Given two vectors $u=(x_1,\ldots,x_n,a_1,\ldots,a_s) \in \{-1,1\}^{n+s}$ and $v=(x_1,\ldots,x_n) \in \{-1,1\}^n$ and a bunch of fixed vectors $c_1,\ldots,c_k \in \{-1,1\}^{n+s}$ and $b_1,\ldots,b_k \in \{-1,1\}^{n}$ and I want to find an instance when all the following (inner product) equations are satisfied: $$c_i \cdot u + 1 = b_i \cdot v,\quad i=1,\ldots,s$$
In general $n$ will be large. So I want to specify the condition that I am only interested in solutions in, say, $\{-1,1\}^{n+s}$ by using the membership of $u$ in Tuples[{-1,1},n+s] instead of putting numerous componentwise integer inequalities into play after specifying Integers as a condition. I will want to do this for varying $n$ and $s.$
Edit: A small example, done componentwise is below. I want to abstract out the equations by using inner products since they will get messy for larger cases. Here $n=s=1$ and I ask for 4 instances. ff allows me to specify +1 and -1 by using ff[x,1].
ff[x_, a_] := (x^2 == a);
FindInstance[
x1*xn + 1 == a1*(x1 + xn) && ff[x1, 1] && ff[xn, 1] &&
ff[a1, 1], {x1, xn, a1}, Integers, 4]
b1
? $\endgroup$a1
by variablesx1
andxn
). The descriptionc_i.u+1==b_i.v
is of a linear problem. $\endgroup$