Axioms for strict and lazy functional programs

We show the adequacy of axioms and proof rules for strict and lazy functional programs. Our basic logic comprises a huge part of what is common to the two styles of functional programming. The logic for call-by-value is obtained by adding the axiom that says that all variables are defined, whereas the logic for call-by-name is obtained by adding the axiom that postulates the existence of an undefined object for each type. To show the correctness of the axiomatization we do not use denotational semantics and the adequacy of the evaluation of programs with respect to the semantics. Instead we use the standard term models based on call-by-value and call-by-name evaluation. We introduce a new method to prove on the syntactical level the monotonicity of the evaluation of functional programs with unbounded recursion. The direct method yields results concerning the proof-theoretic strength of the axiomatization. As a side result we obtain a syntactical proof of the context lemma for simply typed lambda terms with recursion.