We have addressed conjunctive non-determinism by solving subgoals in an arbitrary order, and disjunctive non-determinism by backtracking, capping the depth to which to search. Universal non-determinism corresponds to picking a new parameter name, which is a form of don't care non-determinism (the parameter name does not matter, as long as it is new).
This leaves existential non-determinism, that is, the choice of picking a term t in the right rule for the existential and the left rule for the universal quantifier. We show how this choice can be postponed by a general technique of residuation. Later we show how the residual equality goals can be solved by unification, and how this can be interleaved with the search process. At that point we will have the first full implementation of a theorem prover for intuitionistic logic.