Formalising Type Theory in a modular way for translations between type theories
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A formalization of type theory in Coq

This Coq library formalizes dependent type theory in the style of Per Martin-Löf. The formalization is configurable in the sense that various components can be turned on or off, or be left ambivalent. It is thus easy to instantiate many variants of type theory, such as extensional vs. intensional, with or without universes, etc. The library also formalizes several meta-theorems about type theory.


Requirements for compiling the library

The library works with Coq 8.6, and probably with many other recent versions as well, as it does not use any special features of Coq. You may find out about the best way to install Coq at the Coq web site.

To compile the library you need Coq and make.

Requirements for generating the LaTeX version of rules

In order to generate the LaTeX version of the rules you need Python (either version 2.7 or 3.x) and (obviously) LaTeX, as well as the mathpartir package by Didier Rémy.

Since the mathpartir package is GNU-licensed, and we do not want a viral license, the compilation script attempts to download it using the curl and wget commands. If neither is available, you will have to download mathpartir manually and place it in the latex subfolder of the repository.


The best way to download the library is to clone it from GitHub. If you do not use git (you should) then you can just download the ZIP file of the latest version. Even better, you can fork the repository so that you can easily send us your improvements (see the section on contributions below).


To compile the library and the LaTeX version of the rules run


from the command line. The library is in the src subfolder and the PDF with the rules is latex/rules.pdf.

Specific targets for make are:

  • clean -- clean files
  • latex/rules.pdf -- the rules in PDF, the file can be found in latex/rules.pdf
  • latex/rulesParanoid.pdf -- the rules in PDF, but with preconditions
  • library -- compile the library only

For example, make latex/rules.pdf generates the file latex/rules.pdf.

Using the library

This section briefly describes how the library is structured and how it can be used and extended.

The files

The src folder contains the Coq files. Here is an overview of what is in them:

  • config.v -- configuration options
  • config_tactics.v -- tactics for dealing with configuration options
  • ett.v -- economic type theory
  • ett2ptt.v -- proof that we can pass from economic to paranoid type theory
  • ett_sanity.v -- proof that economic type theory is sane
  • inversion.v -- inversion lemmas
  • negfunext.v -- proof that function extensionality is not provable in MLTT
  • ptt.v -- paranoid type theory
  • ptt2ett.v -- proof that we can pass from paranoid to economic type theory
  • ptt_admissible.v -- various admissibility lemmas for paranoid type theory
  • ptt_inversion.v -- inversion principles for paranoid type theory
  • ptt_sanity.v -- proof that paranoid type theory is sane
  • substitution_elim.v -- an attempt to show that explicit substitutions can be computed
  • syntax.v -- definition of presytnax
  • tactics.v -- tactics for working with the library
  • tt.v -- all the rules of type theory
  • uniqueness.v -- proof of uniqueness of typing

Git branches

The current version of the library is on the master branch, the rest is kept around for legacy:

  • master contains the current status of the translation (sanity is being handled, as well as uniqueness of typing),
  • zero-shift contains another formulation of type theory where substitutions have context annotations (probably a bad idea)
  • simpler-substitutions removes all annotations from substitutions but this results in the loss of uniqueness of typing,
  • faster-magic is its counter-part and maybe should be kept instead of simpler-substitutions,
  • untyped-refl corrresponds to an experiment regarding the removal of typing annotation to refl,
  • inversion corresponds to inversion lemmata (probably subsumed by more recent work),
  • into-coq and into-coq-attempt were branches where we were trying to eliminate reflection by translation from our formalised type theory to Coq directly, but e failed.
  • bool-disjoint is about showing that true = false -> Empty (inside the theory),
  • bool-large-elim is about adding large elimination for Bool.

How the library is structured

(To be written)