NCSimplify1Rational[expr]

Aliases: NCS1R

Description: It applies a collection of relations implemented as rules to $expr$. The goal is to simplify $expr$.

Arguments: $expr$ is an algebraic expression.

Comments / Limitations: WARNING: NCS1R does not first do an ExpandNonCommutativeMultiply. Therefore, it may be the case that one can miss some simplification if $expr$ is not expanded out. The solution, of course, is to call ExpandNonCommutativeMultiply before calling NCS1R. ExpandNonCommutativeMultiply is called from NCSR.

First, NCS1R normalizes $inv[S-a**b]$ to $S^{-1}*inv[1-{(a**b)\over S}]$ provided S is s a commutative expression (only works for numbers S in version 0.2 of NCAlgebra). The the following list of rules are applied.
(0) $inv[-1 + a] \rightarrow - inv[1-a]$
(1) $inv[1-a] (a-b) inv[1-b] \rightarrow inv[1-a] - inv[1-b]$
(2) $inv[1-ab] inv[b] \rightarrow inv[1-ba] a +inv[b]$
(3) $inv[1-ab] a b \rightarrow inv[1-ab] - 1$
(4) $a b inv[1-ab] \rightarrow inv[1-ab] - 1$
(5) $inv[c] inv[1-cb] \rightarrow inv[1-bc] inv[c]$
(6) $b inv[1-ab] \rightarrow inv[1-ba] b$

In the notation of papers [HW], [HSW], these rules implement a superset of the union of the Gröbner basis for EB and the Gröbner basis for RESOL.