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

• f 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, groups, rings, fields 30 479 10149 General Topology 92 413 18313 Algebraic Topology 1 54 1657 Construction of real numbers 15 290 6596 AC in topology 3 14 563 Metamath translation 9 1296 26420 Total 235

3301 78194

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
• n click

Presentation layers

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

• 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

• 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