Programming and proving with classical types

The propositions-as-types correspondence is ordinarily presen- ted as linking the metatheory of typed λ-calculi and the proof theory of intuitionistic logic. Griffin observed that this correspondence could be extended to classical logic through the use of control operators. This observation set off a flurry of further research, leading to the development of Parigot’s λμ-calculus. In this work, we use the λμ-calculus as the foundation for a system of proof terms for classical first-order logic. In particular, we define an extended call-by-value λμ-calculus with a type system in correspondence with full classical logic. We extend the language with polymorphic types, add a host of data types in ‘direct style’, and prove several metatheoretical properties. All of our proofs and definitions are mechanised in Isabelle/HOL, and we automatically obtain an inter- preter for a system of proof terms cum programming language—called μML—using Isabelle’s code generation mechanism. Atop our proof terms, we build a prototype LCF-style interactive theorem prover—called μTP— for classical first-order logic, capable of synthesising μML programs from completed tactic-driven proofs. We present example closed μML programs with classical tautologies for types, including some inexpressible as closed programs in the original λμ-calculus, and some example tactic-driven μTP proofs of classical tautologies

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions

Standalone Tactics using OpenTheory

Abstract. Proof tools in interactive theorem provers are usually developed within and tied to a specific system, which leads to a duplication of effort to make the functionality available in different systems. Many verification projects would benefit from access to proof tools developed in other systems. Using OpenTheory as a language for communicating between systems, we show how to turn a proof tool implemented for one system into a standalone tactic available to many systems via the internet. This enables, for example, LCF-style proof reconstruction efforts to be shared by users of different interactive theorem provers and removes the need for each user to install the external tool being integrated.