80 research outputs found
Proof-checking Euclid
We used computer proof-checking methods to verify the correctness of our
proofs of the propositions in Euclid Book I. We used axioms as close as
possible to those of Euclid, in a language closely related to that used in
Tarski's formal geometry. We used proofs as close as possible to those given by
Euclid, but filling Euclid's gaps and correcting errors. Euclid Book I has 48
propositions, we proved 235 theorems. The extras were partly "Book Zero",
preliminaries of a very fundamental nature, partly propositions that Euclid
omitted but were used implicitly, partly advanced theorems that we found
necessary to fill Euclid's gaps, and partly just variants of Euclid's
propositions. We wrote these proofs in a simple fragment of first-order logic
corresponding to Euclid's logic, debugged them using a custom software tool,
and then checked them in the well-known and trusted proof checkers HOL Light
and Coq.Comment: 53 page
A Synthesis of the Procedural and Declarative Styles of Interactive Theorem Proving
We propose a synthesis of the two proof styles of interactive theorem
proving: the procedural style (where proofs are scripts of commands, like in
Coq) and the declarative style (where proofs are texts in a controlled natural
language, like in Isabelle/Isar). Our approach combines the advantages of the
declarative style - the possibility to write formal proofs like normal
mathematical text - and the procedural style - strong automation and help with
shaping the proofs, including determining the statements of intermediate steps.
Our approach is new, and differs significantly from the ways in which the
procedural and declarative proof styles have been combined before in the
Isabelle, Ssreflect and Matita systems. Our approach is generic and can be
implemented on top of any procedural interactive theorem prover, regardless of
its architecture and logical foundations. To show the viability of our proposed
approach, we fully implemented it as a proof interface called miz3, on top of
the HOL Light interactive theorem prover. The declarative language that this
interface uses is a slight variant of the language of the Mizar system, and can
be used for any interactive theorem prover regardless of its logical
foundations. The miz3 interface allows easy access to the full set of tactics
and formal libraries of HOL Light, and as such has "industrial strength". Our
approach gives a way to automatically convert any procedural proof to a
declarative counterpart, where the converted proof is similar in size to the
original. As all declarative systems have essentially the same proof language,
this gives a straightforward way to port proofs between interactive theorem
provers
A Constructive Algebraic Hierarchy in Coq
AbstractWe describe a framework of algebraic structures in the proof assistant Coq. We have developed this framework as part of the FTA project in Nijmegen, in which a constructive proof of the fundamental theorem of algebra has been formalized in Coq.The algebraic hierarchy that is described here is both abstract and structured. Structures like groups and rings are part of it in an abstract way, defining e.g. a ring as a tuple consisting of a group, a binary operation and a constant that together satisfy the properties of a ring. In this way, a ring automatically inherits the group properties of the additive subgroup. The algebraic hierarchy is formalized in Coq by applying a combination of labelled record types and coercions. In the labelled record types of Coq, one can use dependent types: the type of one label may depend on another label. This allows us to give a type to a dependent-typed tuple like 〈A, f, a〉, where A is a set,f an operation on A and a an element of A. Coercions are functions that are used implicitly (they are inferred by the type checker) and allow, for example, to use the structure A:= 〈A, f, a〉 as a synonym for the carrier set A, as is often done in mathematical practice. Apart from the inheritance and reuse of properties, the algebraic hierarchy has proven very useful for reusing notations
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
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