A Proof Procedure For Separation Logic With Inductive Definitions and Theory Reasoning

TL;DR

This paper introduces a sound and complete proof procedure for entailments in Separation Logic with inductive predicates and theory reasoning, capable of terminating under specific conditions.

Contribution

It presents a novel sequent calculus-based proof procedure that handles inductively defined predicates and theory reasoning in Separation Logic, with proven termination conditions.

Findings

The procedure is sound and complete for the logic.

It terminates when inductive rules on the left terminate.

It produces rational proof trees with doubly exponential size in certain cases.

Abstract

A proof procedure, in the spirit of the sequent calculus, is proposed to check the validity of entailments between Separation Logic formulas combining inductively defined predicates denoted structures of bounded tree width and theory reasoning. The calculus is sound and complete, in the sense that a sequent is valid iff it admits a (possibly infinite) proof tree. We show that the procedure terminates in the two following cases: (i) When the inductive rules that define the predicates occurring on the left-hand side of the entailment terminate, in which case the proof tree is always finite. (ii) When the theory is empty, in which case every valid sequent admits a rational proof tree, where the total number of pairwise distinct sequents occurring in the proof tree is doubly exponential w.r.t.\ the size of the end-sequent.