# Groups are numbers (2). Pattern matching

As in divination, pattern matching. Continues from Groups are numbers (1).

We start from elementary variables, then we define number terms by two operations: substraction and multiplication.

• Variables are terms.
• Substraction (the first line): a is a variable and b is a term, then $a-b$ is a term.
• Multiplication (2nd line): a, b are terms, then $ab$ is a term.

By pattern matching we can prove for example this:

[update: figure replaced, the initial one was wrong by pattern matching only. The difference is that in this correct figure appears “(a-b)d” instead of the wrong “d(a-b)”]

What does it mean? These are just binary trees. Well let’s take a typing convention

where e, x, … are elements of a vector space and the variables are invertible scalars. Moreover take $e = 0$ for simplicity.

Then the previous pattern matching dream says that

$(1-(a-b))c x + (a-b)(c-d) x = (c - (a-b)d)x$

which is true, but from all the irrelevant reasons (vector space, associativity and commutativity  of addition, distributivity, etc):

$(c- ac + bc + ac -ad - bc + bd) x = (c - ad + bd) x = (c - (a-b)d)x$

With the previous typing conventions it reads:

$(c-(a-b))x = (1-b)(c-a)x + b(1- a^{-1})(c-a) x + (bc a^{-1})x$

which is true because the right hand side is:

$((1-b)(c-a) + b(1- a^{-1})(c-a) + bc a^{-1} )x =$

$= (c-a-bc+ab+bc-ab-bca^{-1} +b+bc a^{-1}) x = (c-a+b) x = (c-(a-b))x$

Which is funny because it does not make any sense.