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Logical frameworks like LF allow to specify the syntax and (natural deduction) inference rules for syntax/proof-checking a wide variety of logical systems. A crucial feature that is missing for prototyping logics is a way to specify basic proof automation. We try to alleviate this problem by generating \(\lambda \)Prolog (ELPI) inference predicates from logic specifications and controlling them by logic-independent helper predicates that encapsulate the prover characteristics. We show the feasibility of the approach with three experiments: We directly automate ND calculi, we generate tableau theorem provers and model generators.

International audience; The λ-calculus is a handy formalism to specify the evaluation of higher-order programs. It is not very handy, however, when one interprets the specification as an execution mechanism, because terms can grow exponentially with the number of β-steps. This is why implementations of functional languages and proof assistants always rely on some form of sharing of subterms. These frameworks however do not only evaluate λ-terms, they also have to compare them for equality. In presence of sharing, one is actually interested in equality of the underlying unshared λ-terms. The literature contains algorithms for such a sharing equality, that are polynomial in the sizes of the shared terms. This paper improves the bounds in the literature by presenting the first linear time algorithm. As others before us, we are inspired by Paterson and Wegman's algorithm for first-order unification, itself based on representing terms with sharing as DAGs, and sharing equality as bisimulation of DAGs. Beyond the improved complexity, a distinguishing point of our work is a dissection of the involved concepts. In particular, we show that the algorithm computes the smallest bisimulation between the given DAGs, if any.

We introduce a plugin for the interactive prover Coq to export its libraries to a machine readable XML format. The information exported is the one checked by Coq’s kernel after the input is elaborated, augmented with additional data coming from the elaboration itself. The plugin has been applied to the 49 Coq libraries that have an opam package and that currently compile with the latest version of Coq (8.9.0), generating a large dataset of 1,235,934 compressed XML files organized in 18,780 directories that require 17ÂGB on disk.

International audience; We present a proof-of-concept prototype of a (constructive variant of an) HOL interactive theorem prover written in a Higher Order Logic Programming (HOLP) language, namely an extension of λProlog. The prototype is meant to support the claim, that we reinforce , that HOLP is the class of languages that provides the right abstraction level and programming primitives to obtain concise implementations of theorem provers. We identify and advocate for a programming technique, that we call semi-shallow embedding, while at the same time identifying the reasons why pure λProlog is not sufficient to support that technique, and it needs to be extended.

This paper is a report about the use of Matita, an interactive theorem prover under development at the University of Bologna, for the solution of the POPLmark Challenge, part 1a. We provide three different formalizations, including two direct solutions using pure de Bruijn and locally nameless encodings of bound variables, and a formalization using named variables, obtained by means of a sound translation to the locally nameless encoding. According to this experience, we also discuss some of the proof principles used in our solutions, which have led to the development of a generalized inversion tactic for Matita.

We compare several reduction and conversion strategies for the Calculus of (co)Inductive Constructions by running benchmarks from the library of the Coq proof assistant. All the strategies have been implemented in an independent verifier for the proof objects of Coq that is part of the Matita proof assistant.

We present a new interpreter for $$\lambda $$Prolog that runs consistently faster than the byte code compiled by Teyjus, that is considered the best available implementation of $$\lambda $$Prolog. The key insight is the identification of a fragment of the language, which we call reduction-free fragment $$\mathcal {L}_\lambda ^\beta $$, that occurs quite naturally in $$\lambda $$Prolog programs and that admits constant time reduction and unification rules.

We introduce the notion of nonuniform coercion, which is the promotion of a value of one type to an enriched value of a different type via a nonuniform procedure. Nonuniform coercions are a generalization of the (uniform) coercions known in the literature and they arise naturally when formalizing mathematics in an higher order interactive theorem prover using convenient devices like canonical structures, type classes or unification hints. We also show how nonuniform coercions can be naturally implemented at the user level in an interactive theorem prover that allows unification hints., Comment: In Proceedings TYPES 2009, arXiv:1103.3111

We describe some formal topological results, formalised in Matita 1/2, presented in predicative intuitionistic logic and in terms of Overlap Algebras.Overlap Algebras are new algebraic structures designed to ease reasoning about subsets in an algebraic way within intuitionistic logic. We find that they also ease the formalisation of formal topological results in an interactive theorem prover.Our main result is the existence of a functor between two categories of ‘generalised topological spaces’, one with points (Basic Pairs) and the other point-free (Basic Topologies). This formalisation is part of a wider scientific collaboration with the inventor of the theory, Giovanni Sambin. His goal is to verify in what sense his theory is ‘implementable’, and to discover what problems may arise in the process. We check that all intermediate constructions respect the stringent size requirements imposed by predicative logic. The formalisation is quite unusual, since it has to make explicit size information that is often hidden.We found that the version of Matita used for the formalisation was largely inappropriate. The formalisation drove several major improvements of Matita that will be integrated in the next major release (Matita 1.0). We show some motivating examples, taken directly from the formalisation, for these improvements. We also describe a possibly sub-optimal solution in Matita 1/2, which is exploitable in other similar systems. We briefly discuss a better solution available in Matita 1.0.

It is well known that general recursion cannot be expressed within Martin-Loef's type theory and various approaches have been proposed to overcome this problem still maintaining the termination of the computation of the typable terms. In this work we propose a new approach to this problem based on the use of inductively generated formal topologies., Comment: In Proceedings PAR 2010, arXiv:1012.4555

We present a declarative language inspired by the pseudo-natural language previously used in Matita for the explanation of proof terms. We show how to compile the language to proof terms and how to automatically generate declarative scripts from proof terms. Then we investigate the relationship between the two translations, identifying the amount of proof structure preserved by compilation and re-generation of declarative scripts.

We present a formalisation of a constructive proof of Lebesgue’s Dominated Convergence Theorem given by the Sacerdoti Coen and Zoli in [CSCZ]. The proof is done in the abstract setting of ordered uniformities, also introduced by the two authors as a simplification of Weber’s lattice uniformities given in [Web91, Web93]. The proof is fully constructive, in the sense that it is done in Bishop’s style and, under certain assumptions, it is also fully predicative. The formalisation is done in the Calculus of (Co)Inductive Constructions using the interactive theorem prover Matita [ASTZ07]. It exploits some peculiar features of Matita and an advanced technique to represent algebraic hierarchies previously introduced by the authors in [ST07]. Moreover, we introduce a new technique to cope with duality to halve the formalisation effort.

The disambiguation approach to the input of formulae enables the user to type correct formulae in a terse syntax close to the usual ambiguous mathematical notation. When it comes to incorrect formulae we want to present only errors related to the interpretation meant by the user, hiding errors related to other interpretations (spurious errors). We propose a heuristic to recognize spurious errors, which has been integrated with the disambiguation algorithm of "Efficient Ambiguous Parsing of Mathematical Formulae" (Sacerdoti Coen and Zacchiroli, 2004).

The λμμ-calculus, introduced by Curien and Herbelin, is a calculus isomorphic to (a variant of) the classical sequent calculus LK of Gentzen. As a proof format it has very remarkable properties that we plan to study in future works. In this paper we embed it with a rendering semantics that provides explanations in pseudo-natural language of its proof terms, in the spirit of the work of Yann Coscoy [3] for the λ-calculus. The rendering semantics unveils the richness of the calculus that allows to preserve several proof structures that are identified when encoded in the λ-calculus.

In this paper we analyse the modifications on logical operations --- as proof checking, type inference, reduction and convertibility --- that are required for the identification of a proof assistant environment with a distributed mathematical library, focusing on proof assistants based on the Curry-Howard isomorphism. This identification is aimed at the integration of Mathematical Knowledge Management tools with interactive theorem provers: once the distinction between the proof assistant environment and a mathematical library is blurred, it is possible to exploit Mathematical Knowledge Management rendering, indexing and searching services inside an interactive theorem prover, a first step towards effective loosely-coupled collaborative mathematical environments.

Random Access Machines (RAMs) are a deterministic Turing-complete formalism especially well suited for being encoded in other formalisms. This is due to the fact that RAMs can be defined starting from very primitive concepts and operations, which are unbounded natural numbers, tuples, successor, predecessor and test for equality to zero. Since these concepts are easily available also in theorem-provers and proof-assistants, RAMs are good candidates for proving Turing-completeness of formalisms using a proof-assistant. In this paper we describe an encoding in Coq of RAMs into a Linda Calculus endowed with the Ordered Semantics. We discuss the main difficulties that must be faced and the techniques we adopted to solve them.

Mathematical notation has the characteristic of being ambiguous: operators can be overloaded and information that can be deduced is often omitted. Mathematicians are used to this ambiguity and can easily disambiguate a formula making use of the context and of their ability to find the right interpretation. Software applications that have to deal with formulae usually avoid these issues by fixing an unambiguous input notation. This solution is annoying for mathematicians because of the resulting tricky syntaxes and becomes a show stopper to the simultaneous adoption of tools characterized by different input languages. In this paper we present an efficient algorithm suitable for ambiguous parsing of mathematical formulae. The only requirement of the algorithm is the existence of a "validity" predicate over abstract syntax trees of incomplete formulae with placeholders. This requirement can be easily fulfilled in the applicative area of interactive proof assistants, and in several other areas of Mathematical Knowledge Management.

We prove that (strong) fully-concurrent bisimilarity and causal-net bisimilarity are decidable for finite bounded Petri nets. The proofs are based on a generalization of the ordered marking proof technique that Vogler used to demonstrate that (strong) fully-concurrent bisimilarity (or, equivalently, historypreserving bisimilarity) is decidable on finite safe nets.