This page discusses cases where instantiating of a parameter collapses two generically different functions.

The page is connected to the thread http://groups.google.com/group/fricas-devel/browse_frm/thread/5097633953c6841d/c3cb9671206609f8#c3cb9671206609f8

On Tue, Mar 24, 2009 at 4:24 AM, Ralf Hemmecke wrote:

Aldor's libalgebra has some bugs in it that seem to be really hard to track down. I faintly remember one such bug that I could nearly assign to the coexistence of:

  *: (Integer, %) -> % and *: (R, %) -> %

where R is instantiated with Integer. The problem here is a hidden overloading of implementations. Suppose you implement something like

#include "axiom"

MyWeirdPoly(R: Ring): Ring with {
  *: (Integer, %) -> %;
  *: (R, %) -> %;
  coerce: R -> %;
  -- ...
} == Integer add {
  coerce(r:R):% == r pretend %;
  ((i: Integer) * (x: %)): % == ((i+i)::%)*x;
  ((r: R) * (x: %)): % == (r::%) * x
  -- ...
}

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6603601906039884450-25px001.as
      using Aldor compiler and options 
-O -Fasy -Fao -Flsp -lfricas -Mno-ALDOR_W_WillObsolete -DFriCAS -Y $FRICAS/algebra -I $FRICAS/algebra
      Use the system command )set compiler args to change these 
      options.
   The )library system command was not called after compilation.

)show MyWeirdPoly

   The )show system command is used to display information about types 
      or partial types. For example, )show Integer will show 
      information about Integer .

      MyWeirdPoly is not the name of a known type constructor. If you 
      want to see information about any operations named MyWeirdPoly , 
      issue
                       )display operations MyWeirdPoly

p:=1$MyWeirdPoly(Integer)

   There are no library operations named MyWeirdPoly 
      Use HyperDoc Browse or issue
                            )what op MyWeirdPoly
      to learn if there is any operation containing " MyWeirdPoly " in 
      its name.

   Cannot find a definition or applicable library operation named 
      MyWeirdPoly with argument type(s) 
                                    Type

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Now lets try the same thing in SPAD

)abbrev package MWP2 MyWeirdPoly2
MyWeirdPoly2(R: Ring): Ring with
  _*: (Integer, %) -> %
  _*: (R, %) -> %
  coerce: R -> %
  -- ...
 == Integer add
  coerce(r:R):% == r pretend %
  ((i: Integer) * (x: %)): % == ((i+i)::%)*x
  ((r: R) * (x: %)): % == (r::%) * x
  -- ...

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4436499139492326087-25px004.spad
      using old system compiler.
   MWP2 abbreviates package MyWeirdPoly2 
------------------------------------------------------------------------
   initializing NRLIB MWP2 for MyWeirdPoly2 
   compiling into NRLIB MWP2 
   compiling exported coerce : R -> $
      MWP2;coerce;R$;1 is replaced by r 
Time: 0.01 SEC.

   compiling exported * : (Integer,$) -> $
Time: 0 SEC.

   compiling exported * : (R,$) -> $
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |MyWeirdPoly2| REDEFINED

;;;     ***       |MyWeirdPoly2| REDEFINED
Time: 0 SEC.

   Cumulative Statistics for Constructor MyWeirdPoly2
      Time: 0.01 seconds

--------------non extending category----------------------
.. MyWeirdPoly2(#1) of cat 
(|Join| (|Ring|)
        (CATEGORY |domain| (SIGNATURE * ($ (|Integer|) $))
         (SIGNATURE * ($ |#1| $)) (SIGNATURE |coerce| ($ |#1|))))   has no 
(|IntegerNumberSystem|)    finalizing NRLIB MWP2 
   Processing MyWeirdPoly2 for Browser database:
--->-->MyWeirdPoly2(constructor): Not documented!!!!
--->-->MyWeirdPoly2((* (% (Integer) %))): Not documented!!!!
--->-->MyWeirdPoly2((* (% R %))): Not documented!!!!
--->-->MyWeirdPoly2((coerce (% R))): Not documented!!!!
--->-->MyWeirdPoly2(): Missing Description
; compiling file "/var/aw/var/LatexWiki/MWP2.NRLIB/MWP2.lsp" (written 04 APR 2022 07:14:55 PM):

; /var/aw/var/LatexWiki/MWP2.NRLIB/MWP2.fasl written
; compilation finished in 0:00:00.015
------------------------------------------------------------------------
   MyWeirdPoly2 is now explicitly exposed in frame initial 
   MyWeirdPoly2 will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/MWP2.NRLIB/MWP2

)show MyWeirdPoly2

 MyWeirdPoly2(R: Ring) is a domain constructor
 Abbreviation for MyWeirdPoly2 is MWP2 
 This constructor is exposed in this frame.
------------------------------- Operations --------------------------------

 ?*? : (R, %) -> %                     ?*? : (Integer, %) -> %
 ?*? : (%, %) -> %                     ?*? : (PositiveInteger, %) -> %
 ?+? : (%, %) -> %                     -? : % -> %
 ?-? : (%, %) -> %                     ?=? : (%, %) -> Boolean
 1 : () -> %                           0 : () -> %
 ?^? : (%, PositiveInteger) -> %       annihilate? : (%, %) -> Boolean
 antiCommutator : (%, %) -> %          associator : (%, %, %) -> %
 coerce : R -> %                       coerce : Integer -> %
 coerce : % -> OutputForm              commutator : (%, %) -> %
 hash : % -> SingleInteger             latex : % -> String
 one? : % -> Boolean                   opposite? : (%, %) -> Boolean
 recip : % -> Union(%,"failed")        sample : () -> %
 zero? : % -> Boolean                  ?~=? : (%, %) -> Boolean
 ?*? : (NonNegativeInteger, %) -> %
 ?^? : (%, NonNegativeInteger) -> %
 characteristic : () -> NonNegativeInteger
 hashUpdate! : (HashState, %) -> HashState
 leftPower : (%, NonNegativeInteger) -> %
 leftPower : (%, PositiveInteger) -> %
 leftRecip : % -> Union(%,"failed")
 rightPower : (%, NonNegativeInteger) -> %
 rightPower : (%, PositiveInteger) -> %
 rightRecip : % -> Union(%,"failed")
 subtractIfCan : (%, %) -> Union(%,"failed")

q:=1$MyWeirdPoly2(Integer)

(1)

3*q

(2)

Surprise!

SPAD produces the opposite result from Aldor.