2,937 research outputs found
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
ASMs and Operational Algorithmic Completeness of Lambda Calculus
We show that lambda calculus is a computation model which can step by step
simulate any sequential deterministic algorithm for any computable function
over integers or words or any datatype. More formally, given an algorithm above
a family of computable functions (taken as primitive tools, i.e., kind of
oracle functions for the algorithm), for every constant K big enough, each
computation step of the algorithm can be simulated by exactly K successive
reductions in a natural extension of lambda calculus with constants for
functions in the above considered family. The proof is based on a fixed point
technique in lambda calculus and on Gurevich sequential Thesis which allows to
identify sequential deterministic algorithms with Abstract State Machines. This
extends to algorithms for partial computable functions in such a way that
finite computations ending with exceptions are associated to finite reductions
leading to terms with a particular very simple feature.Comment: 37 page
On sets of terms with a given intersection type
We are interested in how much of the structure of a strongly normalizable
lambda term is captured by its intersection types and how much all the terms of
a given type have in common. In this note we consider the theory BCD
(Barendregt, Coppo, and Dezani) of intersection types without the top element.
We show: for each strongly normalizable lambda term M, with beta-eta normal
form N, there exists an intersection type A such that, in BCD, N is the unique
beta-eta normal term of type A. A similar result holds for finite sets of
strongly normalizable terms for each intersection type A if the set of all
closed terms M such that, in BCD, M has type A, is infinite then, when closed
under beta-eta conversion, this set forms an adaquate numeral system for
untyped lambda calculus. A number of related results are also proved
Building Decision Procedures in the Calculus of Inductive Constructions
It is commonly agreed that the success of future proof assistants will rely
on their ability to incorporate computations within deduction in order to mimic
the mathematician when replacing the proof of a proposition P by the proof of
an equivalent proposition P' obtained from P thanks to possibly complex
calculations. In this paper, we investigate a new version of the calculus of
inductive constructions which incorporates arbitrary decision procedures into
deduction via the conversion rule of the calculus. The novelty of the problem
in the context of the calculus of inductive constructions lies in the fact that
the computation mechanism varies along proof-checking: goals are sent to the
decision procedure together with the set of user hypotheses available from the
current context. Our main result shows that this extension of the calculus of
constructions does not compromise its main properties: confluence, subject
reduction, strong normalization and consistency are all preserved
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