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

Aug 14, 2022 •206 likes •360 views

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

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
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

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