a-[b-c+a-{b-(c-a-a+b)}]

First we look for the innermost grouping symbols, that is,
the grouping symbols which have no grouping symbols within them,

I will color them red:

a-[b-c+a-{b-(c-a-a+b)}]

We combine the "-a-a" by considering it as "-1a-1a" to get "-2a",
and we replace "-a-a" by "-2a" and copy everything else over:

a-[b-c+a-{b-(c-2a+b)}]

Next we remove those innermost grouping symbols by the rule that
since it is preceded by a "-" sign we drop the grouping symbols by
changing all the signs before all the terms.  So we replace 
"-(c-2a+b)" by "-c+2a-b" and copy everything else over:

a-[b-c+a-{b-c+2a-b}]

Next we look for the innermost grouping symbols again, that is,
the grouping symbols which have no grouping symbols within them,

I will color them red:

a-[b-c+a-{b-c+2a-b}]

We see within the { } that thee "b" and the "-b" add to zero, so
we just eliminate them and copy everything else over:

a-[b-c+a-{-c+2a}]

There are no like terms to combine within those grouping symbols,
so we remove those grouping symbols again by the rule that since 
it is preceded by a "-" sign we drop the grouping symbols by 
changing all the signs before all the terms.  So we replace 
"-{-c+2a}" by "+c-2a" and copy everything else over:

a-[b-c+a+c-2a]

We see within the [ ] that the "-c" and the "+c" add to zero, so
we just eliminate them and copy everything else over:

a-[b+a-2a]

We combine the "+a-2a" by considering it as "+1a-2a" to get "-1a",
or just "-a" and we replace "+a-2a" by "-a" and copy everything else 
over:

a-[b-a]

There are no like terms to combine within those final grouping 
symbols, so we remove those final grouping symbols again by the rule 
that since it is preceded by a "-" sign we drop the grouping symbols 
by  changing all the signs before all the terms.  So we replace 
"-[b-a]" by "-b+a" and copy everything else over:

a-b+a

We combine the "a" and "+a" by considering them as "1a" and "1a" to get 
"2a" and we replace them by by "2a" and copy everything else 
over, winding up with

2a-b

Edwin