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.

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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.