Use new notion of Fin to reduce casts.

1 files changed, 66 insertions, 32 deletions

diff --git a/src/Data/Term/Zipper.idr b/src/Data/Term/Zipper.idr
index 3f6668a..608e86c 100644
--- a/src/Data/Term/Zipper.idr
+++ b/src/Data/Term/Zipper.idr
@@ -2,17 +2,53 @@ module Data.Term.Zipper
 
 import Data.DPair
 import Data.Fin.Occurs
-import Data.Vect.Properties.Insert
+import Data.Fin.Strong
 import Data.Vect.Properties.Map
 import Data.Term
 import Syntax.PreorderReasoning
 
+-- Utilities -------------------------------------------------------------------
+
+public export
+insertAt' : SFin k -> a -> Vect (pred k) a -> Vect k a
+insertAt' FZ x xs = x :: xs
+insertAt' (FS i) x (y :: xs) = y :: insertAt' i x xs
+
+public export
+deleteAt' : SFin k -> Vect k a -> Vect (pred k) a
+deleteAt' FZ (x :: xs) = xs
+deleteAt' (FS i) (x :: xs) = x :: deleteAt' i xs
+
+mapInsertAt :
+  (0 f : a -> b) ->
+  (i : SFin k) ->
+  (x : a) ->
+  (xs : Vect (pred k) a) ->
+  map f (insertAt' i x xs) = insertAt' i (f x) (map f xs)
+mapInsertAt f FZ x xs = Refl
+mapInsertAt f (FS i) x (y :: xs) = cong (f y ::) $ mapInsertAt f i x xs
+
+insertAtDeleteAt :
+  (i : SFin k) ->
+  (xs : Vect k a) ->
+  insertAt' i (index (strongToFin i) xs) (deleteAt' i xs) = xs
+insertAtDeleteAt FZ (x :: xs) = Refl
+insertAtDeleteAt (FS i) (x :: xs) = cong (x ::) $ insertAtDeleteAt i xs
+
+indexInsertAt :
+  (i : SFin k) ->
+  (x : a) ->
+  (xs : Vect (pred k) a) ->
+  index (strongToFin i) (insertAt' i x xs) = x
+indexInsertAt FZ x xs = Refl
+indexInsertAt (FS i) x (y :: xs) = indexInsertAt i x xs
+
 -- Definition ------------------------------------------------------------------
 
 public export
 data Zipper : Signature -> Nat -> Type where
   Top : Zipper sig n
-  Op : sig.Operator (S k) -> Fin (S k) -> Vect k (Term sig n) -> Zipper sig n -> Zipper sig n
+  Op : sig.Operator k -> SFin k -> Vect (pred k) (Term sig n) -> Zipper sig n -> Zipper sig n
 
 %name Zipper zip
 
@@ -45,7 +81,7 @@ assoc (Op op i ts zip1) zip2 zip3 = cong (Op op i ts) (assoc zip1 zip2 zip3)
 public export
 (+) : Zipper sig n -> Term sig n -> Term sig n
 Top + t = t
-Op op i ts zip + t = Op op (insertAt i (zip + t) ts)
+Op op i ts zip + t = Op op (insertAt' i (zip + t) ts)
 
 -- Properties
 
@@ -56,12 +92,12 @@ actionAssoc :
   (zip1 ++ zip2) + t = zip1 + (zip2 + t)
 actionAssoc Top zip2 t = Refl
 actionAssoc (Op op i ts zip1) zip2 t =
-  cong (\t => Op op (insertAt i t ts)) $ actionAssoc zip1 zip2 t
+  cong (\t => Op op (insertAt' i t ts)) $ actionAssoc zip1 zip2 t
 
 -- Substitution ----------------------------------------------------------------
 
 export
-(<$>) : (Fin k -> Term sig n) -> Zipper sig k -> Zipper sig n
+(<$>) : (SFin k -> Term sig n) -> Zipper sig k -> Zipper sig n
 f <$> Top = Top
 f <$> Op op i ts zip = Op op i (map (f <$>) ts) (f <$> zip)
 
@@ -69,35 +105,27 @@ f <$> Op op i ts zip = Op op i (map (f <$>) ts) (f <$> zip)
 
 export
 actionHomo :
-  (0 f : Fin k -> Term sig n) ->
+  (0 f : SFin k -> Term sig n) ->
   (zip : Zipper sig k) ->
   (0 t : Term sig k) ->
   f <$> zip + t = (f <$> zip) + (f <$> t)
 actionHomo f Top t = Refl
 actionHomo f (Op op i ts zip) t = cong (Op op) $ Calc $
-  |~ map (f <$>) (insertAt i (zip + t) ts)
-  ~~ insertAt i (f <$> zip + t) (map (f <$>) ts)           ...(mapInsertAt (f <$>) i (zip + t) ts)
-  ~~ insertAt i ((f <$> zip) + (f <$> t)) (map (f <$>) ts) ...(cong (\t => insertAt i t (map (f <$>) ts)) $ actionHomo f zip t)
+  |~ map (f <$>) (insertAt' i (zip + t) ts)
+  ~~ insertAt' i (f <$> zip + t) (map (f <$>) ts)           ...(mapInsertAt (f <$>) i (zip + t) ts)
+  ~~ insertAt' i ((f <$> zip) + (f <$> t)) (map (f <$>) ts) ...(cong (\t => insertAt' i t (map (f <$>) ts)) $ actionHomo f zip t)
 
 -- Cycles ----------------------------------------------------------------------
 
-pivotAt : sig.Operator k -> Fin k -> Vect k (Term sig n) -> Zipper sig n
-pivotAt op i ts =
-  let op' = replace {p = sig.Operator} (snd prf) op in
-  let i' = replace {p = Fin} (snd prf) i in
-  let ts' = replace {p = flip Vect (Term sig n)} (snd prf) ts in
-  Op op' i' (deleteAt i' ts') Top
-  where
-  prf : Exists (\j => k = S j)
-  prf = indexIsSuc i
+pivotAt : sig.Operator k -> SFin k -> Vect k (Term sig n) -> Zipper sig n
+pivotAt op i ts = Op op i (deleteAt' i ts) Top
 
 actionPivotAt :
   (op : sig.Operator k) ->
-  (i : Fin k) ->
+  (i : SFin k) ->
   (ts : Vect k (Term sig n)) ->
-  pivotAt op i ts + index i ts = Op op ts
-actionPivotAt op i@FZ ts = cong (Op op) $ insertAtIndex i ts
-actionPivotAt op i@(FS _) ts = cong (Op op) $ insertAtIndex i ts
+  pivotAt op i ts + index (strongToFin i) ts = Op op ts
+actionPivotAt op i ts = cong (Op op) $ insertAtDeleteAt i ts
 
 pivotNotTop : (zip : Zipper sig k) -> Not (zip ++ pivotAt op i ts = Top)
 pivotNotTop (Op op' j us zip) prf impossible
@@ -108,17 +136,23 @@ noCycles Top t prf = Refl
 noCycles (Op {k} op i ts zip') t@(Op {k = j} op' us) prf =
   void $ pivotNotTop zip' prf''
   where
-  I : Fin j
-  I = replace {p = Fin} (opInjectiveLen prf) i
+  i' : Fin k
+  i' = strongToFin i
+
+  I : SFin j
+  I = replace {p = SFin} (opInjectiveLen prf) i
+
+  I' : Fin j
+  I' = strongToFin I
 
-  prf' : ((zip' ++ pivotAt op' I us) + index I us = index I us)
+  prf' : ((zip' ++ pivotAt op' I us) + index I' us = index I' us)
   prf' = Calc $
-    |~ (zip' ++ pivotAt op' I us) + index I us
-    ~~ zip' + (pivotAt op' I us + index I us)      ...(actionAssoc zip' (pivotAt op' I us) (index I us))
-    ~~ zip' + Op op' us                            ...(cong (zip' +) $ actionPivotAt op' I us)
-    ~~ index i (insertAt i (zip' + Op op' us) ts)  ..<(indexInsertAt i (zip' + Op op' us) ts)
-    ~~ index i (rewrite opInjectiveLen prf in us)  ...(cong (index i) $ opInjectiveTs prf)
-    ~~ index I us                                  ...(rewrite opInjectiveLen prf in Refl)
+    |~ (zip' ++ pivotAt op' I us) + index I' us
+    ~~ zip' + (pivotAt op' I us + index I' us)      ...(actionAssoc zip' (pivotAt op' I us) (index I' us))
+    ~~ zip' + Op op' us                             ...(cong (zip' +) $ actionPivotAt op' I us)
+    ~~ index i' (insertAt' i (zip' + Op op' us) ts) ..<(indexInsertAt i (zip' + Op op' us) ts)
+    ~~ index i' (rewrite opInjectiveLen prf in us)  ...(cong (index i') $ opInjectiveTs prf)
+    ~~ index I' us                                  ...(rewrite opInjectiveLen prf in Refl)
 
   prf'' : zip' ++ pivotAt op' I us = Top
-  prf'' = noCycles (zip' ++ pivotAt op' I us) (assert_smaller t (index I us)) prf'
+  prf'' = noCycles (zip' ++ pivotAt op' I us) (assert_smaller t (index I' us)) prf'