Let $\sigma$ be a permutation. If two positive braids represent $\sigma$ and are of minimal length among the braids representing $\sigma$, then they are equal. From what I could gather, this result is Theorem 9.2.5 in:

Epstein, David, et al. Word processing in groups. AK Peters, Ltd., 1992.

This book refers to a paper by Garside called "The Braid group and other groups", but I can't seem to find the result in Garside's paper.

I was told that this result can be generalised, replacing the positive braids by elements of an Artin monoid, and the permutation group by the associated Coxeter group. Where can this result be found?

I was told that this result appears in "Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras", but I could not find it, probably because I am not familiar enough with the theory to recognise the result that I am looking for.