7,733 research outputs found
Type inference in mathematics
In the theory of programming languages, type inference is the process of
inferring the type of an expression automatically, often making use of
information from the context in which the expression appears. Such mechanisms
turn out to be extremely useful in the practice of interactive theorem proving,
whereby users interact with a computational proof assistant to construct formal
axiomatic derivations of mathematical theorems. This article explains some of
the mechanisms for type inference used by the Mathematical Components project,
which is working towards a verification of the Feit-Thompson theorem
Towards an Intelligent Tutor for Mathematical Proofs
Computer-supported learning is an increasingly important form of study since
it allows for independent learning and individualized instruction. In this
paper, we discuss a novel approach to developing an intelligent tutoring system
for teaching textbook-style mathematical proofs. We characterize the
particularities of the domain and discuss common ITS design models. Our
approach is motivated by phenomena found in a corpus of tutorial dialogs that
were collected in a Wizard-of-Oz experiment. We show how an intelligent tutor
for textbook-style mathematical proofs can be built on top of an adapted
assertion-level proof assistant by reusing representations and proof search
strategies originally developed for automated and interactive theorem proving.
The resulting prototype was successfully evaluated on a corpus of tutorial
dialogs and yields good results.Comment: In Proceedings THedu'11, arXiv:1202.453
A Coq-based Library for Interactive and Automated Theorem Proving in Plane Geometry
International audienceIn this article, we present the development of a library of formal proofs for theorem proving in plane geometry in a pedagogical context. We use the Coq proof assistant. This library includes the basic geometric notions to state theorems and provides a database of theorems to construct interactive proofs more easily. It is an extension of the library of F. Guilhot for interactive theorem proving at the level of high-school geometry, where we eliminate redundant axioms and give formalizations for the geometric concepts using a vector approach. We also enrich this library by offering an automated deduction method which can be used as a complement to interactive proof. For that purpose, we integrate the formalization of the area method which was developed by J. Narboux in Coq
Semi-Automation of Meta-Theoretic Proofs in Beluga
We present a sound and complete focusing calculus for the core of the logic
behind the proof assistant Beluga as well as an overview of its implementation
as a tactic in Beluga's interactive proof environment Harpoon. The focusing
calculus is designed to construct uniform proofs over contextual LF and its
meta-logic in Beluga: a dependently-typed first-order logic with recursive
definitions. The implemented tactic is intended to complete straightforward
sub-cases in proofs allowing users to focus only on the interesting aspects of
their proofs, leaving tedious simple cases to Beluga's theorem prover. We
demonstrate the effectiveness of our work by using the tactic to simplify
proving weak-head normalization for the simply-typed lambda-calculus.Comment: In Proceedings LFMTP 2023, arXiv:2311.0991
A Vernacular for Coherent Logic
We propose a simple, yet expressive proof representation from which proofs
for different proof assistants can easily be generated. The representation uses
only a few inference rules and is based on a frag- ment of first-order logic
called coherent logic. Coherent logic has been recognized by a number of
researchers as a suitable logic for many ev- eryday mathematical developments.
The proposed proof representation is accompanied by a corresponding XML format
and by a suite of XSL transformations for generating formal proofs for
Isabelle/Isar and Coq, as well as proofs expressed in a natural language form
(formatted in LATEX or in HTML). Also, our automated theorem prover for
coherent logic exports proofs in the proposed XML format. All tools are
publicly available, along with a set of sample theorems.Comment: CICM 2014 - Conferences on Intelligent Computer Mathematics (2014
Learning to Prove Theorems via Interacting with Proof Assistants
Humans prove theorems by relying on substantial high-level reasoning and
problem-specific insights. Proof assistants offer a formalism that resembles
human mathematical reasoning, representing theorems in higher-order logic and
proofs as high-level tactics. However, human experts have to construct proofs
manually by entering tactics into the proof assistant. In this paper, we study
the problem of using machine learning to automate the interaction with proof
assistants. We construct CoqGym, a large-scale dataset and learning environment
containing 71K human-written proofs from 123 projects developed with the Coq
proof assistant. We develop ASTactic, a deep learning-based model that
generates tactics as programs in the form of abstract syntax trees (ASTs).
Experiments show that ASTactic trained on CoqGym can generate effective tactics
and can be used to prove new theorems not previously provable by automated
methods. Code is available at https://github.com/princeton-vl/CoqGym.Comment: Accepted to ICML 201
Premise Selection and External Provers for HOL4
Learning-assisted automated reasoning has recently gained popularity among
the users of Isabelle/HOL, HOL Light, and Mizar. In this paper, we present an
add-on to the HOL4 proof assistant and an adaptation of the HOLyHammer system
that provides machine learning-based premise selection and automated reasoning
also for HOL4. We efficiently record the HOL4 dependencies and extract features
from the theorem statements, which form a basis for premise selection.
HOLyHammer transforms the HOL4 statements in the various TPTP-ATP proof
formats, which are then processed by the ATPs. We discuss the different
evaluation settings: ATPs, accessible lemmas, and premise numbers. We measure
the performance of HOLyHammer on the HOL4 standard library. The results are
combined accordingly and compared with the HOL Light experiments, showing a
comparably high quality of predictions. The system directly benefits HOL4 users
by automatically finding proofs dependencies that can be reconstructed by
Metis
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