IsarMathLib a formalized mathematics library for Isabelle/ZF - PowerPoint PPT Presentation

IsarMathLib a formalized mathematics library for Isabelle/ZF Slawomir Kolodynski 11th Conference on Intelligent Computer Mathematics CICM 2018 August 13, 2018 RISC, Hagenberg, Austria

What is IsarMathLib?

What is IsarMathLib? ● Isabelle – a theorem prover developed mostly by University of Cambridge and Technical University of Munich

What is IsarMathLib? ● Isabelle – a theorem prover developed mostly by University of Cambridge and Technical University of Munich ● Isabelle/ZF – one of the object logics provided by the standard Isabelle distribution, implementing Zermelo-Fraenkel set theory

What is IsarMathLib? ● Isabelle – a theorem prover developed mostly by University of Cambridge and Technical University of Munich ● Isabelle/ZF – one of the object logics provided by the standard Isabelle distribution, implementing Zermelo-Fraenkel set theory ● IsarMathLib – an Isabelle/ZF session that can be downloaded from http://download.savannah.nongnu.org/releases/isarmathlib/

What sets IsarMathLib apart?

What sets IsarMathLib apart? ● Zermelo Fraenkel (untyped) set theory as foundation

What sets IsarMathLib apart? ● Zermelo Fraenkel (untyped) set theory as foundation ● Focus on readability

What sets IsarMathLib apart? ● Zermelo Fraenkel (untyped) set theory as foundation ● Focus on readability – By whom: people familiar with standard mathemathical notation and vernacular, but not with any formal proof language

What sets IsarMathLib apart? ● Zermelo Fraenkel (untyped) set theory as foundation ● Focus on readability – By whom: people familiar with standard mathemathical notation and vernacular, but not with any form proof language – Of what: result of the presentation layer processing

What sets IsarMathLib apart? ● Zermelo Fraenkel (untyped) set theory as foundation ● Focus on readability ● A general library of basic facts – not targeted at proving a specific result

Some statistics definitions theorems lines Basics 85 755 14496 Algebra – monoids, 30 479 10149 groups, rings, fields General Topology 92 413 18313 Algebraic Topology 1 54 1657 Construction of real 15 290 6596 numbers AC in topology 3 14 563 Metamath translation 9 1296 26420 Total 235 3301 78194

Presentation layers

Presentation layers ● Isabelle generated proof document ● isarmathlib.org site ● Standard math notation by MathJax ● Interleaved formal and informal narrative ● Folding and unfolding of structured proofs ● Referenced theorems and definitions available on click

Construction of real numbers ● A relatively less known construction of real numbers (Eudoxus reals) is formalized. ● Construction uses the properties of the additive group of integers only, although some ring – specific properties of integers are used in the proofs. ● Axiom of Choice is not used in definitions or proofs.

Construction of real numbers Le t a n alm ost hom om orphism b e a m a p f : Z → Z su ch ● th a t th e se t { f ( n + m )– f ( ) n – f ( m n m ∈ Z } is fin ite W e ) : , . sa y th a t tw o a lm o st h o m o m o rp h ism s f , g a re a lm o st e q u a l if { f ( ) n – g ( ) n : n ∈ Z } is fin ite T h is d e fin e s a n . e q u iva le n ce re la tio n o n th e se t o f a lm o st h o m o m o rp h ism s T h e re a l n u m b e rs a re d e fin e d a s . th e cla sse s o f th is e q u iva le n ce re la tio n A d d itio n o n . re a l n u m b e rs is d e fin e d in a n a tu ra l w a y b y a d d itio n o f a lm o st h o m o m o rp h ism s M u ltip lica tio n . o f re a l n u m b e rs co rre sp o n d s to fu n ctio n a l co m p o sitio n o f a lm o st h o m o m o rp h ism s Po sitive . re a l n u m b e rs a re re p re se n te d b y a lm o st h o m o m o rp h ism s th a t d o n o t h a ve a n u p p e r b o u n d o n Z + .

Metamath translation ● Done by a semi-automatic tool at the syntax level ● About 1300 assertions and 600 proofs have been translated

Thank you