Mendler-style Iso-(Co)inductive predicates: a strongly normalizing approach

TL;DR

This paper extends second-order logic AF2 with Mendler-style iso-(co)inductive predicates, enabling extraction of programs from proofs while ensuring strong normalization through a novel approach that relaxes positivity restrictions.

Contribution

It introduces Mendler-style iso-(co)inductive definitions in AF2, allowing non-monotonic operators and proving strong normalization via realizability semantics.

Findings

Proves strong normalization for the extended logic.

Develops a realizability semantics based on saturated sets.

Enables program extraction from proofs with non-positive operators.

Abstract

We present an extension of the second-order logic AF2 with iso-style inductive and coinductive definitions specifically designed to extract programs from proofs a la Krivine-Parigot by means of primitive (co)recursion principles. Our logic includes primitive constructors of least and greatest fixed points of predicate transformers, but contrary to the common approach, we do not restrict ourselves to positive operators to ensure monotonicity, instead we use the Mendler-style, motivated here by the concept of monotonization of an arbitrary operator on a complete lattice. We prove an adequacy theorem with respect to a realizability semantics based on saturated sets and saturated-valued functions and as a consequence we obtain the strong normalization property for the proof-term reduction, an important feature which is absent in previous related work.