Alternation of a label sequence with strictly increasing indices

Write $[a]=\{0,\ldots,a-1\}$ for $a\in\mathbb N$ (empty when $a=0$). A signed label on $[m]$ is a pair $(\varepsilon,j)$ with $\varepsilon\in\{+,-\}$ and $j\in[m]$; negation reverses its sign. For a sequence $M:[\ell]\to\{+,-\}\times[m]$, write $\operatorname{Alt}_+(M)$ when there exists a strictly increasing $u:[\ell]\to[m]$ such that $\operatorname{im}M=\{((-1)^a,u(a)):a\in[\ell]\}$, and define $\operatorname{Alt}_-(M)$ by reversing all these signs. Here $(-1)^a$ denotes the positive sign for even $a$. These predicates concern the label set ordered by index, not the input order. Let $k,m\in\mathbb N$, let $u:[k]\to[m]$ be strictly increasing, let $s:[k]\to\{+1,-1\}$, and let $L:[k]\to\{+,-\}\times[m]$ satisfy $L(a)=(s_a,u(a))$ for every $a\in[k]$. Then