mechanized reasoning favonia 1 2 2 2 checked! 2 Peace of - PowerPoint PPT Presentation

  mechanized reasoning favonia 1

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checked! 2

Peace of Mind 3 *photo credit: voltamax@pixabay

Dream Everyone can (easily) check their work in computers 4

Notable Projects Four-color theorem Odd-order (Feit-Thompson) theorem Kepler's conjecture (now theorem) 5

Challenges The di � iculty and length of formal proofs are certainly a major cause of the absence of any widespread adoption by mathema- ticians of automated proof checkers. — Mechanizing Proof by Donald MacKenzie 6

Theory Good language (abstraction) 7

Theory Good language (abstraction) + Engineering Good editor E � icient checker 7

Theory Good language (abstraction) + Engineering Good editor E � icient checker + Community 7

Theory Good language (abstraction) + Engineering Good editor E � icient checker + Community 7

Theory Good language (abstraction) 0, 1, 2, 3, ... 8

Theory Good language (abstraction) 0, 1, 2, 3, ... {}, {{}}, {{},{{}}}, {{},{{}},{{},{{}}}}, ... 8

Theory Good language (abstraction) 0, 1, 2, 3, ... {}, {{}}, {{},{{}}}, {{},{{}},{{},{{}}}}, ... (2, 0) {{{},{{}}},{{},{{},{{}}}}} 8

Theory Good language (abstraction) 0, 1, 2, 3, ... {}, {{}}, {{},{{}}}, {{},{{}},{{},{{}}}}, ... (2, 0) {{{},{{}}},{{},{{},{{}}}}} Need structural set theory, type theory, or something Think about Euclid's Elements. 8

classical mathematics (material) set theory 9

classical good lang. for mathematics mechanization (material) set theory 9

classical good lang. for mathematics mechanization (material) set theory 9

classical good lang. for mathematics mechanization insightful (material) even if not for set theory mechanization 9

Homotopy Type Theory 10

classical homotopy homotopy type theory theory (even proofs need to be homotopy-invariant) simplicial sets or some model category 11

Did we address the issues? The di � iculty and length of formal proofs are certainly a major cause of the absence 12 of any widespread adoption by mathema-

Did we address the issues? Many results were mechanized, some done without paper proofs. The di � iculty and length of formal proofs are certainly a major cause of the absence 12 of any widespread adoption by mathema-

Did we address the issues? Many results were mechanized, some done without paper proofs. Mechanized proofs are not longer. The di � iculty and length of formal proofs are certainly a major cause of the absence 12 of any widespread adoption by mathema-

Blakers-Massey in homotopy type theory (2012-13) [Finster, Licata, Lumsdaine] 13

Blakers-Massey in homotopy type theory Full mechanization of (2012-13) [Finster, Licata, Lumsdaine] Blakers-Massey in Agda (2013) [Licata?] [Favonia] 13

Blakers-Massey in homotopy type theory Full mechanization of (2012-13) [Finster, Licata, Lumsdaine] Blakers-Massey in Agda (2013) [Licata?] [Favonia] Un-mechanization into classical theory (2014) unpublished [Rezk] 13

Blakers-Massey in homotopy type theory Full mechanization of (2012-13) [Finster, Licata, Lumsdaine] Blakers-Massey in Agda (2013) [Licata?] [Favonia] Un-mechanization into classical theory (2014) unpublished [Rezk] Generalization [Anel, Biedermann, (2016 or earlier) Finster, Joyal] Generalization available on arXiv (2017) 1703.09050 13

Blakers-Massey in homotopy type theory Full mechanization of (2012-13) [Finster, Licata, Lumsdaine] Blakers-Massey in Agda (2013) [Licata?] [Favonia] Un-mechanization into classical theory (2014) unpublished [Rezk] Generalization Mechanization [Anel, Biedermann, (2016 or earlier) Finster, Joyal] published Generalization (2016) [FFLL] available on arXiv (2017) 1703.09050 13

Blakers-Massey in homotopy type theory Full mechanization of (2012-13) [Finster, Licata, Lumsdaine] Blakers-Massey in Agda (2013) [Licata?] [Favonia] Un-mechanization into classical theory (2014) unpublished [Rezk] Generalization Mechanization [Anel, Biedermann, (2016 or earlier) Finster, Joyal] published Generalization (2016) [FFLL] available on arXiv Mechanization of the (2017) 1703.09050 generalization in Agda? (2017-?) 13

https://plus.google.com/+UrsSchreiber/posts/Zzjd8KM7K5s 14

I will... Mechanize more theorems Design new languages Develop new tools ( RedPRL , ...) 15

I will... Mechanize more theorems Design new languages Develop new tools ( RedPRL , ...) You can... Teach me the theorems you want me to mechanize! 15

I will... Mechanize more theorems Design new languages Develop new tools ( RedPRL , ...) You can... Teach me the theorems you want me to mechanize! Try a modern mechanization tool 15

I will... Mechanize more theorems Design new languages Develop new tools ( RedPRL , ...) You can... Teach me the theorems you want me to mechanize! Try a modern mechanization tool Send me an email by TODAY if you want to play Hanabi 15