Formal proofs of operator identities by a single formal computation

A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two can be added or composed. Algebraically, identities modelled noncommutative polynomials such proves that the polynomial corresponding to lies ideal generated identities. In order prove an identity, however, just membership is enough, ring ignores codomains. We show it suffices additionally verify compatibility this generators with labelled quiver encodes which realized as operators. Then, for every consistent representation category, there exists category instance identity. Moreover, assigning same label several edges quiver, algebraic framework developed allows model different versions indeterminate polynomials.