The same is true for words of the form (w_G(c)) in (G( MM _k)) by the definition of the operation, relations (G1), (G2) and Lemma 4.5.
By Lemma 3.3 W is a subword of a word of the form (w(i epsilon _1,ldots,epsilon _k)) (up to the commutativity relations).
All these words are subwords of words of the form (w(i';epsilon _1,ldots,epsilon _k)) such that the configurations ((i epsilon _1,ldots,epsilon _k)) and ((i',epsilon _1',ldots,epsilon _k')) of M are equivalent.
The vertex set V consists of the following: all generators y i ( 1 ≤ i ≤ n ) and x of P M, words of the form y 1 α i 1 y 2 α i 2 ⋯ y n α i n ( 1 ≤ i ≤ n ) in the presentation (1), words of the form y i y i + 1 ( 1 ≤ i ≤ n − 1 ) in (1).
all generators y i ( 1 ≤ i ≤ n ) and x of P M, words of the form y 1 α i 1 y 2 α i 2 ⋯ y n α i n ( 1 ≤ i ≤ n ) in the presentation (1), words of the form y i y i + 1 ( 1 ≤ i ≤ n − 1 ) in (1).
Every word W in the generators of (S( MM _k)) that is not equal to 0 in (S( MM _k)) is, modulo the commutativity relations, a subword of some word of the form (w(i epsilon _1,ldots,epsilon _k)).
(a) Indeed every non-zero element of length (le n) of (check{S}) is represented (modulo the commutativity relations) by a word of the form (q_ia_1^{m_1}ldots A_k^{epsilon^{epsilon _k}ldots A_k^{epsilon _k}) where (m_jle n), (epsilon _jin {0,1}).
(Here, we omitted the words of the remaining format. In other words, we do not take y i y i + 2, y i y i + 3, etc. as a vertex in our set), word of the form x k if l ≠ 1 in the presentation (1).
In his translation of De interpretatione, Boethius preserves Aristotle's wording of the O form as "Not every man is white".
We wished to minimize any expectation of SMT resulting from the wording of the consent form.
