I was wondering what kind of tools are available (if any) for avoiding pages of bracket manipulations in proving associativity properties.


To be concrete, a (more easily stated) analogue of the particular problem I have in mind is:


We have $M$ a magma (a set with a binary operation) and six elements $a, b, c, x, y, z$. I want to prove that $(xy)z = x(yz)$ and I know that


i) right composition with any of $a, b, c$ is injective (e.g. the map $M \to M$ given by $m \mapsto ma$ is injective),


ii) we have the associativity relation for any triple that does not contain two consecutive expressions that contain one of $x, y, z$, (e.g. $z(ab) = (za)b$ and $y((ab)z) = (y(ab))z$),


iii) any pair consisting of one element of $a, b, c$ and one from $z, y, x$ is commutative (e.g. $ax = xa$).


The idea is to convert $x(y(z(a(bc))))$ into $a(x(b(y(cz))))$ where we can then rearrange the brackets.


It feels to me like there should exist a framework for doing this cleanly without having to dirty my hands or the readers eyes with pages of equations (which by the way, I have already written out for my version of the problem and am dreading converting into LaTeX).


A potential framework to my mind is to write such a nested bracketed expression like this as a tree with the $x, y, z, a, b, c$ as leaves. Then the relations i, ii, iii correspond to operations converting one tree to another. In a perfect world, one can prove that my relations are sufficient to generate all possible trees with $a, b, c, x, y, z$ as leaves.



Proving associativity relations by using a group acting on a set of trees?
[Thompson's group $F$](http://arxiv.org/abs/math/0505481) is the group "responsible" for associativity relations. It does act on binary trees. I am not sure it will help you much because you would probably have to explain to the readers what Thompson's group is, but in fact proving associativity relation amounts to manipulating with elements of $F$. 


