With respect to a free basis, the word length of an element is the length of its reduced word

Given: The hypotheses of the Statement.

The word length $\mathrm{\mid }g{\mathrm{\mid }}_S$ is the least $n$ such that $g$ is a product of $n$ elements of $S\cup S^{-1}$ (Word length of a group element with respect to a generating set).

Word length is defined on every element and satisfies $\mathrm{\mid }gh{\mathrm{\mid }}_S\le \mathrm{\mid }g{\mathrm{\mid }}_S+\mathrm{\mid }h{\mathrm{\mid }}_S$, $\mathrm{\mid }{g}^{-1}{\mathrm{\mid }}_S=\mathrm{\mid }g{\mathrm{\mid }}_S$, and $\mathrm{\mid }g{\mathrm{\mid }}_S=0$ exactly when $g$ is the identity (Word length is defined on every element and satisfies the subadditivity, inversion and vanishing laws).

The Cayley graph of a free group with respect to a free basis is a tree (The Cayley graph of a free group with respect to a free basis is a tree).

An elementary cancellation deletes two adjacent letters $x{x}^{-1}$ or $x^{-1}x$. A word is reduced if no elementary cancellation applies. (Words in an alphabet with formal inverses, elementary cancellation, and reduced words).

The reduced words on $X\bigsqcup X^{-1}$ form a group when the product of reduced words is their concatenation followed by free reduction. (Reduced words form the free group on an alphabet).

Every class in $W(X)\mathrm{/}\sim$ contains exactly one reduced word. (Every class in $W(X)\mathrm{/}\sim$ contains exactly one reduced word).

The subset $B$ is a free basis of $F$ if $(F,i)$ is a free group on the set $B$ in the sense of. (A free basis of a group).