Hammer for Coq: Automation for Dependent Type Theory

Abstract

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.

Relaxed Weighted Path Order in Theorem Proving

“…This section provides an evaluation of our experimental implementation of WPO in E Prover. 1 We use a single good-performing E strategy with the different term orders. The strategy was randomly selected and is provided in "Appendix A".…”

“…• We evaluate WPO against existing orderings in E Prover on parts of the TPTP library, the proofs stemming from the AIM conjecture [10], and on the CoqHammer proofs [1] in Sect. 6.…”

Meta-F $$^\star $$ : Proof Automation with SMT, Tactics, and Metaprograms

“…Finally, ITPs are seeing increasing use of "hammers" such as Sledgehammer [15,16,55] in Isabelle/HOL, and similar tools for HOL Light and HOL4 [44], and Mizar [45], to interface with ATPs. This technique is similar to Meta-F , which, given its support for a dependently typed logic is especially related to a recent hammer for Coq [27]. Unlike these hammers, Meta-F does not aim to reconstruct SMT proofs, gaining efficiency at the cost of trusting the SMT solver.…”

Verifying Bit-vector Invertibility Conditions in Coq (Extended Abstract)

“…In this section we provide specific details about proving invertibility equivalences in Coq. In addition to the bit-vector library described in Section 4, in several proofs of invertibility equivalences we benefited from CoqHammer [4], a plug-in that aims at extending the automation in Coq by combining machine 1 Theorem bvashr_ult2_rtl : forall (n : N), forall (s t : bitvector n), 2 (exists (x : bitvector n), (bv_ult (bv_ashr_a s x) t = true)) -> 3 (((bv_ult s t = true) \/ (bv_slt s (zeros n)) = false) /\ 4 (bv_eq t (zeros n)) = false). 5 Proof.…”