fully allow partial get functions

By choosing gl₁ = suc and gl₂ = id, the tail function can now be
bidirectionalized.

2 files changed, 32 insertions, 25 deletions

diff --git a/Bidir.agda b/Bidir.agda
index e0960f5..7add8fd 100644
--- a/Bidir.agda
+++ b/Bidir.agda
@@ -18,6 +18,9 @@ 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 Function.Equality using (_⟶_ ; _⟨$⟩_)
+open import Function.Injection using (module Injection) renaming (Injection to _↪_)
+open Injection using (to)
 open import Relation.Binary.Core using (refl ; _≡_)
 open import Relation.Binary.Indexed using (_at_) renaming (Setoid to ISetoid)
 open import Relation.Binary.PropositionalEquality using (cong ; sym ; inspect ; [_] ; trans ; cong₂ ; decSetoid ; module ≡-Reasoning) renaming (setoid to EqSetoid)
@@ -25,13 +28,13 @@ 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.PartialVecVec using (get-type ; free-theorem)
 open import Generic using (mapMV ; mapMV-cong ; mapMV-purity ; sequenceV ; sequence-map ; VecISetoid)
 open import FinMap
 import CheckInsert
 open CheckInsert A
 import BFF
-open BFF.VecBFF A using (assoc ; enumerate ; denumerate ; bff)
+open BFF.PartialVecBFF A using (assoc ; enumerate ; denumerate ; bff)
 open Setoid using () renaming (_≈_ to _∋_≈_)
 open module A = DecSetoid A using (Carrier) renaming (_≟_ to deq)
 
@@ -125,13 +128,13 @@ 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 : {I : Setoid ℓ₀ ℓ₀} → (gl₁ : I ↪ EqSetoid ℕ) → (gl₂ : I ⟶ EqSetoid ℕ) → (get : get-type gl₁ gl₂) → {i : Setoid.Carrier I} → (s : Vec Carrier (to gl₁ ⟨$⟩ i)) → bff gl₁ gl₂ get s (get s) ≡ just s
+theorem-1 gl₁ gl₂ get s = begin
+  bff gl₁ gl₂ get s (get s)
+    ≡⟨ cong (bff gl₁ gl₂ get s ∘ get) (sym (lemma-map-denumerate-enumerate s)) ⟩
+  bff gl₁ gl₂ get s (get (map (denumerate s) (enumerate s)))
+    ≡⟨ cong (bff gl₁ gl₂ get s) (free-theorem gl₁ gl₂ get (denumerate s) (enumerate s)) ⟩
+  bff gl₁ gl₂ get 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))) ⟩
@@ -189,8 +192,8 @@ 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 <$> mapMV f v ≡ mapMV f (get v)
-lemma-get-mapMV {f = f} {v = v} p get = let w , pw = lemma-mapM-successful v p in begin
+lemma-get-mapMV : {A B : Set} {f : A → Maybe B} {I : Setoid ℓ₀ ℓ₀} → (gl₁ : I ↪ EqSetoid ℕ) → (gl₂ : I ⟶ EqSetoid ℕ) → {i : Setoid.Carrier I} {v : Vec A (to gl₁ ⟨$⟩ i)} {r : Vec B (to gl₁ ⟨$⟩ i)} → mapMV f v ≡ just r → (get : get-type gl₁ gl₂) → get <$> mapMV f v ≡ mapMV f (get v)
+lemma-get-mapMV {f = f} gl₁ gl₂ {v = v} p get = let w , pw = lemma-mapM-successful v p in begin
   get <$> mapMV f v
     ≡⟨ cong (_<$>_ get) (sym (sequence-map f v)) ⟩
   get <$> (sequenceV (map f v))
@@ -200,11 +203,11 @@ lemma-get-mapMV {f = f} {v = v} p get = let w , pw = lemma-mapM-successful v p i
   get <$> 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 gl₁ gl₂ get 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 gl₁ gl₂ get f v) ⟩
   sequenceV (map f (get v))
     ≡⟨ sequence-map f (get v) ⟩
   mapMV f (get v) ∎
@@ -217,16 +220,16 @@ 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 : {I : Setoid ℓ₀ ℓ₀} → (gl₁ : I ↪ EqSetoid ℕ) → (gl₂ : I ⟶ EqSetoid ℕ) → (get : get-type gl₁ gl₂) → {i : Setoid.Carrier I} → (v : Vec Carrier (gl₂ ⟨$⟩ i)) → (s u : Vec Carrier (to gl₁ ⟨$⟩ i)) → bff gl₁ gl₂ get s v ≡ just u → VecISetoid A.setoid at _ ∋ get u ≈ v
+theorem-2 gl₁ gl₂ get v s u p with (lemma->>=-just ((flip union (delete-many (get (enumerate s)) (fromFunc (denumerate s)))) <$> (assoc (get (enumerate s)) v)) p)
+theorem-2 gl₁ gl₂ get v s u p | h′ , ph′ with (lemma-<$>-just (assoc (get (enumerate s)) v) ph′)
+theorem-2 gl₁ gl₂ get 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 gl₁ gl₂ get 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 gl₁ gl₂ (trans (cong (flip _>>=_ h′↦r ∘ _<$>_ h↦h′) (sym ph)) p) get ⟩
   mapMV (flip lookupM (h↦h′ h)) (get s′)
     ≡⟨ sym (sequence-map (flip lookupM (h↦h′ h)) (get s′)) ⟩
   sequenceV (map (flip lookupM (h↦h′ h)) (get s′))
diff --git a/Precond.agda b/Precond.agda
index 19329b5..a6f2871 100644
--- a/Precond.agda
+++ b/Precond.agda
@@ -6,7 +6,7 @@ open import Data.Nat using (ℕ)
 open import Data.Fin using (Fin ; zero ; suc)
 open import Data.Fin.Props using (_≟_)
 open import Data.List using (List ; [] ; _∷_)
-import Level
+open import Level using () renaming (zero to ℓ₀)
 import Category.Monad
 import Category.Functor
 open import Data.Maybe using (Maybe ; nothing ; just ; maybe′)
@@ -20,7 +20,11 @@ open Data.List.Any.Membership-≡ using (_∉_)
 open import Data.Maybe using (just)
 open import Data.Product using (∃ ; _,_ ; proj₂)
 open import Function using (flip ; _∘_ ; id)
-open import Relation.Binary.PropositionalEquality using (refl ; cong ; inspect ; [_] ; sym ; decSetoid)
+open import Function.Equality using (_⟶_ ; _⟨$⟩_)
+open import Function.Injection using (module Injection) renaming (Injection to _↪_)
+open Injection using (to)
+open import Relation.Binary using (Setoid)
+open import Relation.Binary.PropositionalEquality using (refl ; cong ; inspect ; [_] ; sym ; decSetoid) renaming (setoid to EqSetoid)
 open Relation.Binary.PropositionalEquality.≡-Reasoning using (begin_ ; _≡⟨_⟩_ ; _∎)
 open import Relation.Nullary using (yes ; no)
 
@@ -31,7 +35,7 @@ open CheckInsert (decSetoid deq) using (checkInsert ; lemma-checkInsert-new ; le
 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)
+open BFF.PartialVecBFF (decSetoid deq) using (get-type ; 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 +73,9 @@ 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 : {I : Setoid ℓ₀ ℓ₀} → (gl₁ : I ↪ EqSetoid ℕ) → (gl₂ : I ⟶ EqSetoid ℕ) → (get : get-type gl₁ gl₂) → {i : Setoid.Carrier I} → (s : Vec Carrier (to gl₁ ⟨$⟩ i)) → (v : Vec Carrier (gl₂ ⟨$⟩ i)) → ∃ (λ h → assoc (get (enumerate s)) v ≡ just h) → ∃ λ u → bff gl₁ gl₂ get s v ≡ just u
+assoc-enough gl₁ gl₂ get s v (h , p) = let w , pw = lemma-union-delete-fromFunc (lemma-assoc-domain (get s′) v h p) in _ , (begin
+  bff gl₁ gl₂ get 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′) ⟩