Sharing a Library between Proof Assistants: Reaching out to the HOL Family

We observe today a large diversity of proof systems. This diversity has the negative consequence that a lot of theorems are proved many times. Unlike programming languages, it is difficult for these systems to co-operate because they do not implement the same logic. Logical frameworks are a class of theorem provers that overcome this issue by their capacity of implementing various logics. In this work, we study the STTforall logic, an extension of Simple Type Theory that has been encoded in the logical framework Dedukti. We present a translation from this logic to OpenTheory, a proof system and interoperability tool between provers of the HOL family. We have used this translation to export an arithmetic library containing Fermat's little theorem to OpenTheory and to two other proof systems that are Coq and Matita.Comment: In Proceedings LFMTP 2018, arXiv:1807.0135

Exporter une librairie d'arithmétique depuis Dedukti vers HOL

Today, we observe a large diversity of proof systems. This diversity has the negative consequence that a lot of theorems are proved many times. Unlike programming languages, it is difficult for these systems to cooperate because they do not implement the same logic. Logical frameworks are a class of theorems provers that overcome this issue by their capacity of implementing various logics. In this work, we study the STT∀ βδ logic, an extension of the Simple Type Theory that has been encoded in the logical framework Dedukti. We show that this new logic is a good candidate to export proofs to other provers. As an example, we show how this logic has been encoded into Dedukti and how we used it to export proofs to the HOL family provers via OpenTheory

Internship report MPRI 2 Reverse engineering on arithmetic proofs

International audiencededukti is a logical framework that implements the λΠ− modulo theory, an extension of the simply typed lambda calculus with dependent types and rewriting rules. It aims to be a back-end for other proof checkers by compiling proofs from these proof checkers to dedukti. This may also increase re-usability of proofs between proof checkers. However if a logic is more powerful than an other, a theorem in the first logic may not be a theorem in the second. During this internship, we consider arithmetic theorems since many proof checker are able to check arithmetic proofs. One problem that we study in this master thesis is to translate arithmetic proofs coming from a powerful proof checker, -- in our casematita -- to a less powerful proof checker -- HOL-- . This translation needs to modify the logic used in proofs and that is why dedukti is handy here. But a lot of arithmetic theorems are proved also by automatic provers. Indeed, today a lot of easy arithmetic theorems are proved by this kind of tool. But most of them do not give a proof if it claims to prove a theorem. Since for these kind of tool, constructing a full proof may be tiresome, they prefer to give a certificate , a sketch of a proof. However, any automatic prover can implement its own certificate format. To answer this problem, Zakaria Chihani & Dale Miller proposed a certificate framework: Foundational Proof Certificate (FPC) [CMR13]. This framework aims to provide a certificate format shared by many automatic provers so that from the latter, a full proof might be reconstructed.However, for now, no certificate format is given for arithmetic proofs. A second problem addressed in this internship is to answer what kind of certificate is needed for arithmetic proofs (arithmetic without multiplication)

Some axioms for type theories

The $\lambda\Pi$-calculus modulo theory is a logical framework in which many type systems can be expressed as theories. We present such a theory, the theory $\mathcal{U}$, where proofs of several logical systems can be expressed. Moreover, we identify a sub-theory of $\mathcal{U}$ corresponding to each of these systems, and prove that, when a proof in $\mathcal{U}$ uses only symbols of a sub-theory, then it is a proof in that sub-theory

L'intéropérabilité entre des preuves d'arithmétique en utilisan Dedukti

International audienc

Tabletop imaging of structural evolutions in chemical reactions

The introduction of femto-chemistry has made it a primary goal to follow the nuclear and electronic evolution of a molecule in time and space as it undergoes a chemical reaction. Using Coulomb Explosion Imaging we have shot the first high-resolution molecular movie of a to and fro isomerization process in the acetylene cation. So far, this kind of phenomenon could only be observed using VUV light from a Free Electron Laser [Phys. Rev. Lett. 105, 263002 (2010)]. Here we show that 266 nm ultrashort laser pulses are capable of initiating rich dynamics through multiphoton ionization. With our generally applicable tabletop approach that can be used for other small organic molecules, we have investigated two basic chemical reactions simultaneously: proton migration and C=C bond-breaking, triggered by multiphoton ionization. The experimental results are in excellent agreement with the timescales and relaxation pathways predicted by new and definitively quantitative ab initio trajectory simulations

Cumulative Types Systems and Levels

International audienceCumulative Typed Systems (CTS), extend Pure Type Systems with a subtyping relation on universes. We introduce LCTS, a CTS enriched with a notion of level. LCTS has subject reduction (reduction preserves types) but lacks a strong reduction property that levels are also preserved. We show that this strong subject reduction property implies two famous conjectures on CTS: Expansion postponement and the equivalence between explicit and implicit conversion. The former is an open conjecture in the general case for PTS/CTS. The latter has been proved by Siles [5] for PTS only and is still a conjecture for CTS. We rephrase this notion of level using a well-founded order on derivation trees. We show that the existence of such well-founded order implies a type system with the strong subject reduction property. Hence, these two conjectures is a direct consequence of the existence of such well-founded order. Yet, it is not known if such well-founded order exists in general

Cumulative Types Systems and Levels

International audienceCumulative Typed Systems (CTS), extend Pure Type Systems with a subtyping relation on universes. We introduce LCTS, a CTS enriched with a notion of level. LCTS has subject reduction (reduction preserves types) but lacks a strong reduction property that levels are also preserved. We show that this strong subject reduction property implies two famous conjectures on CTS: Expansion postponement and the equivalence between explicit and implicit conversion. The former is an open conjecture in the general case for PTS/CTS. The latter has been proved by Siles [5] for PTS only and is still a conjecture for CTS. We rephrase this notion of level using a well-founded order on derivation trees. We show that the existence of such well-founded order implies a type system with the strong subject reduction property. Hence, these two conjectures is a direct consequence of the existence of such well-founded order. Yet, it is not known if such well-founded order exists in general

Logipedia: a multi-system encyclopedia of formal proofs

Libraries of formal proofs are an important part of our mathematical heritage, but their usability and sustainability is poor. Indeed, each library is specific to a proof system, sometimes even to some version of this system. Thus, a library developed in one system cannot, in general, be used in another and when the system is no more maintained, the library may be lost. This impossibility of using a proof developed in one system in another has been noted for long and a remediation has been proposed: as we have empirical evidence that most of the formal proofs developed in one of these systems can also be developed in another, we can develop a standard language, in which these proofs can be translated, and then used in all systems supporting this standard. Logipedia is an attempt to build such a multi-system online encyclopedia of formal proofs expressed in such as standard language. It is based on two main ideas: the use of a logical framework and of reverse mathematics

Logipedia: a multi-system encyclopedia of formal proofs

Libraries of formal proofs are an important part of our mathematical heritage, but their usability and sustainability is poor. Indeed, each library is specific to a proof system, sometimes even to some version of this system. Thus, a library developed in one system cannot, in general, be used in another and when the system is no more maintained, the library may be lost. This impossibility of using a proof developed in one system in another has been noted for long and a remediation has been proposed: as we have empirical evidence that most of the formal proofs developed in one of these systems can also be developed in another, we can develop a standard language, in which these proofs can be translated, and then used in all systems supporting this standard. Logipedia is an attempt to build such a multi-system online encyclopedia of formal proofs expressed in such as standard language. It is based on two main ideas: the use of a logical framework and of reverse mathematics