Inductive proof search modulo

Abstract

We present an original narrowing-based proof search method for inductive theorems in equational rewrite theories given by a rewrite system R and a set E of equalities. It has the specificity to be grounded on deduction modulo and to rely on narrowing to provide both induction variables and instantiation schemas. Whenever the equational rewrite system (R, E) has good properties of termination, sufficient completeness, and when E is constructor and variable preserving, narrowing at defined-innermost positions leads to consider only unifiers which are constructor substitutions. This is especially interesting for associative and associative-commutative theories for which the general proof search system is refined. The method is shown to be sound and refutationally correct and complete. A major feature of our approach is to provide a constructive proof in deduction modulo for each successful instance of the proof search procedure.

Narrowing Based Inductive Proof Search

“…Our goal will be to achieve this as a way to mechanize proof search in a proof assistant based on type theory and the rewriting calculus [2,26]. Along these lines, a first prototype has been written in collaboration with Paul Brauner and is described in [24]. Moreover our approach provides the ability to use an induction principle based on Noetherian rewrite systems, therefore strongly enhancing over the structural induction principle which is, in practice, used in most of the current proof assistants.…”

“…The genericity of narrowing modulo may enlighten and ease the use of such class rewrite systems to base inductive proof search. This has been explored in [24]. The third one concerns inductive proof by consistency which is indeed at the source of the use of rewrite techniques for induction [22,15,6,25].…”

“…Denoting PE ind (Q) the canonical proposition associated to RE ind (Q, <), this leads to: Refutational completeness of the proof search system IndNarrow, i.e. proving that if a sequent has no proof in deduction modulo, then there is a derivation leading to Refutation, is not detailed here, but this result can be derived from the proof given in [24] in the more complex context where associative commutative theories are considered.…”

Integrating Implicit Induction Proofs into Certified Proof Environments

“…On the other hand, [12] identifies explicit induction schemas from the proof steps that instantiate variables. More recently, Nahon et al [15] proposed a theoretical foundation based on deduction modulo that permits automated construction of inductive proofs into the sequent calculus, ready for insertion into proof assistants. As Brotherston has already shown in his PhD thesis [6], the idea is to perform 'Descente Infinie'-style proofs using an extension of the sequent calculus with explicit inductive schemas that define conjectures in terms of induction hypotheses and conclusions linked by shared variables.…”

Automated Certification of Implicit Induction Proofs

“…For this reason, we followed the method ii), that has been firstly tested manually in [24], then automatically on toy examples in [23]. The method directly translates every step of a Spike proof into Coq scripts, which distinguishes it from previous methods based on proof reconstruction techniques [7,10,13] that mainly transform implicit into explicit induction proofs.…”