Given a set $A \subseteq \{1,2,3,\dots\}$, decide if $\sum_{x \in A} x \neq \sum_{y \in B} y$ for every $B$ that is a combination with repeated usage of $A$.
We define $B$ to be a combination with repeated usage of set $A$ if $B$ is a multiset; every element of $B$ is an element of $A$; and there exists an element in $B$ that occurs with multiplicity 2 or more.
Is this problem coNP-complete?
Observe:
$B$ could consist of elements occurring multiple times. For example $a$ could occur two times and $b$ could occur 7 times: $B = \{a,a,b,b,b,b,b,b,b,\dots\}$.
$B$ might not have all the elements in $A$, e.g., $A = \{2,4\}$ and $B=\{2,2,2\}$. Both sum up to $6$.
When $A = \{2,4\}$, $B=\{2,2,2\}$ sums up to the same as $A$, thus the answer is False in this case.
