rewrite main theorems to using Vec instead of List

1 files changed, 40 insertions, 52 deletions

diff --git a/Bidir.agda b/Bidir.agda
index cbac029..1c09d1d 100644
--- a/Bidir.agda
+++ b/Bidir.agda
@@ -4,14 +4,13 @@ open import Data.Nat using (ℕ)
 open import Data.Fin using (Fin)
 open import Data.Fin.Props using (_≟_)
 open import Data.Maybe using (Maybe ; nothing ; just ; maybe′)
-open import Data.List using (List ; [] ; _∷_ ; map ; length)
-open import Data.List.Properties using (map-cong ; ∷-injective) renaming (map-compose to map-∘)
+open import Data.List using (List)
 open import Data.List.Any using (Any ; any ; here ; there)
 open import Data.List.All using (All)
 open Data.List.Any.Membership-≡ using (_∈_ ; _∉_)
-open import Data.Vec using (toList ; fromList ; tabulate) renaming (lookup to lookupVec ; _∷_ to _∷V_)
-open import Data.Vec.Properties using (tabulate-∘ ; lookup∘tabulate)
-open import Data.Product using (∃ ; _,_ ; proj₁ ; proj₂)
+open import Data.Vec using (Vec ; [] ; _∷_ ; toList ; fromList ; map ; tabulate) renaming (lookup to lookupVec)
+open import Data.Vec.Properties using (tabulate-∘ ; lookup∘tabulate ; map-cong ; map-∘)
+open import Data.Product using (∃ ; _×_ ; _,_ ; proj₁ ; proj₂)
 open import Data.Empty using (⊥-elim)
 open import Function using (id ; _∘_ ; flip)
 open import Relation.Nullary using (yes ; no ; ¬_)
@@ -25,22 +24,22 @@ open import CheckInsert
 
 open import BFF using (_>>=_ ; fmap)
 
-open BFF.ListBFF using (assoc ; enumerate ; denumerate ; bff)
+open BFF.VecBFF using (assoc ; enumerate ; denumerate ; bff)
 
-lemma-1 : {τ : Set} {n : ℕ} → (eq : EqInst τ) → (f : Fin n → τ) → (is : List (Fin n)) → assoc eq is (map f is) ≡ just (restrict f is)
+lemma-1 : {τ : Set} {m n : ℕ} → (eq : EqInst τ) → (f : Fin n → τ) → (is : Vec (Fin n) m) → assoc eq is (map f is) ≡ just (restrict f (toList is))
 lemma-1 eq f []        = refl
 lemma-1 eq f (i ∷ is′) = begin
   assoc eq (i ∷ is′) (map f (i ∷ is′))
     ≡⟨ refl ⟩
   assoc eq is′ (map f is′) >>= checkInsert eq i (f i)
     ≡⟨ cong (λ m → m >>= checkInsert eq i (f i)) (lemma-1 eq f is′) ⟩
-  just (restrict f is′) >>= (checkInsert eq i (f i))
+  just (restrict f (toList is′)) >>= (checkInsert eq i (f i))
     ≡⟨ refl ⟩
-  checkInsert eq i (f i) (restrict f is′)
-    ≡⟨ lemma-checkInsert-restrict eq f i is′ ⟩
-  just (restrict f (i ∷ is′)) ∎
+  checkInsert eq i (f i) (restrict f (toList is′))
+    ≡⟨ lemma-checkInsert-restrict eq f i (toList is′) ⟩
+  just (restrict f (toList (i ∷ is′))) ∎
 
-lemma-lookupM-assoc : {A : Set} {n : ℕ} → (eq : EqInst A) → (i : Fin n) → (is : List (Fin n)) → (x : A) → (xs : List A) → (h : FinMapMaybe n A) → assoc eq (i ∷ is) (x ∷ xs) ≡ just h → lookupM i h ≡ just x
+lemma-lookupM-assoc : {A : Set} {m n : ℕ} → (eq : EqInst A) → (i : Fin n) → (is : Vec (Fin n) m) → (x : A) → (xs : Vec A m) → (h : FinMapMaybe n A) → assoc eq (i ∷ is) (x ∷ xs) ≡ just h → lookupM i h ≡ just x
 lemma-lookupM-assoc eq i is x xs h    p with assoc eq is xs
 lemma-lookupM-assoc eq i is x xs h    () | nothing
 lemma-lookupM-assoc eq i is x xs h    p | just h' = apply-checkInsertProof eq i x h' record
@@ -59,15 +58,13 @@ lemma-lookupM-assoc eq i is x xs h    p | just h' = apply-checkInsertProof eq i
   ; wrong = λ x' x≢x' lookupM≡justx' → lemma-just≢nothing (trans (sym p) (lemma-checkInsert-wrong eq i x h' x' x≢x' lookupM≡justx'))
   }
 
-lemma-∉-lookupM-assoc : {A : Set} {n : ℕ} → (eq : EqInst A) → (i : Fin n) → (is : List (Fin n)) → (xs : List A) → (h : FinMapMaybe n A) → assoc eq is xs ≡ just h → (i ∉ is) → lookupM i h ≡ nothing
+lemma-∉-lookupM-assoc : {A : Set} {m n : ℕ} → (eq : EqInst A) → (i : Fin n) → (is : Vec (Fin n) m) → (xs : Vec A m) → (h : FinMapMaybe n A) → assoc eq is xs ≡ just h → (i ∉ toList is) → lookupM i h ≡ nothing
 lemma-∉-lookupM-assoc eq i []         []         h ph i∉is = begin
   lookupM i h
     ≡⟨ cong (lookupM i) (sym (lemma-from-just ph)) ⟩
   lookupM i empty
     ≡⟨ lemma-lookupM-empty i ⟩
   nothing ∎
-lemma-∉-lookupM-assoc eq i []         (x' ∷ xs') h () i∉is
-lemma-∉-lookupM-assoc eq i (i' ∷ is') []         h () i∉is
 lemma-∉-lookupM-assoc eq i (i' ∷ is') (x' ∷ xs') h ph i∉is with i ≟ i'
 lemma-∉-lookupM-assoc eq i (i' ∷ is') (x' ∷ xs') h ph i∉is | yes p = contradiction (here p) i∉is
 lemma-∉-lookupM-assoc eq i (i' ∷ is') (x' ∷ xs') h ph i∉is | no ¬p with assoc eq is' xs' | inspect (assoc eq is') xs'
@@ -93,10 +90,8 @@ lemma-∉-lookupM-assoc eq i (i' ∷ is') (x' ∷ xs') h ph i∉is | no ¬p | ju
 _in-domain-of_ : {n : ℕ} {A : Set} → (is : List (Fin n)) → (FinMapMaybe n A) → Set
 _in-domain-of_ is h = All (λ i → ∃ λ x → lookupM i h ≡ just x) is
 
-lemma-assoc-domain : {n : ℕ} {A : Set} → (eq : EqInst A) → (is : List (Fin n)) → (xs : List A) → (h : FinMapMaybe n A) → assoc eq is xs ≡ just h → is in-domain-of h
-lemma-assoc-domain eq []  []  h  ph  = Data.List.All.[]
-lemma-assoc-domain eq [] (x' ∷ xs') h ()
-lemma-assoc-domain eq (i' ∷ is') [] h ()
+lemma-assoc-domain : {m n : ℕ} {A : Set} → (eq : EqInst A) → (is : Vec (Fin n) m) → (xs : Vec A m) → (h : FinMapMaybe n A) → assoc eq is xs ≡ just h → (toList is) in-domain-of h
+lemma-assoc-domain eq []         []         h ph = Data.List.All.[]
 lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h ph with assoc eq is' xs' | inspect (assoc eq is') xs'
 lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h () | nothing | ph'
 lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h ph | just h' | Reveal_is_.[_] ph' = apply-checkInsertProof eq i' x' h' record {
@@ -111,7 +106,7 @@ lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h ph | just h' | Reveal_is_.[_]
   ; wrong = λ x'' x'≢x'' lookupM-i'-h'≡just-x'' → lemma-just≢nothing (trans (sym ph) (lemma-checkInsert-wrong eq i' x' h' x'' x'≢x'' lookupM-i'-h'≡just-x''))
   }
 
-lemma-map-lookupM-insert : {n : ℕ} {A : Set} → (eq : EqInst A) → (i : Fin n) → (is : List (Fin n)) → (x : A) → (h : FinMapMaybe n A) → ¬(i ∈ is) → is in-domain-of h → map (flip lookupM (insert i x h)) is ≡ map (flip lookupM h) is
+lemma-map-lookupM-insert : {m n : ℕ} {A : Set} → (eq : EqInst A) → (i : Fin n) → (is : Vec (Fin n) m) → (x : A) → (h : FinMapMaybe n A) → ¬(i ∈ (toList is)) → (toList is) in-domain-of h → map (flip lookupM (insert i x h)) is ≡ map (flip lookupM h) is
 lemma-map-lookupM-insert eq i []         x h i∉is ph = refl
 lemma-map-lookupM-insert eq i (i' ∷ is') x h i∉is ph = begin
   lookupM i' (insert i x h) ∷ map (flip lookupM (insert i x h)) is'
@@ -120,11 +115,9 @@ lemma-map-lookupM-insert eq i (i' ∷ is') x h i∉is ph = begin
     ≡⟨ cong (_∷_ (lookupM i' h)) (lemma-map-lookupM-insert eq i is' x h (i∉is ∘ there) (Data.List.All.tail ph)) ⟩
   lookupM i' h ∷ map (flip lookupM h) is' ∎
 
-lemma-map-lookupM-assoc : {n : ℕ} {A : Set} → (eq : EqInst A) → (i : Fin n) → (is : List (Fin n)) → (x : A) → (xs : List A) → (h : FinMapMaybe n A) → (h' : FinMapMaybe n A) → assoc eq is xs ≡ just h' → checkInsert eq i x h' ≡ just h → map (flip lookupM h) is ≡ map (flip lookupM h') is
+lemma-map-lookupM-assoc : {m n : ℕ} {A : Set} → (eq : EqInst A) → (i : Fin n) → (is : Vec (Fin n) m) → (x : A) → (xs : Vec A m) → (h : FinMapMaybe n A) → (h' : FinMapMaybe n A) → assoc eq is xs ≡ just h' → checkInsert eq i x h' ≡ just h → map (flip lookupM h) is ≡ map (flip lookupM h') is
 lemma-map-lookupM-assoc eq i []         x []         h h' ph' ph = refl
-lemma-map-lookupM-assoc eq i []         x (x' ∷ xs') h h' ()  ph
-lemma-map-lookupM-assoc eq i (i' ∷ is') x []         h h' ()  ph
-lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph with any (_≟_ i) (i' ∷ is')
+lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph with any (_≟_ i) (toList (i' ∷ is'))
 lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p with Data.List.All.lookup (lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h' ph') p
 lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p | (x'' , p') with lookupM i h' 
 lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | yes p | (x'' , refl) | .(just x'') with eq x x''
@@ -138,10 +131,8 @@ lemma-map-lookupM-assoc eq i (i' ∷ is') x (x' ∷ xs') h h' ph' ph | no ¬p |
     ≡⟨ lemma-map-lookupM-insert eq i (i' ∷ is') x h' ¬p (lemma-assoc-domain eq (i' ∷ is') (x' ∷ xs') h' ph') ⟩
   map (flip lookupM h') (i' ∷ is') ∎
 
-lemma-2 : {τ : Set} {n : ℕ} → (eq : EqInst τ) → (is : List (Fin n)) → (v : List τ) → (h : FinMapMaybe n τ) → assoc eq is v ≡ just h → map (flip lookupM h) is ≡ map just v
+lemma-2 : {τ : Set} {m n : ℕ} → (eq : EqInst τ) → (is : Vec (Fin n) m) → (v : Vec τ m) → (h : FinMapMaybe n τ) → assoc eq is v ≡ just h → map (flip lookupM h) is ≡ map just v
 lemma-2 eq []       []       h p = refl
-lemma-2 eq []       (x ∷ xs) h ()
-lemma-2 eq (x ∷ xs) []       h ()
 lemma-2 eq (i ∷ is) (x ∷ xs) h p with assoc eq is xs | inspect (assoc eq is) xs
 lemma-2 eq (i ∷ is) (x ∷ xs) h () | nothing | _
 lemma-2 eq (i ∷ is) (x ∷ xs) h p | just h' | Reveal_is_.[_] ir = begin
@@ -163,28 +154,24 @@ lemma-2 eq (i ∷ is) (x ∷ xs) h p | just h' | Reveal_is_.[_] ir = begin
   map just (x ∷ xs) ∎
 
 postulate
-  free-theorem-list-list : {β γ : Set} → (get : {α : Set} → List α → List α) → (f : β → γ) → get ∘ map f ≗ map f ∘ get
+  free-theorem-list-list : {β γ : Set} → {getlen : ℕ → ℕ} → (get : {α : Set} {n : ℕ} → Vec α n → Vec α (getlen n)) → (f : β → γ) → {m : ℕ } → (v : Vec β m) → get (map f v) ≡ map f (get v)
 
-toList-map-commutes : {A B : Set} {n : ℕ} → (f : A → B) → (v : Data.Vec.Vec A n) → (toList (Data.Vec.map f v)) ≡ map f (toList v)
-toList-map-commutes f Data.Vec.[] = refl
-toList-map-commutes f (x ∷V xs) = cong (_∷_ (f x)) (toList-map-commutes f xs)
-
-lemma-map-denumerate-enumerate : {A : Set} → (as : List A) → map (denumerate as) (enumerate as) ≡ as
-lemma-map-denumerate-enumerate [] = refl
+lemma-map-denumerate-enumerate : {m : ℕ} {A : Set} → (as : Vec A m) → map (denumerate as) (enumerate as) ≡ as
+lemma-map-denumerate-enumerate []       = refl
 lemma-map-denumerate-enumerate (a ∷ as) = cong (_∷_ a) (begin
-  map (flip lookupVec (a ∷V (fromList as))) (toList (tabulate Fin.suc))
-    ≡⟨ cong (map (flip lookupVec (a ∷V (fromList as))) ∘ toList) (tabulate-∘ Fin.suc id) ⟩
-  map (flip lookupVec (a ∷V (fromList as))) (toList (Data.Vec.map Fin.suc (tabulate id)))
-    ≡⟨ cong (map (flip lookupVec (a ∷V fromList as))) (toList-map-commutes Data.Fin.suc (tabulate id)) ⟩
-  map (flip lookupVec (a ∷V fromList as)) (map Fin.suc (enumerate as))
-    ≡⟨ sym (map-∘ (enumerate as)) ⟩
-  map (flip lookupVec (a ∷V (fromList as)) ∘ Fin.suc) (enumerate as)
+  map (flip lookupVec (a ∷ as)) (tabulate Fin.suc)
+    ≡⟨ cong (map (flip lookupVec (a ∷ as))) (tabulate-∘ Fin.suc id) ⟩
+  map (flip lookupVec (a ∷ as)) (map Fin.suc (tabulate id))
+    ≡⟨ refl ⟩
+  map (flip lookupVec (a ∷ as)) (map Fin.suc (enumerate as))
+    ≡⟨ sym (map-∘ _ _ (enumerate as)) ⟩
+  map (flip lookupVec (a ∷ as) ∘ Fin.suc) (enumerate as)
     ≡⟨ refl ⟩
   map (denumerate as) (enumerate as)
     ≡⟨ lemma-map-denumerate-enumerate as ⟩
   as ∎)
 
-theorem-1 : (get : {α : Set} → List α → List α) → {τ : Set} → (eq : EqInst τ) → (s : List τ) → bff get eq s (get s) ≡ just s
+theorem-1 : {getlen : ℕ → ℕ} → (get : {α : Set} {n : ℕ} → Vec α n → Vec α (getlen n)) → {m : ℕ} {τ : Set} → (eq : EqInst τ) → (s : Vec τ m) → bff get eq s (get s) ≡ just s
 theorem-1 get eq s = begin
   bff get eq s (get s)
     ≡⟨ cong (bff get eq s ∘ get) (sym (lemma-map-denumerate-enumerate s)) ⟩
@@ -194,10 +181,10 @@ theorem-1 get eq s = begin
     ≡⟨ refl ⟩
   fmap (flip map (enumerate s) ∘ flip lookup) (fmap (flip union (fromFunc (denumerate s))) (assoc eq (get (enumerate s)) (map (denumerate s) (get (enumerate s)))))
     ≡⟨ cong (fmap (flip map (enumerate s) ∘ flip lookup) ∘ fmap (flip union (fromFunc (denumerate s)))) (lemma-1 eq (denumerate s) (get (enumerate s))) ⟩
-  fmap (flip map (enumerate s) ∘ flip lookup) (fmap (flip union (fromFunc (flip lookupVec (fromList s)))) (just (restrict (denumerate s) (get (enumerate s)))))
+  fmap (flip map (enumerate s) ∘ flip lookup) (fmap (flip union (fromFunc (flip lookupVec s))) (just (restrict (denumerate s) (toList (get (enumerate s))))))
     ≡⟨ refl ⟩
-  just ((flip map (enumerate s) ∘ flip lookup) (union (restrict (denumerate s) (get (enumerate s))) (fromFunc (denumerate s))))
-    ≡⟨ cong just (cong (flip map (enumerate s) ∘ flip lookup) (lemma-union-restrict (denumerate s) (get (enumerate s)))) ⟩
+  just ((flip map (enumerate s) ∘ flip lookup) (union (restrict (denumerate s) (toList (get (enumerate s)))) (fromFunc (denumerate s))))
+    ≡⟨ cong just (cong (flip map (enumerate s) ∘ flip lookup) (lemma-union-restrict (denumerate s) (toList (get (enumerate s))))) ⟩
   just ((flip map (enumerate s) ∘ flip lookup) (fromFunc (denumerate s)))
     ≡⟨ refl ⟩
   just (map (flip lookup (fromFunc (denumerate s))) (enumerate s))
@@ -210,14 +197,15 @@ lemma-fmap-just : {A B : Set} → {f : A → B} {b : B} → (ma : Maybe A) → f
 lemma-fmap-just (just x) fmap-f-ma≡just-b = x , refl
 lemma-fmap-just nothing  ()
 
-lemma-from-map-just : {A : Set} → {xs ys : List A} → map Maybe.just xs ≡ map Maybe.just ys → xs ≡ ys
-lemma-from-map-just {xs = []}      {ys = []}      p  = refl
-lemma-from-map-just {xs = []}      {ys = y ∷ ys'} ()
-lemma-from-map-just {xs = x ∷ xs'} {ys = []}      ()
-lemma-from-map-just {xs = x ∷ xs'} {ys = y ∷ ys'} p with ∷-injective p
+∷-injective : {A : Set} {n : ℕ} {x y : A} {xs ys : Vec A n} → (x ∷ xs) ≡ (y ∷ ys) → x ≡ y × xs ≡ ys
+∷-injective refl = refl , refl
+
+lemma-from-map-just : {A : Set} {n : ℕ} → {xs ys : Vec A n} → map Maybe.just xs ≡ map Maybe.just ys → xs ≡ ys
+lemma-from-map-just {xs = []}      {ys = []}       p  = refl
+lemma-from-map-just {xs = x ∷ xs'} {ys = y ∷ ys'}  p with ∷-injective p
 lemma-from-map-just {xs = x ∷ xs'} {ys = .x ∷ ys'} p | refl , p' = cong (_∷_ x) (lemma-from-map-just p')
 
-lemma-union-not-used : {n : ℕ} {A : Set} (h : FinMapMaybe n A) → (h' : FinMap n A) → (is : List (Fin n)) → is in-domain-of h → map just (map (flip lookup (union h h')) is) ≡ map (flip lookupM h) is
+lemma-union-not-used : {m n : ℕ} {A : Set} (h : FinMapMaybe n A) → (h' : FinMap n A) → (is : Vec (Fin n) m) → (toList is) in-domain-of h → map just (map (flip lookup (union h h')) is) ≡ map (flip lookupM h) is
 lemma-union-not-used h h' []        p = refl
 lemma-union-not-used h h' (i ∷ is') p with Data.List.All.head p
 lemma-union-not-used h h' (i ∷ is') p | x , lookupM-i-h≡just-x = begin
@@ -236,7 +224,7 @@ lemma-union-not-used h h' (i ∷ is') p | x , lookupM-i-h≡just-x = begin
     ≡⟨ cong (_∷_ (lookupM i h)) (lemma-union-not-used h h' is' (Data.List.All.tail p)) ⟩
   lookupM i h ∷ map (flip lookupM h) is' ∎
 
-theorem-2 : (get : {α : Set} → List α → List α) → {τ : Set} → (eq : EqInst τ) → (v s u : List τ) → bff get eq s v ≡ just u → get u ≡ v
+theorem-2 : {getlen : ℕ → ℕ} (get : {α : Set} {n : ℕ} → Vec α n → Vec α (getlen n)) → {τ : Set} {m : ℕ} → (eq : EqInst τ) → (v : Vec τ (getlen m)) → (s u : Vec τ m) → bff get eq s v ≡ just u → get u ≡ v
 theorem-2 get eq v s u p with lemma-fmap-just (assoc eq (get (enumerate s)) v) (proj₂ (lemma-fmap-just (fmap (flip union (fromFunc (denumerate s))) (assoc eq (get (enumerate s)) v)) p))
 theorem-2 get eq v s u p | h , ph = begin
   get u