Abstract

A cyclic proof system gives us another way of representing inductive and coinductive definitions and efficient proof search. Podelski-Rybalchenko termination theorem is important for program termination analysis. This paper first shows that Heyting arithmetic HA proves Kleene-Brouwer theorem for induction and Podelski-Rybalchenko theorem for induction. Then by using this theorem this paper proves the equivalence between the provability of the intuitionistic cyclic proof system and that of the intuitionistic system of Martin-Lof’s inductive definitions when both systems contain HA.

Highlights

  • This paper studies two subjects: intuitionistic Podelski-Rybalchenko theorem for induction, and equivalence between intuitionistic system of Martin-Lof’s inductive definitions and an intuitionistic cyclic proof system

  • We say Podelski-Rybalchenko theorem for induction when we replace well-foundedness by induction principle in Podelski-Rybalchenko theorem. [3] shows Podelski-Rybalchenko theorem for induction is provable in intuitionistic second-order logic. [5] shows that this theorem for induction is provable in Peano arithmetic, by using the fact that Peano arithmetic can formalize Ramsey theorem

  • This paper studies the equivalence for intuitionistic logic, namely, the provability of the intuitionistic cyclic proof system, called CLJIDω, is the same as that of the intuitionistic system of Martin-Lof’s inductive definitions, called LJID

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Summary

Introduction

This paper studies two subjects: intuitionistic Podelski-Rybalchenko theorem for induction, and equivalence between intuitionistic system of Martin-Lof’s inductive definitions and an intuitionistic cyclic proof system. This paper studies the equivalence for intuitionistic logic, namely, the provability of the intuitionistic cyclic proof system, called CLJIDω, is the same as that of the intuitionistic system of Martin-Lof’s inductive definitions, called LJID. This question is theoretically interesting, and answers will potentially give new techniques of theorem proving by cyclic proofs to type theories with inductive/coinductive types and program extraction by constructive proofs.

HA-Provable Podelski-Rybalchenko Theorem for Induction
Kleene-Brouwer Theorem
Proof Ideas for Podelski-Rybalchenko Theorem for Induction
Proof of Podelski-Rybalchenko Theorem for Induction
Intuitionistic Martin-Lof ’s Inductive Definition System LJID
Cyclic Proof System CLJIDω
Cyclic Proofs for Coinductive Predicates
Countermodel and Addition of Heyting Arithmetic
Equivalence Theorem
Proof Transformation
Stage Numbers for Inductive Definitions
Path Relation
Related Work
Conclusion
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