lean2/hott/port.md

We port a lot of algebra and number systems (nat, int, ...) files from the standard library to the HoTT library.

Port instructions:

• use the script port.pl in scripts/ to port the file. e.g. execute the following in the scripts folder: ./port.pl ../library/algebra/lattice.lean ../hott/algebra/lattice.hlean
• remove imports starting with data. or logic. (sometimes you need to replace a data. import by the corresponding types. import)
• Open namespaces eq and algebra if needed
• (optional) add option set_option class.force_new true
• fix all remaining errors. Typical errors include
  • Replacing "and" by "prod" in comments
  • and.intro is replaced by prod.intro, which should be prod.mk.

Currently, the following differences exist between the two libraries, relevant to porting:

• All of the algebraic hierarchy is in the algebra namespace in the HoTT library (on top-level in the standard library).
• The projections "zero" and "one" are reducible in HoTT. This was needed to allow type class inferences to infer

H : is_trunc 0 A |- is_trunc (succ (-1)) A
H : is_trunc 1 A |- is_trunc (succ 0) A

• Projections of most algebraic structures are definitions instead of theorems in HoTT
• Basic properties of nat.add have a simpler proof in HoTT (so that it computes better)