Cover set induction is known as a proof method that keeps the advantages of explicit induction and proof by consistency. Most implicit induction proof procedures are defined in a cover set induction framework. Contextual cover set (CCS) is a new concept that fully characterizes explicit induction schemes, such as the cover sets, and many simplification techniques as those specific to the "proof by consistency" approach. Firstly, we present an abstract inference system uniformly defined in terms of contextual cover sets as our general framework to build implicit induction provers. Then, we show that it generalizes existing cover set induction procedures.This paper also contributes to the general problem of assembling reasoning systems in a sound manner. Elementary CCSs are generated by reasoning modules that implement various simplification techniques defined for a large class of deduction mechanisms such as rewriting, conditional rewriting and resolution-based methods for clauses. We present a generic and sound integration schema of reasoning modules inside our procedure together with a simple methodology for improvements and incremental sound extensions of the concrete proof procedures. As a case study, the inference system of the SPIKE theorem prover has been shown to be an instance of the abstract inference system integrating reasoning modules based on rewriting techniques defined for conditional theories. Our framework allows for modular and incremental sound extensions of SPIKE when new reasoning techniques are proposed. An extension of the prover, incorporating inductive semantic subsumption techniques, has proved the correctness of the MJRTY algorithm by performing a combination of arithmetic and inductive reasoning.

Induction is a powerful proof technique adapted to reason on setswith an unbounded number of elements. In a first-order setting, twodifferent methods are distinguished: the conventional induction,based on explicit induction schemas, and the implicit induction,based on reductive procedures. We propose a new cycle-basedinduction method that keeps their best features, i.e. i) performslazy induction, ii) naturally fits for mutual induction, and iii) isfree of reductive constraints. The heart of the method is a proofstrategy that identifies in the proof script the subset of formulascontributing to validate the application of induction hypotheses.The conventional and implicit induction are particular cases of ourmethod.

No abstract

CLKID ω is a sequent-based cyclic inference system able to reason on first-order logic with inductive definitions. The current approach for verifying the soundness of CLKID ω proofs is based on expensive model-checking techniques leading to an explosion in the number of states. We propose proof strategies that guarantee the soundness of a class of CLKID ω proofs if some ordering and derivability constraints are satisfied. They are inspired from previous works about cyclic well-founded induction reasoning, known to provide effective sets of ordering constraints. A derivability constraint can be checked in linear time. Under certain conditions, one can build proofs that implicitly satisfy the ordering constraints.

No abstract

Abstract. The Available Bit Rate protocol (ABR) for ATM networks is well-adapted to data traffic by providing minimum rate guarantees and low cell loss to the ABR source end system. An ABR conformance algorithm for controlling the source rates through an interface has been defined by ATM Forum and a more efficient version of it has been designed in [13]. We present in this work the first complete mechanical verification of the equivalence between these two algorithms. The proof is involved and has been supported by the PVS theorem-prover. It has required many lemmas, case analysis and induction reasoning for the manipulation of unbounded scheduling lists. Some ABR conformance protocols have been verified in previous works. However these protocols are approximations of the one we consider here. For instance, the algorithms mechanically proved in [10] and [5] consider scheduling lists with only two elements.

No abstract

In first-order logic, the formula-based instances of the Noetherian induction principle allow to perform effectively simultaneous, mutual and lazy induction reasoning. Compared to the termbased Noetherian induction instances, they are not directly supported by the current proof assistants. We provide general formal tools for certifying formula-based Noetherian induction proofs by the Coq proof assistant, then show how to apply them to certify proofs of conjectures about conditional specifications, built with: i) a reductive rewrite-based induction system, and ii) a reductive-free cyclic induction system. The generation of reductive proofs and their certification process can be easily automatised, without requiring additional definitions or proof transformations, but may involve many ordering constraints to be checked during the certification process. On the other hand, the reductive-free proofs generate fewer ordering constraints, may involve more general specifications and the certification process is more effective. However, their proof generation is less automatic and the generated proofs need to be normalised before being certified. The methodology for certifying reductive-free cyclic induction proofs related to conditional specifications extends a previous approach used for implicit induction proofs and it can be easily adapted to certify any formula-based Noetherian induction reasoning. In practice, the methodology has been implemented to automatically certify implicit induction proofs generated by the SPIKE theorem prover as well as reductive-free cyclic proofs built by the same system but in a less automatic way.