Hammers provide most powerful general purpose automation for proof assistants based on HOL and set theory today. Despite the gaining popularity of the more advanced versions of type theory, such as those based on the Calculus of Inductive Constructions, the construction of hammers for such foundations has been hindered so far by the lack of translation and reconstruction components. In this paper, we present an architecture of a full hammer for dependent type theory together with its implementation for the Coq proof assistant. A key component of the hammer is a proposed translation from the Calculus of Inductive Constructions, with certain extensions introduced by Coq, to untyped first-order logic. The translation is “sufficiently” sound and complete to be of practical use for automated theorem provers. We also introduce a proof reconstruction mechanism based on an eauto-type algorithm combined with limited rewriting, congruence closure and some forward reasoning. The algorithm is able to re-prove in the Coq logic most of the theorems established by the ATPs. Together with machine-learning based selection of relevant premises this constitutes a full hammer system. The performance of the whole procedure is evaluated in a bootstrapping scenario emulating the development of the Coq standard library. For each theorem in the library only the previous theorems and proofs can be used. We show that 40.8% of the theorems can be proved in a push-button mode in about 40 s of real time on a 8-CPU system.

We present a new and formal coinductive proof of confluence and normalisation of Böhm reduction in infinitary lambda calculus. The proof is simpler than previous proofs of this result. The technique of the proof is new, i.e., it is not merely a coinductive reformulation of any earlier proofs. We formalised the proof in the Coq proof assistant.

We show a model construction for a system of higher-order illative combinatory logic Iω, thus establishing its strong consistency. We also use a variant of this construction to provide a complete embedding of first-order intuitionistic predicate logic with second-order propositional quantifiers into the system I0 of Barendregt, Bunder and Dekkers, which gives a partial answer to a question posed by these authors.This paper is a revised version of [Cza13] which appeared in the Journal of Symbolic Logic, vol. 78, issue 3,. An error in Section 5 and some minor mistakes in Section 4 are corrected. Also, the construction in Section 4 is slightly simplified.

Hammers are tools for employing external automated theorem provers (ATPs) to improve automation in formal proof assistants. Strong automation can greatly ease the task of developing formal proofs. An essential component of any hammer is the translation of the logic of a proof assistant to the format of an ATP. Formalisms of state-of-the-art ATPs are usually first-order, so some method for eliminating lambda-abstractions is needed. We present an experimental comparison of several combinatory abstraction algorithms for HOL(y)Hammer-a hammer for HOL Light. The algorithms are compared on problems involving non-trivial lambdaabstractions selected from the HOL Light core library and a library for multivariate analysis. We succeeded in developing algorithms which outperform both lambda-lifting and the simple Schönfinkel's algorithm used in Sledgehammer for Isabelle/HOL. This increases the ATPs' success rate on translated problems, thus enhancing automation in proof assistants.

Hammers are tools that provide general purpose automation for formal proof assistants. Despite the gaining popularity of the more advanced versions of type theory, there are no hammers for such systems. We present an extension of the various hammer components to type theory: (i) a translation of a significant part of the Coq logic into the format of automated proof systems; (ii) a proof reconstruction mechanism based on a Ben-Yelles-type algorithm combined with limited rewriting, congruence closure and a first-order generalization of the left rules of Dyckhoff's system LJT.