* algebra/taylor.spad.pamphlet (InnerTaylorSeries): Now satisfies

	BiModule(Coef,Coef).  Remove duplicate exports.  Use rep and per
	operators. 

1 files changed, 25 insertions, 35 deletions

diff --git a/src/algebra/taylor.spad.pamphlet b/src/algebra/taylor.spad.pamphlet
index 48f29f56..3f4aa0ef 100644
--- a/src/algebra/taylor.spad.pamphlet
+++ b/src/algebra/taylor.spad.pamphlet
@@ -14,7 +14,7 @@
 )abbrev domain ITAYLOR InnerTaylorSeries
 ++ Author: Clifton J. Williamson
 ++ Date Created: 21 December 1989
-++ Date Last Updated: 25 February 1989
+++ Date Last Updated: October 26, 2009
 ++ Basic Operations:
 ++ Related Domains: UnivariateTaylorSeries(Coef,var,cen)
 ++ Also See:
@@ -36,7 +36,7 @@ InnerTaylorSeries(Coef): Exports == Implementation where
   ST  ==> Stream Coef
   STT ==> StreamTaylorSeriesOperations Coef
 
-  Exports ==> Ring with
+  Exports == Join(Ring, BiModule(Coef,Coef)) with
     coefficients: % -> Stream Coef
       ++\spad{coefficients(x)} returns a stream of ring elements.
       ++ When x is a univariate series, this is a stream of Taylor
@@ -59,41 +59,29 @@ InnerTaylorSeries(Coef): Exports == Implementation where
       ++ i.e. the degree of the first non-zero term of the series.
     order: (%,NNI) -> NNI
       ++\spad{order(x,n)} returns the minimum of n and the order of x.
-    * : (Coef,%)->%
-      ++\spad{c*x} returns the product of c and the series x.
-    * : (%,Coef)->%
-      ++\spad{x*c} returns the product of c and the series x.
     * : (%,Integer)->%
       ++\spad{x*i} returns the product of integer i and the series x.
     if Coef has IntegralDomain then IntegralDomain
       --++ An IntegralDomain provides 'exquo'
 
-  Implementation ==> add
-
-    Rep := Stream Coef
+  Implementation == add
 
---% declarations
-    x,y: %
-
---% definitions
+    Rep == ST
 
     -- In what follows, we will be calling operations on Streams
     -- which are NOT defined in the package Stream.  Thus, it is
     -- necessary to explicitly pass back and forth between Rep and %.
-    -- This will be done using the functions 'stream' and 'series'.
 
-    stream : % -> Stream Coef
-    stream x  == x pretend Stream(Coef)
-    series st == st pretend %
+    series st == per st
 
-    0 == coerce(0)$STT
-    1 == coerce(1)$STT
+    0 == per coerce(0)$STT
+    1 == per coerce(1)$STT
 
     x = y ==
       -- tests if two power series are equal
       -- difference must be a finite stream of zeroes of length <= n + 1,
       -- where n = $streamCount$Lisp
-      st : ST := stream(x - y)
+      st : ST := rep(x - y)
       n : I := _$streamCount$Lisp
       for i in 0..n repeat
         empty? st => return true
@@ -101,26 +89,26 @@ InnerTaylorSeries(Coef): Exports == Implementation where
         st := rst st
       empty? st
 
-    coefficients x == stream x
+    coefficients x == rep x
 
-    x + y            == stream(x) +$STT stream(y)
-    x - y            == stream(x) -$STT stream(y)
-    (x:%) * (y:%)    == stream(x) *$STT stream(y)
-    - x              == -$STT (stream x)
-    (i:I) * (x:%)    == (i::Coef) *$STT stream x
-    (x:%) * (i:I)    == stream(x) *$STT (i::Coef)
-    (c:Coef) * (x:%) == c *$STT stream x
-    (x:%) * (c:Coef) == stream(x) *$STT c
+    x + y            == per(rep x +$STT rep y)
+    x - y            == per(rep x -$STT rep y)
+    (x:%) * (y:%)    == per(rep x *$STT rep y)
+    - x              == per(-$STT rep x)
+    (i:I) * (x:%)    == per((i::Coef) *$STT rep x)
+    (x:%) * (i:I)    == per(rep x *$STT (i::Coef))
+    (c:Coef) * (x:%) == per(c *$STT rep x)
+    (x:%) * (c:Coef) == per(rep x *$STT c)
 
     recip x ==
-      (rec := recip$STT stream x) case "failed" => "failed"
-      series(rec :: ST)
+      (rec := recip$STT rep x) case "failed" => "failed"
+      per(rec :: ST)
 
     if Coef has IntegralDomain then
 
       x exquo y ==
-        (quot := stream(x) exquo$STT stream(y)) case "failed" => "failed"
-        series(quot :: ST)
+        (quot := rep x exquo$STT rep y) case "failed" => "failed"
+        per(quot :: ST)
 
     x:% ** n:NNI ==
       n = 0 => 1
@@ -135,7 +123,7 @@ InnerTaylorSeries(Coef): Exports == Implementation where
       zero? frst st => iOrder(rst st,n + 1,n0)
       n
 
-    order(x,n) == iOrder(stream x,0,n)
+    order(x,n) == iOrder(rep x,0,n)
 
     iOrder2: (ST,NNI) -> NNI
     iOrder2(st,n) ==
@@ -143,7 +131,9 @@ InnerTaylorSeries(Coef): Exports == Implementation where
       zero? frst st => iOrder2(rst st,n + 1)
       n
 
-    order x == iOrder2(stream x,0)
+    order x == iOrder2(rep x,0)
+
+    zero? x == empty? rep x
 
 @
 \section{domain UTS UnivariateTaylorSeries}