Intrinsic reasoning about functional programs I: first order theories

Abstract

We propose a rudimentary formal framework for reasoning about recursion equations over inductively generated data. Our formalism admits all equational programs (including diverging terms), and yet singles out none. While being simple, this framework has numerous extensions and applications. Here we (1) lay out the basic concepts and definitions; (2) show that (for the natural numbers) the deductive power of our formalism is similar to that of Peano's Arithmetic; (3) prove a strong normalization theorem; and (4) exhibit a mapping from natural deduction derivations to an applied λ-calculus, à la Curry–Howard, which provides a transparent proof of Gödel's result that the provably recursive functions of PA are definable by primitive recursion in finite types. Also of interest is the extent, uncommon among similar constructions, to which our mapping is oblivious to first order terms, including first order quantification and equality. Additional applications and extensions of the method will be presented in sequel papers. In particular, the methodology introduced here lends itself to systematic links between natural sub-formalisms and computational complexity classes. Since these sub-formalisms are largely transparent, allowing one to reason formally in the full system while leaving to automatic software tools the task of verifying that restrictions are complied with, the framework is particularly attractive for the development of Feasible Mathematics.