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Simulating the behaviour of Plus and Times.

• To: mathgroup at smc.vnet.net
• Subject: [mg75858] Simulating the behaviour of Plus and Times.
• From: Szabolcs <szhorvat at gmail.com>
• Date: Sat, 12 May 2007 03:03:17 -0400 (EDT)
• Organization: University of Bergen

Is it possible to define two functions whose behaviour is the same as 
that of Plus, Times and Minus in circumstances similar to the ones 
illustrated below?

In[1]:= 1+2
Out[1]= 3

In[2]:= a + a
Out[2]= 2 a

In[3]:= 3a + 5a
Out[3]= 8 a

In[4]:= x a + y a
Out[4]= a x + a y

In[5]:= 3(a+b) + 2(a+b)
Out[5]= 5 (a+b)

In[6]:= 0a
Out[6]= 0

In[7]:= a-a
Out[7]= 0

Could someone show me how to define two such functions, plus and times, 
that operate on integers represented as num[someNumber]? Example:

In := plus[times[num[2], plus[a, b]], times[plus[a, b], num[3]]]
Out = times[num[5], plus[a, b]]

In := plus[times[num[-1], a], a]
Out = num[0]