Complete Proof Systems for Algebraic Simply-Typed Terms.

We show that reasoning by case analysis (on whether subprograms diverge or converge) is complete for proving PCF observational congruences of algebraic terms. The latter are applicative combinations of first-order variables and a constant Ω denoting a diverging program of base type. A restricted version of the logic is complete for proving equality of algebraic terms in the full continuous type hierarchy (equivalently, observational congruence in PCF with parallel conditional). We show that the provability in the latter logic is in co-NP. We also give complete equational proof systems for a subclass of algebraic terms; provability in these systems is in linear time.