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# Isabelle/HoTT
Isabelle/HoTT is an experimental implementation of [homotopy type theory](https://en.wikipedia.org/wiki/Homotopy_type_theory) in the [Isabelle](https://isabelle.in.tum.de/) interactive theorem prover.
It largely follows the development of the theory in the [Homotopy Type Theory book](https://homotopytypetheory.org/book/), but aims to be general enough to eventually support cubical and other kinds of homotopy type theories.
Work is slowly ongoing to develop the logic into a fully-featured proof environment in which one can formulate and prove mathematical statements, in the style of the univalent foundations school.
While Isabelle has provided support for object logics based on Martin-Löf type theory since the beginning, these have largely been ignored in favor of Isabelle/HOL.
Thus this project is also an experiment in creating a viable framework, based on dependent type theory, inside the simple type theoretic foundations of Isabelle/Pure.
**Caveat prover**: *This project is under active experimentation and is not yet guaranteed fit for any particular purpose.*
### Usage
Isabelle/HoTT is compatible with Isabelle2020.
To use, add the Isabelle/HoTT folder path to `.isabelle/Isabelle2020/ROOTS` (on Mac/Linux/cygwin installations):
```
$ echo path/to/Isabelle/HoTT >> ~/.isabelle/Isabelle2020/ROOTS
```
### To-do list
In no particular order. Some of the following might(?) require changes to the Isabelle prover itself.
- [ ] Dedicated type information data
- [ ] Tactic-based term elaboration has (at least) two problems:
1. `assume(s)` clauses don't accept schematic vars, and
2. it often results in overly-flexible subgoals that the typechecker doesn't solve.
Will need an elaborator integrated into Isabelle's syntax checking.
- [ ] Proper handling of universes.
- [ ] Inductive type definitions.
- [ ] Recursive function definitions.
- [ ] Higher inductive type definitions.
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