Homotopy Type Theory in Agda 17|7|7 1 Goal synthetic homotopy - PowerPoint PPT Presentation

Homotopy Type Theory in Agda 17|7|7 1

Goal synthetic homotopy theory in Agda + other needed theories 2

Goal synthetic homotopy theory in Agda + other needed theories Agda and Coq were the only two immediately usable systems for HoTT 2

Decentralized Dev. HoTT/Agda-HoTT favonia/homotopy [obsolete] nicolaikraus/HoTT-Agda [fork] dlicata335/hott-agda guillaumebrunerie/JamesConstruction ... 3

Decentralized Dev. HoTT/Agda-HoTT favonia/homotopy [obsolete] nicolaikraus/HoTT-Agda [fork] dlicata335/hott-agda guillaumebrunerie/JamesConstruction ... porting theorems and forking are common 3

HoTT/Agda-HoTT - generalized Blakers-Massey (WIP) - total space of Hopf, 3x3 lemma - Seifert-van Kampen theorem - Mayer–Vietoris sequences - cubical reasoning - Freudenthal suspension theorem - Eilenberg-MacLane spaces K(G,n) - ... Guillaume Brunerie, Kuen-Bang Hou (Favonia), 4 Evan Cavallo, Eric Finster, Jesper Cockx, Christian Sattler, Chris Jeris and Michael Shulman

Used Features - MLTT-style logic/programming languages - inductive-inductive & inductive-recursive - powerful mixfix parser - pattern matching - universe polymorphism - ... 5

Used Features - MLTT-style logic/programming languages - inductive-inductive & inductive-recursive - powerful mixfix parser - pattern matching - universe polymorphism - ... Used Automation - higher-order unification - literal overloading - FEW tactics 5

Higher Inductive Types? Simulated by rewriting rules in HoTT-Agda 6

Higher Inductive Types? Simulated by rewriting rules in HoTT-Agda postulate S¹ : Type ₀ base : S¹ loop : base == base module S¹Elim {l}{P : S¹ → Type l} (base* : P base) (loop* : base* == base* [ P ↓ loop ]) where postulate f : Π S¹ P * effectively base- β : f base ↦ base* {-# REWRITE base- β #-} the same as postulate Dan's trick loop- β : apd f loop == loop* 6

Semantics of Agda - NOT well-understood (as a whole) - Many individual features proved 7

Semantics of Agda - NOT well-understood (as a whole) - Many individual features proved Mode of Usage - Highly experimental 7

Structures and Stats core/ [10520 code + 1024 comments] basic synthetic homotopy theory theorems/ [16107 code + 1577 comments] interesting results continuous integration through travis the entire codebase can be checked in 20-30 mins 8

Homotopy Type Theory in Agda 17|7|7 1 Goal synthetic homotopy - PowerPoint PPT Presentation

Homotopy Type Theory in Agda 17|7|7 1 Goal synthetic homotopy theory in Agda + other needed theories 2 Goal synthetic homotopy theory in Agda + other needed theories Agda and Coq were the only two immediately usable systems for HoTT 2

An Invitation to Homotopy Type Theory Tingxiang Zou Type Theory Formal Systems: Churchs

The Seifert-van Kampen Theorem in Homotopy Type Theory [ CSL 2016 ] * Favonia , Carnegie Mellon

Homotopy canonicity of homotopy type theory Christian Sattler jww. Chris Kapulkin University of

Seifert-van Kampen Theorem in Homotopy Type Theory [ Toronto version ] Michael Shulman @ U of

Higher Algebra in Homotopy Type Theory Ulrik Buchholtz TU Darmstadt Formal Methods in

Homotopy type theory Simon Huber University of Gothenburg Summer School on Types, Sets and

Sets in Homotopy type theory Egbert Rijke, Bas Spitters jww Univalent Foundations Program Sept

Towards a realizability model of homotopy type theory Jonas Frey joint work (in progress) with

Recent progress in Homotopy type theory Univalent Foundations team July 22nd, 2013 Supported by

An Introduction to Homotopy Type Theory Morteza Moniri Department of Mathematics Shahid

Programming in Homotopy Type Theory Dan Licata Institute for Advanced Study Joint work with

Homotopy and Directed Type Theory: a Sample Dan Doel October 24, 2011 Type Theory Overview

Computations in homotopy type theory Guillaume Brunerie MLoC 2019, University of Stockholm

Cubical Type Theory Goal: provide a computational justification for Homotopy Type Theory and

Relating the Effective Topos to Homotopy Type Theory Giuseppe Rosolini Universit di Genova

Homotopy-theoretic aspects of Martin-L of type theory Nicola Gambino University of Palermo

Ultra-Strong Machine Learning Comprehensibility of Programs Learned with Inductive Logic

Recursion and Induction

2nd International Workshop on Argument for Agreement and Assurance (AAA 2015), Kanagawa Japan,

USING DOMAIN KNO WLEDGE IN MA CHINE LEARNING Inductiv e vs Analytical Learning

Induction & Recursion Weiss: ch 7.1 Recursion a programming strategy for solving

15-150 Fall 2020 Stephen Brookes Lecture 4 Proving correctness Last time Specification

A Relational Logic for Higher-Order Programs Alejandro Aguirre 1 Gilles Barthe 1 Marco Gaboardi 2

Inductive Program Synthesis for Bidirectional Transformations Tobias G odderz, Helmut Grohne,

Homotopy Type Theory in Agda 17|7|7 1 Goal synthetic homotopy - PowerPoint PPT Presentation

Homotopy Type Theory in Agda 17|7|7 1 Goal synthetic homotopy theory in Agda + other needed theories 2 Goal synthetic homotopy theory in Agda + other needed theories Agda and Coq were the only two immediately usable systems for HoTT 2

An Invitation to Homotopy Type Theory Tingxiang Zou Type Theory Formal Systems: Churchs

The Seifert-van Kampen Theorem in Homotopy Type Theory [ CSL 2016 ] * Favonia , Carnegie Mellon

Homotopy canonicity of homotopy type theory Christian Sattler jww. Chris Kapulkin University of

Seifert-van Kampen Theorem in Homotopy Type Theory [ Toronto version ] Michael Shulman @ U of

Higher Algebra in Homotopy Type Theory Ulrik Buchholtz TU Darmstadt Formal Methods in

Homotopy type theory Simon Huber University of Gothenburg Summer School on Types, Sets and

Sets in Homotopy type theory Egbert Rijke, Bas Spitters jww Univalent Foundations Program Sept

Towards a realizability model of homotopy type theory Jonas Frey joint work (in progress) with

Recent progress in Homotopy type theory Univalent Foundations team July 22nd, 2013 Supported by

An Introduction to Homotopy Type Theory Morteza Moniri Department of Mathematics Shahid

Programming in Homotopy Type Theory Dan Licata Institute for Advanced Study Joint work with

Homotopy and Directed Type Theory: a Sample Dan Doel October 24, 2011 Type Theory Overview

Computations in homotopy type theory Guillaume Brunerie MLoC 2019, University of Stockholm

Cubical Type Theory Goal: provide a computational justification for Homotopy Type Theory and

Relating the Effective Topos to Homotopy Type Theory Giuseppe Rosolini Universit di Genova

Homotopy-theoretic aspects of Martin-L of type theory Nicola Gambino University of Palermo

Ultra-Strong Machine Learning Comprehensibility of Programs Learned with Inductive Logic

Recursion and Induction

2nd International Workshop on Argument for Agreement and Assurance (AAA 2015), Kanagawa Japan,

USING DOMAIN KNO WLEDGE IN MA CHINE LEARNING Inductiv e vs Analytical Learning

Induction &amp; Recursion Weiss: ch 7.1 Recursion a programming strategy for solving

15-150 Fall 2020 Stephen Brookes Lecture 4 Proving correctness Last time Specification

A Relational Logic for Higher-Order Programs Alejandro Aguirre 1 Gilles Barthe 1 Marco Gaboardi 2

Inductive Program Synthesis for Bidirectional Transformations Tobias G odderz, Helmut Grohne,

Induction & Recursion Weiss: ch 7.1 Recursion a programming strategy for solving