Combining typing and size constraints for checking the termination of higher-order conditional rewrite systems

TL;DR

This paper extends size-based termination techniques to higher-order conditional rewrite systems, incorporating constraints and conditions to more precisely analyze and verify termination of complex functions.

Contribution

It introduces a novel type-checking algorithm with constraint solving and a new termination criterion that accounts for conditions in higher-order rewriting.

Findings

First to incorporate conditions into higher-order termination analysis

Developed a type-checking algorithm based on Presburger arithmetic constraints

Enabled more precise termination checking for complex higher-order functions

Abstract

In a previous work, the first author extended to higher-order rewriting and dependent types the use of size annotations in types, a termination proof technique called type or size based termination and initially developed for ML-like programs. Here, we go one step further by considering conditional rewriting and explicit quantifications and constraints on size annotations. This allows to describe more precisely how the size of the output of a function depends on the size of its inputs. Hence, we can check the termination of more functions. We first give a general type-checking algorithm based on constraint solving. Then, we give a termination criterion with constraints in Presburger arithmetic. To our knowledge, this is the first termination criterion for higher-order conditional rewriting taking into account the conditions in termination.