sum x / y $$\mathtt {sum}(x, y)$$ for $$x>y$$ defined as $$x + \mathtt {sum}(x+1, y)$$ or $$y + \mathtt {sum}(x, y-1)$$
all / geq / leq the conjecture holds for all xy where $$x \le y$$, or only for $$x \le y = c$$, or only for $$c = x \le y$$; where $$c\in \mathbb Z$$ is an interpreted constant
val declared / defined $$\mathtt {val}$$ was either not defined, only declared and axiomatized (as in (6)), or defined as a total computable function (as in (14))
ax-fin/ax-all/ ax-leq/ax-geq the axiom holds for integers in an interval $$[c, c')$$, or for all $$x\in \mathbb Z$$, or only for $$x \le c$$, or only for $$x \ge c$$; where $$c, c'\in \mathbb Z$$ are constants
conj-fin/conj-all/conj-leq /conj-geq the conjecture holds for integers in an interval $$[c, c']$$, or for all integers, or only for integers $$\le c$$, or only for integers $$\ge c$$; where $$c, c' \in \mathbb Z$$ are constants
power 0 / 1 $$\mathtt {power}$$ defined starting with $$\mathtt {power}(x, 0) = 1$$ or $$\mathtt {power}(x, 1) = x$$
all / pos / neg the conjecture holds either for all xy, or only for $$x, y \ge 0$$, or only for $$x, y \le 0$$