1 files changed, 91 insertions, 5 deletions

diff --git a/src/Encoded/Arith.idr b/src/Encoded/Arith.idr
index d2f83bc..5097664 100644
--- a/src/Encoded/Arith.idr
+++ b/src/Encoded/Arith.idr
@@ -1,9 +1,14 @@
 module Encoded.Arith
 
+import Data.Nat
 import Encoded.Bool
 import Encoded.Pair
+import Syntax.PreorderReasoning
+import Term.Semantics
 import Term.Syntax
 
+-- Utilities -------------------------------------------------------------------
+
 export
 rec : {ty : Ty} -> Term (N ~> ty ~> (N * ty ~> ty) ~> ty) ctx
 rec = Abs $ Abs $ Abs $
@@ -32,10 +37,14 @@ pred = Abs'
   (\n => App
     (Rec n
       (Const Zero)
-      (Abs' (\f =>
-        Rec (App f [<Zero])
-          (Abs' (\k => Rec k (Suc Zero) (Const Zero)))
-          (Const $ Abs' (\k => Rec k (Suc Zero) (Abs' Suc . shift f))))))
+      (Abs $ Abs $
+        let f = Var (There Here) in
+        let k = Var Here in
+        Rec k
+          (Suc Zero)
+          (Const $ Rec (App f [<Zero])
+            Zero
+            (Const $ Suc $ App f [<Lit 1]))))
     [<Lit 1])
 
 export
@@ -76,6 +85,83 @@ divmod = Abs $ Abs $
     (App pair [<Zero, Zero])
     (Abs' (\p =>
       App if'
-        [<App lte [<shift d, App snd [<p]]
+        [<App lte [<shift d, App (Abs' Suc . snd) [<p]]
         , App pair [<Suc (App fst [<p]), Zero]
         , App mapSnd [<Abs' Suc, p]]))
+
+-- Properties ------------------------------------------------------------------
+
+-- Redefinition in Idris
+
+namespace Sem
+  public export
+  recHelper : Nat -> a -> ((Nat, a) -> a) -> (Nat, a)
+  recHelper k z s = rec k (0, z) (\p => (S (fst p), s p))
+
+  public export
+  rec' : Nat -> a -> ((Nat, a) -> a) -> a
+  rec' k z s = snd (recHelper k z s)
+
+  public export
+  slowPred : Nat -> Nat
+  slowPred n = rec' n 0 fst
+
+  public export
+  predHelper : Nat -> Nat -> Nat
+  predHelper n =
+    rec n
+      (const 0)
+      (\f, k =>
+        rec k
+          1
+          (const $ rec (f 0)
+            0
+            (const $ S $ f 1)))
+
+  public export
+  pred' : Nat -> Nat
+  pred' n = predHelper n 1
+
+  public export
+  divmodStep : Nat -> (Nat, Nat) -> (Nat, Nat)
+  divmodStep d (div, mod) =
+    if d <= S mod
+    then
+      (S div, 0)
+    else
+      (div, S mod)
+
+  public export
+  divmod : Nat -> Nat -> (Nat, Nat)
+  divmod n d = rec n (0, 0) (divmodStep d)
+
+-- Proofs
+
+export
+fstRecHelper : (k : Nat) -> (0 z : a) -> (0 s : (Nat, a) -> a) -> fst (recHelper k z s) = k
+fstRecHelper 0 z s = Refl
+fstRecHelper (S k) z s = cong S $ fstRecHelper k z s
+
+export
+slowPredCorrect : (k : Nat) -> slowPred k = pred k
+slowPredCorrect 0 = Refl
+slowPredCorrect (S k) = fstRecHelper k 0 fst
+
+export
+predCorrect : (k : Nat) -> pred' k = pred k
+predCorrect 0 = Refl
+predCorrect 1 = Refl
+predCorrect (S (S k)) = cong S $ predCorrect (S k)
+
+-- divmodSmall :
+--   (d, div, mod : Nat) ->
+--   {auto 0 ok : NonZero d} ->
+  -- mod <= d
+
+export
+divmodCorrect :
+  (n, d : Nat) ->
+  {auto 0 ok : NonZero d} ->
+  n = fst (divmod n d) * d + snd (divmod n d)
+divmodCorrect 0 d = Refl
+divmodCorrect (S k) d = ?divmodCorrect_rhs_1