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A cyclic proof system gives us another way of representing inductive definitions and efficient proof search. In 2011 Brotherston and Simpson conjectured the equivalence between the provability of the classical cyclic proof system and that of the classical system of Martin-Lof's inductive definitions. This paper studies the conjecture for intuitionistic logic. This paper first points out that the countermodel of FOSSACS 2017 paper by the same authors shows the conjecture for intuitionistic logic is false in general. Then this paper shows the conjecture for intuitionistic logic is true under arithmetic, namely, the provability of the intuitionistic cyclic proof system is the same as that of the intuitionistic system of Martin-Lof's inductive definitions when both systems contain Heyting arithmetic HA. For this purpose, this paper also shows that HA proves Podelski-Rybalchenko theorem for induction and Kleene-Brouwer theorem for induction. These results immediately give another proof to the conjecture under arithmetic for classical logic shown in LICS 2017 paper by the same authors. IntroductionAn inductive definition is a way to define a predicate by an expression which may contain the predicate itself. The predicate is interpreted by the least fixed point of the defining equation. Inductive definitions are important in computer science, since they can define useful recursive data structures such as lists and trees. Inductive definitions are important also in mathematical logic, since they increase the proof theoretic strength. Martin-Löf's system of inductive definitions given in [11] is one of the most popular systems of inductive definitions. This system has production rules for an inductive predicate, and the production rule determines the introduction rules and the elimination rules for the predicate.Brotherston and Simpson [5,8] proposed an alternative formalization of inductive definitions, called a cyclic proof system. A proof, called a cyclic proof, is defined by proof search, going upwardly in a proof figure. If we encounter the same sequent (called a bud) as some sequent we already passed (called a companion) we can stop. The induction rule is replaced by a case rule, for this purpose. The soundness is guaranteed by some additional condition, called a global trace condition, which can show the case rule decreases some measure of a bud from that of the companion. In general, for proof search, a cyclic proof system can find an induction formula in a more efficient way than Martin-Löf's system, since a cyclic proof system does not have to choose fixed induction formulas in advance. A cyclic proof system enables us efficient implementation of theorem provers with inductive definitions [4,6,7,9].Brotherston and Simpson [8] investigated Martin-Löf's system LKID of inductive definitions in classical logic for the first-order language, and the cyclic proof system CLKID ω for the same language, showed the provability of CLKID ω includes that of LKID, and conjectured the equivalence.By 2017, the equivalence was le...

Static analysis of multi-staged programs is challenging because the basic assumption of conventional static analysis no longer holds: the program text itself is no longer a fixed static entity, but rather a dynamically constructed value. This article presents a semanticpreserving translation of multi-staged call-by-value programs into unstaged programs and a static analysis framework based on this translation. The translation is semantic-preserving in that every small-step reduction of a multi-staged program is simulated by the evaluation of its unstaged version. Thanks to this translation we can analyze multi-staged programs with existing static analysis techniques that have been developed for conventional unstaged programs: we first apply the unstaging translation, then we apply conventional static analysis to the unstaged version, and finally we cast the analysis results back in terms of the original staged program. Our translation handles staging constructs that have been evolved to be useful in practice (typified in Lisp's quasi-quotation): open code as values, unrestricted operations on references and intentional variable-capturing substitutions. This article omits references for which we refer the reader to our companion technical report.

Abstract. We consider the satisfiability problem for a fragment of separation logic including inductive predicates with shape and arithmetic properties. We show that the fragment is decidable if the arithmetic properties can be represented as semilinear sets. Our decision procedure is based on a novel algorithm to infer a finite representation for each inductive predicate which precisely characterises its satisfiability. Our analysis shows that the proposed algorithm runs in exponential time in the worst case. We have implemented our decision procedure and integrated it into an existing verification system. Our experiment on benchmarks shows that our procedure helps to verify the benchmarks effectively.

This paper analyses the computational behaviour of λ-term applications. The properties we are interested in are weak normalisation (i.e. there is a terminating reduction) and strong normalisation (i.e. all reductions are terminating).One can prove that the application of a λ-term M to a fixed number n of copies of the same arbitrary strongly normalising λ-term is strongly normalising if and only if the application of M to n different arbitrary strongly normalising λ-terms is strongly normalising. I.e. one has that M X . . . X n is strongly normalising, for an arbitrary strongly normalising X, if and only if MX 1 . . . X n is strongly normalising for arbitrary strongly normalising X 1 , . . . , X n . The analogous property holds when replacing strongly normalising by weakly normalising.As an application of the result on strong normalisation the λ-terms whose interpretation is the top element (in the environment which associates the top element to all variables) of the Honsell-Lenisa model turn out to be exactly the λ-terms which, applied to an arbitrary number of strongly normalising λ-terms, always produces strongly normalising λ-terms. This proof uses a finitary logical description of the model by means of intersection types. This answers an open question stated by Dezani, Honsell and Motohama.