a0 ∧ ... ∧ an ⊢c0 ∧ ... ∧ cm
It discharges each consequent given what it knows by inspecting the antecedents. Antecedents and consequents may be anything; e.g., kind equalities, type equalities, instantiation requests, etc. While proving a goal new consequents, sometimes an antecedent, may be introduced (trivially).
Thing is, I don't trust the time complexity of the algorithm. Checking for a consequent whether an antecedent exists is linear (times the term size of the antecedent), substitution is linear (times the term size), almost every step the prover takes is two of these linear, possibly quadratic, operations. As far as I know, proving in this general manner might be cubic.
So, I am working in the unknown here. Looks to me I want to keep the number of antecedents as far down as possible to keep the number of lookups and substitutions down, might also be that it is cubic but pragmatically that doesn't matter since you only operate on small numbers of small terms, lastly, since I sometimes am doing a 'bisimulation' proof, I might actually need to do it this way.