TL;DR
A proof checker called Nörgler is introduced that builds upon and extends the established approach pioneered by GDV and supports checking propositional, (untyped and typed) first-order, and higher-order refutations represented in TSTP.
In this paper we continue assessing the feasibility of the approach to the mechanization of type theory by using classical syntax and Stoughton's multiple substitutions and report some substantial progress. We present formal proofs of confluence for beta-reduction and by using Takahashi's revision of Tait and Martin-L\"of's proof, subject reduction for the entire family of the Pure Type Systems and consistency for some impredicative subclass, assuming normalization. As to the proof of confluence, we also develop a theory of alpha-commutative relations which, in our view, entails a clearer presentation and treatment of the problem than in similar developments. Finally, we assess general merits and drawbacks of the approach. The whole development has been machine-checked using Agda.
FLEX is presented, a foundational Constrained Horn Clause (CHC) solver implemented in LEAN, that reduces the trusted base to the kernel alone, and allows using LEAN's entire proof ecosystem to verify low-level systems code, via three contributions.
Term rewriting systems are a common tool in automated reasoning and semantics of programming languages, and many practical applications require these systems to be convergent. While automated tools and theory exist to establish convergence, this paper is concerned with a practical method for testing it to quickly find useful counterexamples. Standard property-based testing approaches struggle here: exhaustively computing all normal forms is fundamentally flawed and too slow, while generating random normal forms makes counterexample minimization (shrinking) fragile due to dependencies on earlier generated test data. To solve this, we introduce a QuickCheck testing method based on generating and shrinking random execution traces. By checking if the first and last terms of a generated trace share the same deterministic normal form, we remove the data dependency between generators. This approach yields a property that efficiently finds counterexamples and enables fast, robust shrinking. We demonstrate the effectiveness of this method on various examples, ranging from group theory equations to distributed process calculus.
This paper introduces the strategy language of Pgeon, a meta-prover with a clear separation between inference rules and proof search. We give the semantics of strategies as functions over proof states, and of the operators that are used to combine them, allowing for sequential composition, choice, repetition and interleaving of strategies. This language is designed to handle the challenge of fair proof search in semi-decidable logics, where simple depth-first exploration of the proof space is not guaranteed to achieve completeness. We showcase the expressiveness and effectiveness of the approach through case studies in first-order and modal logics.