pass get functions as records

This allows passing both getlen and get as a single parameter. It also
allows to make the free theorem a prerequisite instead of a postulate.

5 files changed, 52 insertions, 30 deletions

diff --git a/BFF.agda b/BFF.agda
index 61eeefd..79f3b3d 100644
--- a/BFF.agda
+++ b/BFF.agda
@@ -19,7 +19,7 @@ import CheckInsert
 import FreeTheorems
 
 module VecBFF (A : DecSetoid ℓ₀ ℓ₀) where
-  open FreeTheorems.VecVec public using (get-type)
+  open FreeTheorems.VecVec public using (Get)
   open module A = DecSetoid A using (Carrier) renaming (_≟_ to deq)
   open CheckInsert A
 
@@ -33,9 +33,9 @@ module VecBFF (A : DecSetoid ℓ₀ ℓ₀) where
   denumerate : {n : ℕ} → Vec Carrier n → Fin n → Carrier
   denumerate = flip lookupV
 
-  bff : {getlen : ℕ → ℕ} → (get-type getlen) → ({n : ℕ} → Vec Carrier n → Vec Carrier (getlen n) → Maybe (Vec Carrier n))
-  bff get s v = let s′ = enumerate s
-                    t′ = get s′
+  bff : (G : Get) → ({n : ℕ} → Vec Carrier n → Vec Carrier (Get.getlen G n) → Maybe (Vec Carrier n))
+  bff G s v = let   s′ = enumerate s
+                    t′ = Get.get G s′
                     g  = fromFunc (denumerate s)
                     g′ = delete-many t′ g
                     h  = assoc t′ v
diff --git a/Bidir.agda b/Bidir.agda
index e0960f5..42f8821 100644
--- a/Bidir.agda
+++ b/Bidir.agda
@@ -25,7 +25,7 @@ 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 FreeTheorems.VecVec using (Get ; module Get)
 open import Generic using (mapMV ; mapMV-cong ; mapMV-purity ; sequenceV ; sequence-map ; VecISetoid)
 open import FinMap
 import CheckInsert
@@ -125,13 +125,13 @@ lemma-map-denumerate-enumerate (a ∷ as) = cong (_∷_ a) (begin
   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
-  bff get s (get s)
-    ≡⟨ cong (bff get s ∘ get) (sym (lemma-map-denumerate-enumerate s)) ⟩
-  bff get s (get (map (denumerate s) (enumerate s)))
-    ≡⟨ cong (bff get s) (free-theorem get (denumerate s) (enumerate s)) ⟩
-  bff get s (map (denumerate s) (get (enumerate s)))
+theorem-1 : (G : Get) → {m : ℕ} → (s : Vec Carrier m) → bff G s (Get.get G s) ≡ just s
+theorem-1 G s = begin
+  bff G s (get s)
+    ≡⟨ cong (bff G s ∘ get) (sym (lemma-map-denumerate-enumerate s)) ⟩
+  bff G s (get (map (denumerate s) (enumerate s)))
+    ≡⟨ cong (bff G s) (free-theorem (denumerate s) (enumerate s)) ⟩
+  bff G s (map (denumerate s) (get (enumerate s)))
     ≡⟨ refl ⟩
   (h′↦r ∘ h↦h′) (assoc (get (enumerate s)) (map (denumerate s) (get (enumerate s))))
     ≡⟨ cong (h′↦r ∘ h↦h′) (lemma-1 (denumerate s) (get (enumerate s))) ⟩
@@ -151,6 +151,7 @@ theorem-1 get s = begin
     ≡⟨ cong just (lemma-map-denumerate-enumerate s) ⟩
   just s ∎
     where open ≡-Reasoning
+          open Get G
           h↦h′ = _<$>_ (flip union (delete-many (get (enumerate s)) (fromFunc (denumerate s))))
           h′↦r = flip _>>=_ (flip mapMV (enumerate s) ∘ flip lookupVec)
 
@@ -189,8 +190,8 @@ lemma-mapM-successful         (x ∷ xs) p  | just y | just ys | [ p′ ] with l
 lemma-mapM-successful         (x ∷ xs) p  | just y | just ys | [ p′ ] | w , pw = y ∷ w , cong (_∷_ (just y)) pw
 
 
-lemma-get-mapMV : {A B : Set} {f : A → Maybe B} {n : ℕ} {v : Vec A n} {r : Vec B n} → mapMV f v ≡ just r → {getlen : ℕ → ℕ} (get : get-type getlen) → get <$> mapMV f v ≡ mapMV f (get v)
-lemma-get-mapMV {f = f} {v = v} p get = let w , pw = lemma-mapM-successful v p in begin
+lemma-get-mapMV : {A B : Set} {f : A → Maybe B} {n : ℕ} {v : Vec A n} {r : Vec B n} → mapMV f v ≡ just r → (get : Get) → Get.get get <$> mapMV f v ≡ mapMV f (Get.get get v)
+lemma-get-mapMV {f = f} {v = v} p G = let w , pw = lemma-mapM-successful v p in begin
   get <$> mapMV f v
     ≡⟨ cong (_<$>_ get) (sym (sequence-map f v)) ⟩
   get <$> (sequenceV (map f v))
@@ -200,15 +201,16 @@ lemma-get-mapMV {f = f} {v = v} p get = let w , pw = lemma-mapM-successful v p i
   get <$> just w
     ≡⟨ sym (lemma-just-sequence (get w)) ⟩
   sequenceV (map just (get w))
-    ≡⟨ cong sequenceV (sym (free-theorem get just w)) ⟩
+    ≡⟨ cong sequenceV (sym (free-theorem just w)) ⟩
   sequenceV (get (map just w))
     ≡⟨ cong (sequenceV ∘ get) (sym pw) ⟩
   sequenceV (get (map f v))
-    ≡⟨ cong sequenceV (free-theorem get f v) ⟩
+    ≡⟨ cong sequenceV (free-theorem f v) ⟩
   sequenceV (map f (get v))
     ≡⟨ sequence-map f (get v) ⟩
   mapMV f (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 _))
@@ -217,16 +219,16 @@ sequence-cong {S} {m₁ = just x ∷ xs} {m₂ = just y ∷ ys} (just x≈y VecE
 sequence-cong {S} {m₁ = just x ∷ xs} {m₂ = just y ∷ ys} (just x≈y VecEq.∷-cong xs≈ys) | nothing | nothing | nothing = Setoid.refl (MaybeSetoid (VecISetoid S at _))
 sequence-cong {S}                                       (nothing VecEq.∷-cong xs≈ys) = Setoid.refl (MaybeSetoid (VecISetoid S at _))
 
-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 → VecISetoid A.setoid at _ ∋ get u ≈ v
-theorem-2 get v s u p with (lemma->>=-just ((flip union (delete-many (get (enumerate s)) (fromFunc (denumerate s)))) <$> (assoc (get (enumerate s)) v)) p)
-theorem-2 get v s u p | h′ , ph′ with (lemma-<$>-just (assoc (get (enumerate s)) v) ph′)
-theorem-2 get v s u p | h′ , ph′ | h , ph = drop-just (begin⟨ MaybeSetoid (VecISetoid A.setoid at _) ⟩
+theorem-2 : (G : Get) → {m : ℕ} → (v : Vec Carrier (Get.getlen G m)) → (s u : Vec Carrier m) → bff G s v ≡ just u → VecISetoid A.setoid at _ ∋ Get.get G u ≈ v
+theorem-2 G v s u p with (lemma->>=-just ((flip union (delete-many (Get.get G (enumerate s)) (fromFunc (denumerate s)))) <$> (assoc (Get.get G (enumerate s)) v)) p)
+theorem-2 G v s u p | h′ , ph′ with (lemma-<$>-just (assoc (Get.get G (enumerate s)) v) ph′)
+theorem-2 G v s u p | h′ , ph′ | h , ph = drop-just (begin⟨ MaybeSetoid (VecISetoid A.setoid at _) ⟩
   get <$> (just u)
     ≡⟨ cong (_<$>_ get) (sym p) ⟩
-  get <$> (bff get s v)
+  get <$> (bff G s v)
     ≡⟨ cong (_<$>_ get ∘ flip _>>=_ h′↦r ∘ _<$>_ h↦h′) ph ⟩
   get <$> mapMV (flip lookupM (h↦h′ h)) s′
-    ≡⟨ lemma-get-mapMV (trans (cong (flip _>>=_ h′↦r ∘ _<$>_ h↦h′) (sym ph)) p) get ⟩
+    ≡⟨ lemma-get-mapMV (trans (cong (flip _>>=_ h′↦r ∘ _<$>_ h↦h′) (sym ph)) p) G ⟩
   mapMV (flip lookupM (h↦h′ h)) (get s′)
     ≡⟨ sym (sequence-map (flip lookupM (h↦h′ h)) (get s′)) ⟩
   sequenceV (map (flip lookupM (h↦h′ h)) (get s′))
@@ -237,6 +239,7 @@ theorem-2 get v s u p | h′ , ph′ | h , ph = drop-just (begin⟨ MaybeSetoid
     ≡⟨ lemma-just-sequence v ⟩
   just v ∎)
     where open SetoidReasoning
+          open Get G
           s′   = enumerate s
           g    = fromFunc (denumerate s)
           g′   = delete-many (get s′) g
diff --git a/FreeTheorems.agda b/FreeTheorems.agda
index f37cada..d2ba3ef 100644
--- a/FreeTheorems.agda
+++ b/FreeTheorems.agda
@@ -10,12 +10,29 @@ module ListList where
   get-type : Set₁
   get-type = {A : Set} → List A → List A
 
+  record Get : Set₁ where
+    field
+      get : {A : Set} → List A → List A
+      free-theorem : {α β : Set} → (f : α → β) → get ∘ map f ≗ map f ∘ get
+
   postulate
     free-theorem : (get : get-type) → {α β : Set} → (f : α → β) → get ∘ map f ≗ map f ∘ get
 
+  assume-get : get-type → Get
+  assume-get get = record { get = get; free-theorem = free-theorem get }
+
 module VecVec where
   get-type : (ℕ → ℕ) → Set₁
   get-type getlen = {A : Set} {n : ℕ} → Vec A n → Vec A (getlen n)
 
+  record Get : Set₁ where
+    field
+      getlen : ℕ → ℕ
+      get : {A : Set} {n : ℕ} → Vec A n → Vec A (getlen n)
+      free-theorem : {α β : Set} (f : α → β) {n : ℕ} → get {_} {n} ∘ mapV f ≗ mapV f ∘ get
+
   postulate
     free-theorem : {getlen : ℕ → ℕ} → (get : get-type getlen) → {α β : Set} → (f : α → β) → {n : ℕ} → get {_} {n} ∘ mapV f ≗ mapV f ∘ get
+
+  assume-get : {getlen : ℕ → ℕ} → (get : get-type getlen) → Get
+  assume-get {getlen} get = record { getlen = getlen; get = get; free-theorem = free-theorem get }
diff --git a/LiftGet.agda b/LiftGet.agda
index 31a632e..2199aff 100644
--- a/LiftGet.agda
+++ b/LiftGet.agda
@@ -13,8 +13,8 @@ open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨
 
 import FreeTheorems
 open import Generic using (length-replicate ; subst-cong ; subst-fromList ; subst-subst ; toList-fromList ; toList-subst)
-open FreeTheorems.ListList using (get-type ; free-theorem)
-open FreeTheorems.VecVec using () renaming (get-type to getV-type)
+open FreeTheorems.ListList using (get-type ; free-theorem) renaming (Get to GetL)
+open FreeTheorems.VecVec using () renaming (get-type to getV-type ; Get to GetV)
 
 getVec-to-getList : {getlen : ℕ → ℕ} → (getV-type getlen) → get-type
 getVec-to-getList get = toList ∘ get ∘ fromList
diff --git a/Precond.agda b/Precond.agda
index 19329b5..a1f1081 100644
--- a/Precond.agda
+++ b/Precond.agda
@@ -30,8 +30,9 @@ import CheckInsert
 open CheckInsert (decSetoid deq) using (checkInsert ; lemma-checkInsert-new ; lemma-lookupM-checkInsert-other)
 import BFF
 open import Bidir (decSetoid deq) using (_in-domain-of_ ; lemma-assoc-domain ; lemma-just-sequence)
-
-open BFF.VecBFF (decSetoid deq) using (get-type ; assoc ; enumerate ; denumerate ; bff)
+import FreeTheorems
+open FreeTheorems.VecVec using (Get ; module Get)
+open BFF.VecBFF (decSetoid deq) using (assoc ; enumerate ; denumerate ; bff)
 
 lemma-lookup-map-just : {n : ℕ} (f : Fin n) {A : Set} (v : Vec A n) → lookup f (map Maybe.just v) ≡ Maybe.just (lookup f v)
 lemma-lookup-map-just zero    (x ∷ xs) = refl
@@ -69,9 +70,9 @@ lemma-union-delete-fromFunc {n = n} {is = i ∷ is} {h = h} {g = g} ((x , px) Da
           maybe′ just (lookupM i (delete-many is (map just g))) (lookup i h) ∎
         inner f | no f≢i = cong (flip (maybe′ just) (lookup f h)) (lemma-lookupM-delete (delete-many is (map just g)) f≢i)
 
-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) = let w , pw = lemma-union-delete-fromFunc (lemma-assoc-domain (get s′) v h p) in _ , (begin
-  bff get s v
+assoc-enough : (G : Get) → {m : ℕ} → (s : Vec Carrier m) → (v : Vec Carrier (Get.getlen G m)) → ∃ (λ h → assoc (Get.get G (enumerate s)) v ≡ just h) → ∃ λ u → bff G s v ≡ just u
+assoc-enough G s v (h , p) = let w , pw = lemma-union-delete-fromFunc (lemma-assoc-domain (get s′) v h p) in _ , (begin
+  bff G s v
     ≡⟨ cong (flip _>>=_ (flip mapMV s′ ∘ flip lookupM) ∘ _<$>_ (flip union g′)) p ⟩
   mapMV (flip lookupM (union h g′)) s′
     ≡⟨ sym (sequence-map (flip lookupM (union h g′)) s′) ⟩
@@ -84,7 +85,8 @@ assoc-enough get s v (h , p) = let w , pw = lemma-union-delete-fromFunc (lemma-a
   sequenceV (map Maybe.just (map (flip lookup w) s′))
     ≡⟨ lemma-just-sequence (map (flip lookup w) s′) ⟩
   just (map (flip lookup w) s′) ∎)
-  where s′ = enumerate s
+  where open Get G
+        s′ = enumerate s
         g  = fromFunc (denumerate s)
         g′ = delete-many (get s′) g