mechanized reasoning - ias school of mathematicsfavonia/files/mechanization-ias2017... · 6 the...
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favonia
mechanized
reasoning
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checked!
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Peace of Mind*photo credit: voltamax@pixabay
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DreamEveryone can (easily) check
their work in computers
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Notable ProjectsFour-color theorem
Odd-order (Feit-Thompson) theoremKepler's conjecture (now theorem)
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The di�iculty and length of formal proofsare certainly a major cause of the absenceof any widespread adoption by mathema-ticians of automated proof checkers.
— Mechanizing Proof by Donald MacKenzie
Challenges
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Good language (abstraction)Theory
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Good language (abstraction)Theory
Good editorE�icient checker
Engineering+
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Good language (abstraction)Theory
Good editorE�icient checker
Engineering+
Community+
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Good language (abstraction)Theory
Good editorE�icient checker
Engineering+
Community+
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Good language (abstraction)Theory
0, 1, 2, 3, ...
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Good language (abstraction)Theory
0, 1, 2, 3, ...{}, {{}}, {{},{{}}}, {{},{{}},{{},{{}}}}, ...
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Good language (abstraction)Theory
0, 1, 2, 3, ...{}, {{}}, {{},{{}}}, {{},{{}},{{},{{}}}}, ...
(2, 0){{{},{{}}},{{},{{},{{}}}}}
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Good language (abstraction)Theory
0, 1, 2, 3, ...{}, {{}}, {{},{{}}}, {{},{{}},{{},{{}}}}, ...
(2, 0){{{},{{}}},{{},{{},{{}}}}}
Need structural set theory,type theory, or something
Think about Euclid's Elements.
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(material)set theory
classicalmathematics
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(material)set theory
classicalmathematics
good lang. formechanization
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(material)set theory
classicalmathematics
good lang. formechanization
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(material)set theory
classicalmathematics
good lang. formechanization
insightfuleven if not formechanization
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Homotopy Type Theory
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simplicial sets orsome model category
classicalhomotopy
theory
homotopytype theory
(even proofs need to behomotopy-invariant)
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Did we address the issues?
The di�iculty and length of formal proofsare certainly a major cause of the absenceof any widespread adoption by mathema-
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Many results were mechanized,some done without paper proofs.
Did we address the issues?
The di�iculty and length of formal proofsare certainly a major cause of the absenceof any widespread adoption by mathema-
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Many results were mechanized,some done without paper proofs.
Did we address the issues?
Mechanized proofs are not longer.
The di�iculty and length of formal proofsare certainly a major cause of the absenceof any widespread adoption by mathema-
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Blakers-Massey inhomotopy type theory
(2012-13)[Finster, Licata, Lumsdaine]
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Blakers-Massey inhomotopy type theory Full mechanization of
Blakers-Massey in Agda(2012-13)
(2013)
[Finster, Licata, Lumsdaine]
[Licata?] [Favonia]
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Blakers-Massey inhomotopy type theory Full mechanization of
Blakers-Massey in AgdaUn-mechanization
into classical theory
(2012-13)
(2013)
(2014)unpublished [Rezk]
[Finster, Licata, Lumsdaine]
[Licata?] [Favonia]
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Blakers-Massey inhomotopy type theory Full mechanization of
Blakers-Massey in AgdaUn-mechanization
into classical theory
Generalization
(2012-13)
(2013)
(2014)
(2016 or earlier)
Generalizationavailable on arXiv
(2017)1703.09050
unpublished [Rezk]
[Finster, Licata, Lumsdaine]
[Licata?] [Favonia]
[Anel, Biedermann,Finster, Joyal]
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Blakers-Massey inhomotopy type theory Full mechanization of
Blakers-Massey in AgdaUn-mechanization
into classical theory
Generalization
(2012-13)
(2013)
(2014)
(2016 or earlier)
Generalizationavailable on arXiv
(2017)
Mechanizationpublished
(2016)
1703.09050
unpublished [Rezk]
[Finster, Licata, Lumsdaine]
[Licata?] [Favonia]
[FFLL]
[Anel, Biedermann,Finster, Joyal]
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Blakers-Massey inhomotopy type theory Full mechanization of
Blakers-Massey in AgdaUn-mechanization
into classical theory
Generalization
Mechanization of thegeneralization in Agda?
(2012-13)
(2013)
(2014)
(2016 or earlier)
Generalizationavailable on arXiv
(2017)
Mechanizationpublished
(2016)
1703.09050
(2017-?)
unpublished [Rezk]
[Finster, Licata, Lumsdaine]
[Licata?] [Favonia]
[FFLL]
[Anel, Biedermann,Finster, Joyal]
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https://plus.google.com/+UrsSchreiber/posts/Zzjd8KM7K5s
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I will...Mechanize more theorems
Develop new tools (RedPRL, ...)Design new languages
15
You can...Teach me the theorems
you want me to mechanize!
I will...Mechanize more theorems
Develop new tools (RedPRL, ...)Design new languages
15
You can...Teach me the theorems
you want me to mechanize!Try a modern mechanization tool
I will...Mechanize more theorems
Develop new tools (RedPRL, ...)Design new languages
15
You can...Teach me the theorems
you want me to mechanize!Try a modern mechanization tool
Send me an email by TODAYif you want to play Hanabi
I will...Mechanize more theorems
Develop new tools (RedPRL, ...)Design new languages