lean2/library/standard/hott/hott.md

standard.hott

A library for Homotopy Type Theory, which avoid the use of prop. Many standard types are imported from standard.data, but then theorems are proved about them using predicate versions of the logical operations. For example, we use the path type, products, sums, sigmas, and the empty type, rather than equality, and, or, exists, and false. These operations take values in Type rather than Prop.

We view Homotopy Theory Theory as compatible with the axiomatic framework of Lean, simply ignoring the impredicative, proof irrelevant Prop. This is o.k. as long as we do not import additional axioms like propositional extensionality or Hilbert choice, which are incompatible with HoTT.

• path : the path type and operations on paths
• equiv : equivalence of types
• trunc : truncatedness of types
• funext : the functional extensionality axiom
• inhabited : a predicative version of the inhabited class