A Refinement of de Bruijn's Formal Language of Mathematics

@article{Kamareddine2004ARO,
  title={A Refinement of de Bruijn's Formal Language of Mathematics},
  author={Fairouz Kamareddine and R. P. Nederpelt},
  journal={Journal of Logic, Language and Information},
  year={2004},
  volume={13},
  pages={287-340}
}
We provide a syntax and a derivation system fora formal language of mathematics called Weak Type Theory (WTT). We give the metatheory of WTT and a number of illustrative examples.WTT is a refinement of de Bruijn's Mathematical Vernacular (MV) and hence:– WTT is faithful to the mathematician's language yet isformal and avoids ambiguities.– WTT is close to the usualway in which mathematicians express themselves in writing.– WTT has a syntaxbased on linguistic categories instead of set/type… 
Towards an interactive mathematical proof mode
TLDR
The approach in this paper is to approximate a proof language by writing proof-sketches, a notion by Wiedijk, with the aim that they should eventually be verifiable by a proof-checker.
Textbook Proofs Meet Formal Logic - The Problem of Underspecification and Granularity
TLDR
This work defines a calculus-independent representation language for formal mathematics that allows for underspecified parts and provides two systems of rules that check if a proof is correct and at an acceptable level of granularity.
Flexible Encoding of Mathematics on the Computer
This paper reports on refinements and extensions to the MathLang framework that add substantial support for natural language text. We show how the extended framework supports multiple views of
A MathLang Path into a Coq Proof Skeleton
TLDR
New reasoning methods are added to MathLang’s Document Rhetorical aspect (DRa) and a new path from an annotated text into a proof skeleton for the Coq proof assistant is developed.
Gradual computerisation and verification of mathematics : MathLang's path into Mizar
TLDR
This thesis presents a full path of computerisation and formalisation of mathematical documents into the Mizar proof checker using the MathLang framework and develops the third level of the gradual path, which aims at capturing the rhetorical structure of mathematical Documents.
Reasoning inside a formula and ontological correctness of a formal mathematical text
TLDR
The notion of a locally valid statement is introduced, a statement that can be considered true at a given position inside a firstorder formula, and a formal definition of ontological correctness is given for a text written in a special formal language called ForTheL.
Weak type theory
ions can be recognized in natural language by the word 'some'. It indicates that it doesn't really matter whichever element we choose from a particular class or domain; we can just pick a general
Gradual Computerisation/Formalisation of Mathematical Texts into Mizar
TLDR
The gradual computerisation process of an ordinary mathematical text into more formal versions ending with a fully formalised Mizar text is explained in this paper using Barendregt's version of the proof of Pythagoras' theorem.
Towards Flexiformal Mathematics
TLDR
Flexiformalizing mathematics is proposed which addresses the bottlenecks discussed above in two ways: first, by allowing content of flexible formality, it minimizes the starting cost of flexiformalization compared to formalization and second, by co-representing the narration, structure and meaning of mathematical knowledge it forms a basis for not only machine processing but also for building practical, human-oriented applications.
The Space of Mathematical Software Systems - A Survey of Paradigmatic Systems
TLDR
A novel conceptualization of mathematical software that focuses on five aspects, each with their own communities, challenges, and successes is devised to give researchers a guide to this space of systems.
...
...

References

SHOWING 1-10 OF 30 REFERENCES
The mathematical language AUTOMATH, its usage, and some of its extensions
TLDR
The possibilities of superimposed languages, automatic theorem proving, and extensions of Automath are described in this chapter.
A Comparison of Mizar and Isar
TLDR
A list of differences between Mizar and Isar is presented, highlighting the strengths of both systems from the perspective of end-users, and some key differences of the internal mechanisms of structured proof processing in either system are pointed out.
A survey of the Theorema project
TLDR
The present early-prototype version of the Theorems software system is implemented in Mathetnatica 3.0 and consists of a general higher-order predicate logic prover and a collection of special provers that call each other depending on the particular proof situations.
From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931
The fundamental texts of the great classical period in modern logic, some of them never before available in English translation, are here gathered together for the first time. Modern logic, heralded
On the structure of Mizar types
  • G. Bancerek
  • Computer Science, Mathematics
    Electron. Notes Theor. Comput. Sci.
  • 2003
An Overview of the MIZAR Project
TLDR
The Mizar project is a long-term eeort aimed at developing software to support a working mathematician in preparing papers by designing a language for writing formal mathematics based on the Tarski-Grothendieck set theory.
Isabelle/HOL: A Proof Assistant for Higher-Order Logic
TLDR
This presentation discusses Functional Programming in HOL, which aims to provide students with an understanding of the programming language through the lens of Haskell.
Pure Type Systems with Definitions
TLDR
The main result is a proof that for many PTS's, including the Calculus of Constructions, this extension preserves strong normalisation.
...
...