) → β → {l : List α} → PairVec α β l → PairVec α β (x ∷L l)
+
+PairVecFirstShaped : (α : Set) → Shaped (List α) (PairVec α)
+PairVecFirstShaped α = record
+  { arity = length
+  ; content = content
+  ; fill = fill
+  ; isShaped = record
+    { content-fill = content-fill
+    ; fill-content = fill-content
+    } }
+  where content : {β : Set} {s : List α} → PairVec α β s → Vec β (length s)
+        content []P          = []
+        content (a , b ∷P p) = b ∷ content p
+
+        fill : {β : Set} → (s : List α) → Vec β (length s) → PairVec α β s
+        fill []L      v       = []P
+        fill (a ∷L s) (b ∷ v) = a , b ∷P fill s v
+
+        content-fill : {β : Set} {s : List α} → (p : PairVec α β s) → fill s (content p) ≡ p
+        content-fill []P          = refl
+        content-fill (a , b ∷P p) = cong (_,_∷P_ a b) (content-fill p)
+
+        fill-content : {β : Set} → (s : List α) → (v : Vec β (length s)) → content (fill s v) ≡ v
+        fill-content []L      []      = refl
+        fill-content (a ∷L s) (b ∷ v) = cong (_∷_ b) (fill-content s v)
diff --git a/GetTypes.agda b/GetTypes.agda
index eb72cea..f23d154 100644
--- a/GetTypes.agda
+++ b/GetTypes.agda
@@ -12,6 +12,8 @@ open import Relation.Binary using (Setoid)
 open Injection using (to)
 
 open import Generic using (≡-to-Π)
+open import Structures using (Shaped ; module Shaped)
+open import Instances using (VecShaped)
 
 module ListList where
   record Get : Set₁ where
@@ -49,3 +51,43 @@ VecVec-to-PartialVecVec G = record
   ; get = get
   ; free-theorem = free-theorem
   } where open VecVec.Get G
+
+module PartialShapeShape where
+  record Get : Set₁ where
+    field
+      SourceShape : Set
+      SourceContainer : Set → SourceShape → Set
+      SourceShapeT : Shaped SourceShape SourceContainer
+
+      ViewShape : Set
+      ViewContainer : Set → ViewShape → Set
+      ViewShapeT : Shaped ViewShape ViewContainer
+
+      I : Setoid ℓ₀ ℓ₀
+      gl₁ : I ↪ EqSetoid SourceShape
+      gl₂ : I ⟶ EqSetoid ViewShape
+
+    |I|   = Setoid.Carrier I
+    |gl₁| = _⟨$⟩_ (to gl₁)
+    |gl₂| = _⟨$⟩_ gl₂
+
+    open Shaped SourceShapeT using () renaming (fmap to fmapS)
+    open Shaped ViewShapeT using () renaming (fmap to fmapV)
+
+    field
+      get : {A : Set} {i : |I|} → SourceContainer A (|gl₁| i) → ViewContainer A (|gl₂| i)
+      free-theorem : {α β : Set} → (f : α → β) → {i : |I|} → get {_} {i} ∘ fmapS f ≗ fmapV f ∘ get
+
+    open Shaped SourceShapeT public using () renaming (fmap to fmapS)
+    open Shaped ViewShapeT public using () renaming (fmap to fmapV)
+
+PartialVecVec-to-PartialShapeShape : PartialVecVec.Get → PartialShapeShape.Get
+PartialVecVec-to-PartialShapeShape G = record
+  { SourceShapeT = VecShaped
+  ; ViewShapeT   = VecShaped
+  ; I            = I
+  ; gl₁          = gl₁
+  ; gl₂          = gl₂
+  ; get          = get
+  ; free-theorem = free-theorem
+  } where open PartialVecVec.Get G
diff --git a/Instances.agda b/Instances.agda
new file mode 100644
index 0000000..2e4ed3e
--- /dev/null
+++ b/Instances.agda
@@ -0,0 +1,73 @@
+module Instances where
+
+open import Level using () renaming (zero to ℓ₀)
+open import Category.Functor using (RawFunctor)
+open import Data.Maybe as M using (Maybe)
+open import Data.Nat using (ℕ)
+open import Data.Product using (_×_ ; _,_ ; proj₁ ; proj₂)
+open import Data.Vec using (Vec)
+import Data.Vec.Equality
+open Data.Vec.Equality.PropositionalEquality using () renaming (to-≡ to VecEq-to-≡)
+open import Function using (_∘_ ; id)
+open import Relation.Binary using (Setoid ; module Setoid)
+open import Relation.Binary.Indexed using (_at_) renaming (Setoid to ISetoid)
+open import Relation.Binary.PropositionalEquality as P using (_≡_ ; _≗_ ; module ≡-Reasoning)
+
+open Setoid using () renaming (_≈_ to _∋_≈_)
+
+open import Generic using (VecISetoid)
+open import Structures using (Functor ; Shaped ; module 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  = P.refl
+
+        identity : {α : Set} → _<$>_ {α} id ≗ id
+        identity (M.just x) = P.refl
+        identity M.nothing  = P.refl
+
+        composition : {α β γ : Set} → (g : β → γ) → (h : α → β) → _<$>_ (g ∘ h) ≗ _<$>_ g ∘ _<$>_ h
+        composition g h (M.just x) = P.refl
+        composition g h M.nothing  = P.refl
+
+VecShaped : Shaped ℕ Vec
+VecShaped = record
+  { arity = id
+  ; content = id
+  ; fill = λ _ → id
+  ; isShaped = record
+    { content-fill = λ _ → P.refl
+    ; fill-content = λ _ _ → P.refl
+    } }
+
+ShapedISetoid : (S : Setoid ℓ₀ ℓ₀) {C : Set → (Setoid.Carrier S) → Set} → Shaped (Setoid.Carrier S) C → Setoid ℓ₀ ℓ₀ → ISetoid (Setoid.Carrier S) ℓ₀ ℓ₀
+ShapedISetoid S {C} ShapeT α = record
+  { Carrier = C (Setoid.Carrier α)
+  ; _≈_ = λ {s₁} {s₂} c₁ c₂ → S ∋ s₁ ≈ s₂ × ISetoid._≈_ (VecISetoid α) (content c₁) (content c₂)
+  ; isEquivalence = record
+    { refl = Setoid.refl S , ISetoid.refl (VecISetoid α)
+    ; sym = λ p → Setoid.sym S (proj₁ p) , ISetoid.sym (VecISetoid α) (proj₂ p)
+    ; trans = λ p q → Setoid.trans S (proj₁ p) (proj₁ q) , ISetoid.trans (VecISetoid α) (proj₂ p) (proj₂ q)
+    } } where open Shaped ShapeT
+
+Shaped≈-to-≡ : {S : Set} {C : Set → S → Set} → (ShapeT : Shaped S C) → (α : Set) → {s : S} {x y : C α s} → ShapedISetoid (P.setoid S) ShapeT (P.setoid α) at s ∋ x ≈ y → x ≡ y
+Shaped≈-to-≡ ShapeT α {s} {x} {y} (shape≈ , content≈) = begin
+  x
+    ≡⟨ P.sym (content-fill x) ⟩
+  fill s (content x)
+    ≡⟨ P.cong (fill s) (VecEq-to-≡ content≈) ⟩
+  fill s (content y)
+    ≡⟨ content-fill y ⟩
+  y ∎
+  where open ≡-Reasoning
+        open Shaped ShapeT
diff --git a/Precond.agda b/Precond.agda
index a76fddc..85c955c 100644
--- a/Precond.agda
+++ b/Precond.agda
@@ -25,6 +25,7 @@ open import Relation.Binary.PropositionalEquality using (refl ; cong ; inspect ;
 open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎)
 open import Relation.Nullary using (yes ; no)
 
+open import Structures using (IsFunctor ; Shaped)
 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)
@@ -32,8 +33,8 @@ 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)
+open GetTypes.PartialShapeShape using (Get ; module Get)
+open BFF.PartialShapeBFF (decSetoid deq) using (assoc ; enumerate ; denumerate ; bff)
 
 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
@@ -62,32 +63,32 @@ lemma-union-delete-fromFunc {n = n} {is = i ∷ is} {h = h} {g = g} (Data.List.A
           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
+assoc-enough : (G : Get) → {i : Get.|I| G} → (j : Get.|I| G) → (s : Get.SourceContainer G Carrier (Get.|gl₁| G i)) → (v : Get.ViewContainer G Carrier (Get.|gl₂| G j)) → ∃ (λ h → assoc (Shaped.content (Get.ViewShapeT G) (Get.get G (enumerate (Get.SourceShapeT G) (Get.|gl₁| G j)))) (Shaped.content (Get.ViewShapeT G) v) ≡ just h) → ∃ λ u → bff G j s v ≡ just u
+assoc-enough G {i} j s v (h , p) = _ , cong (_<$>_ ((λ f → fmapS f t) ∘ flip lookupM) ∘ _<$>_ (flip union (reshape g′ (Shaped.arity SourceShapeT (|gl₁| j))))) p
   where open Get G
-        g′ = delete-many (get (enumerate s)) (fromFunc (denumerate s))
-        t  = enumeratel (|gl₁| j)
+        g′ = delete-many (Shaped.content ViewShapeT (get (enumerate SourceShapeT (|gl₁| i)))) (fromFunc (denumerate SourceShapeT s))
+        t  = enumerate SourceShapeT (|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 : Get) → {i : Get.|I| G} → (s : Get.SourceContainer G Carrier (Get.|gl₁| G i)) → (v : Get.ViewContainer G Carrier (Get.|gl₂| G i)) → ∃ (λ h → assoc (Shaped.content (Get.ViewShapeT G) (Get.get G (enumerate (Get.SourceShapeT G) (Get.|gl₁| G i)))) (Shaped.content (Get.ViewShapeT G) v) ≡ just h) → ∃ λ u → bff G i s v ≡ just (Get.fmapS G 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)
+  just (fmapS (flip lookupM (union h (reshape g′ (Shaped.arity SourceShapeT (|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) ∎) ⟩ _ ∎)
+        ≡⟨ cong ((λ f → fmapS f t) ∘ flip lookupM ∘ union h) (lemma-reshape-id g′) ⟩
+      fmapS (flip lookupM (union h g′)) t
+        ≡⟨ cong ((λ f → fmapS f t) ∘ flip lookupM) (proj₂ wp) ⟩
+      fmapS (flip lookupM (fromFunc (proj₁ wp))) t
+        ≡⟨ IsFunctor.cong (Shaped.isFunctor SourceShapeT (|gl₁| i)) (lemma-lookupM-fromFunc (proj₁ wp)) t ⟩
+      fmapS (Maybe.just ∘ proj₁ wp) t
+        ≡⟨ IsFunctor.composition (Shaped.isFunctor SourceShapeT (|gl₁| i)) just (proj₁ wp) t ⟩
+      fmapS Maybe.just (fmapS (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)
+        s′ = enumerate SourceShapeT (|gl₁| i)
+        g  = fromFunc (denumerate SourceShapeT s)
+        g′ = delete-many (Shaped.content ViewShapeT (get s′)) g
+        t  = enumerate SourceShapeT (|gl₁| i)
+        wp = lemma-union-delete-fromFunc (lemma-assoc-domain (Shaped.content ViewShapeT (get t)) (Shaped.content ViewShapeT v) h p)
 
 data All-different {A : Set} : List A → Set where
   different-[] : All-different []
diff --git a/Structures.agda b/Structures.agda
new file mode 100644
index 0000000..10abd42
--- /dev/null
+++ b/Structures.agda
@@ -0,0 +1,87 @@
+module Structures where
+
+open import Category.Functor using (RawFunctor ; module RawFunctor)
+open import Category.Monad using (module RawMonad)
+open import Data.Maybe using (Maybe) renaming (monad to MaybeMonad)
+open import Data.Nat using (ℕ)
+open import Data.Vec as V using (Vec)
+import Data.Vec.Properties as VP
+open import Function using (_∘_ ; flip ; id)
+open import Function.Equality using (_⟶_ ; _⇨_ ; _⟨$⟩_)
+open import Relation.Binary using (_Preserves_⟶_)
+open import Relation.Binary.PropositionalEquality as P using (_≗_ ; _≡_ ; refl ; module ≡-Reasoning)
+
+open import Generic using (sequenceV)
+
+record IsFunctor (F : Set → Set) (f : {α β : Set} → (α →  β) → F α → F β) : Set₁ where
+  field
+    cong : {α β : Set} → f {α} {β} Preserves _≗_ ⟶ _≗_
+    identity : {α : Set} → f {α} id ≗ id
+    composition : {α β γ : Set} → (g : β → γ) → (h : α → β) →
+                  f (g ∘ h) ≗ f g ∘ f h
+
+  isCongruence : {α β : Set} → (P.setoid α ⇨ P.setoid β) ⟶ P.setoid (F α) ⇨ P.setoid (F β)
+  isCongruence {α} {β} = record
+    { _⟨$⟩_ = λ g → record
+      { _⟨$⟩_ = f (_⟨$⟩_ g)
+      ; cong = P.cong (f (_⟨$⟩_ g))
+      }
+    ; cong = λ {g} {h} g≗h {x} x≡y → P.subst (λ z → f (_⟨$⟩_ g) x ≡ f (_⟨$⟩_ h) z) x≡y (cong (λ _ → g≗h refl) x)
+    }
+
+record Functor (f : Set → Set) : Set₁ where
+  field
+    rawfunctor : RawFunctor f
+    isFunctor : IsFunctor f (RawFunctor._<$>_ rawfunctor)
+
+  open RawFunctor rawfunctor public
+  open IsFunctor isFunctor public
+
+record IsShaped (S : Set)
+                (C : Set → S → Set)
+                (arity : S → ℕ)
+                (content : {α : Set} {s : S} → C α s → Vec α (arity s))
+                (fill : {α : Set} → (s : S) → Vec α (arity s) → C α s)
+                : Set₁ where
+  field
+    content-fill : {α : Set} {s : S} → (c : C α s) → fill s (content c) ≡ c
+    fill-content : {α : Set} → (s : S) → (v : Vec α (arity s)) → content (fill s v) ≡ v
+
+  fmap : {α β : Set} → (f : α → β) → {s : S} → C α s → C β s
+  fmap f {s} c = fill s (V.map f (content c))
+
+  isFunctor : (s : S) → IsFunctor (flip C s) (λ f → fmap f)
+  isFunctor s = record
+    { cong = λ g≗h c → P.cong (fill s) (VP.map-cong g≗h (content c))
+    ; identity = λ c → begin
+        fill s (V.map id (content c))
+          ≡⟨ P.cong (fill s) (VP.map-id (content c)) ⟩
+        fill s (content c)
+          ≡⟨ content-fill c ⟩
+        c ∎
+    ; composition = λ g h c → P.cong (fill s) (begin
+        V.map (g ∘ h) (content c)
+          ≡⟨ VP.map-∘ g h (content c) ⟩
+        V.map g (V.map h (content c))
+          ≡⟨ P.cong (V.map g) (P.sym (fill-content s (V.map h (content c)))) ⟩
+        V.map g (content (fill s (V.map h (content c)))) ∎)
+    } where open ≡-Reasoning
+
+  fmap-content : {α β : Set} → (f : α → β) → {s : S} → content {β} {s} ∘ fmap f ≗ V.map f ∘ content
+  fmap-content f c = fill-content _ (V.map f (content c))
+  fill-fmap : {α β : Set} → (f : α → β) → (s : S) → fmap f ∘ fill s ≗ fill s ∘ V.map f
+  fill-fmap f s v = P.cong (fill s ∘ V.map f) (fill-content s v)
+
+  sequence : {α : Set} {s : S} → C (Maybe α) s → Maybe (C α s)
+  sequence {s = s} c = fill s <$> sequenceV (content c)
+    where open RawMonad MaybeMonad
+
+record Shaped (S : Set) (C : Set → S → Set) : Set₁ where
+  field
+    arity : S → ℕ
+    content : {α : Set} {s : S} → C α s → Vec α (arity s)
+    fill : {α : Set} → (s : S) → Vec α (arity s) → C α s
+
+    isShaped : IsShaped S C arity content fill
+
+  open IsShaped isShaped public