https://github.com/annenkov/two-level
Two-Level Type Theory
https://github.com/annenkov/two-level
formalization homotopy-type-theory lean proof-assistant type-theory
Last synced: about 2 months ago
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Two-Level Type Theory
- Host: GitHub
- URL: https://github.com/annenkov/two-level
- Owner: annenkov
- Created: 2017-05-06T19:02:56.000Z (about 8 years ago)
- Default Branch: master
- Last Pushed: 2019-12-13T20:39:18.000Z (over 5 years ago)
- Last Synced: 2024-12-25T04:26:24.120Z (6 months ago)
- Topics: formalization, homotopy-type-theory, lean, proof-assistant, type-theory
- Language: Lean
- Size: 151 KB
- Stars: 27
- Watchers: 5
- Forks: 1
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# Lean Formalisation of Two-Level Type Theory
Files in the `2ltt` directory:
| File | |
|--------------------|:--------------------------------------------------------------------:
| fibrant.lean | implementation of the two-level type theory
| finite.lean | facts about finite sets and categories with a finite set of objects
| limit.lean | definition of limits and construction of limits in category of pretypes
| inverse.lean | definition of inverse categories (Section 4.2)
| pullbacks.lean | definition of a pullback, constructed explicitly and using the limit of a diagram along with a proof these definitions are equivalent. Proof of that fibrations are closed under pullbacks.
| matching.lean | definition of the matching object (Section 4.3)
| matching_facts.lean | facts about the matching object from the category C with one object removed
| fibrantlimits_aux.lean | auxiliary lemmas for the proof of the fibrant limits theorem including equivalences forming the core of the proof
| fibrantlimits.lean | a proof of the fibrant limit theorem (Section 4.3, Theorem 4.8.)
| simplicial.lean | initial definition of semi-simplicial types (work in progress)
| facts.lean | some auxiliary lemmas which we could not find in the standard library
| types/* | some examples of reasoning in the inner (fibrant) theory. Mainly, ported from the Lean 2 HoTT library.
## Compilation
Requires Lean 2 (version 0.2.0) to compile.
Lean 2 installation instructions in Ubuntu : https://github.com/leanprover/lean2/blob/master/doc/make/ubuntu-12.04-detailed.md
After installing Lean 2, in the terminal, go to the directory containing the development and use ```make``` to compile the project, or run go to the `2ltt` folder and run ```linja``` directly.
Use your favorite editor to navigate the source code (see some hints about emacs lean-mode blow).
### Using Docker
Alternatively, use `Docker` image. The image is based on Ubuntu and contains Lean 2, our development and Emacs with `lean-mode`, allowing to navigate through the development and compile it.
Steps to build a Docker image:
* In the terminal, go to the directory containing the development and run `make docker`
* After building step is complete (this might take some time), run `make run-image` to connect your current terminal session to the container.
* In the container, type `emacs /root/2ltt/fibrantlimits.lean` to start editing `fibrantlimits.lean` with the lean-mode in Emacs.
### Interacting with the Lean development in the Docker image
As we have said before, the docker image comes with Emacs and lean-mode. The lean-mode provides syntax highlighting and allows to interactively develop proofs.
Use C-c C-g to show goal in tactic proof, C-c C-p to print information about identifier, C-c C-t to show type, and C-c C-l to comple a lean file.
Alternatively, one can use the command line to compile the project or individual files:
* After executing `make run-image`, change directory (in the Ubuntu's command line) to our Lean development `cd /root/2ltt`.
* Run `make clean` to clean previously compiled files
* Run `make` again to rebuild the project.
* Individual files can be compiled using `lean `.
The image comes with the preinstalled `nano` editor that can be used to view/edit the source files (but without syntax highlighting). Other editors can be installed, if required, using the `apt-get` command. For example, `apt-get install vim`.