# A Formalization of the Semantics of Functional-Logic Programming in Isabelle

## TL;DR

This paper formalizes the CRWL semantics of non-strict non-deterministic functional-logic programming within Isabelle/HOL, proving key properties and offering new insights into rule linearity requirements.

## Contribution

It provides a formal Isabelle/HOL proof of CRWL semantics, demonstrating properties like closedness, polarity, and compositionality, and reveals that left linearity of rules is unnecessary.

## Key findings

- Proved CRWL properties in Isabelle/HOL
- Showed left linearity is not required for key properties
- Enhanced understanding of semantics in functional-logic programming

## Abstract

Modern functional-logic programming languages like Toy or Curry feature non-strict non-deterministic functions that behave under call-time choice semantics. A standard formulation for this semantics is the CRWL logic, that specifies a proof calculus for computing the set of possible results for each expression. In this paper we present a formalization of that calculus in the Isabelle/HOL proof assistant. We have proved some basic properties of CRWL: closedness under c-substitutions, polarity and compositionality. We also discuss some insights that have been gained, such as the fact that left linearity of program rules is not needed for any of these results to hold.