lean2/hott/hott.md

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The Lean Homotopy Type Theory Library

The Lean Homotopy Type Theory library consists of the following directories:

  • init : constants and theorems needed for low-level system operations
  • types : concrete datatypes and type constructors
  • hit: higher inductive types
  • algebra : algebraic structures
  • cubical: cubical types

The following files don't fit in any of the subfolders:

  • hprop_trunc: in this file we prove that is_trunc n A is a mere proposition. We separate this from types.trunc to avoid circularity in imports.
  • eq2: coherence rules for the higher dimensional structure of equality
  • function: embeddings, (split) surjections, retractions
  • arity : equality theorems about functions with arity 2 or higher

See book.md for an overview of the sections of the HoTT book which have been covered.

Lean's homotopy type theory kernel is a version of Martin-Löf Type Theory with:

  • universe polymorphism
  • a non-cumulative hierarchy of universes, Type 0, Type 1, ...
  • inductively defined types
  • Two HITs: n-truncation and quotients.

Note that there is no proof-irrelevant or impredicative universe.

By default, the univalence axiom is declared on initialization.

See also the standard library.