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Agda is a proof assistant based on MLTT with a countable hierarchy of explicitly polymorphic universes, very powerful inductive-inductive types? and inductive-recursive types, and a flexible pattern-matching? syntax.
Homotopy type theory has already been formalized in agda HoTT-Agda repository. You can contribute to it.
In case you want to independently formalizing HoTT in agda, you need to be careful about certain issues.
Consider using the latest version of agda. Some older versions of agda are known to have bugs when handling absurd patterns.
You should always use --without-K
to compile agda source code. An easy way to ensure this is to use the compiler pragma {-# OPTIONS
--without-K #-}
at the top of your agda source file.
What is wrong with axiom K? One of the key ideas of Homotopy type theory is to regard the identity Id x y
as the path between the points x
and y
. Axiom K? allows one to prove uniqueness of elements in the identity type which essentially means that non-trivial topological spaces like the loop will be contracted to a point. For details see the post on Identity Elimination.
Agda does not support higher inductive types (HIT) yet. However, by carefully hiding the constructors of the type and exposing only the induction principle for the type one can simulate it.
update: there is now {-# OPTIONS --cubical #-}
and HITs
Some relevant reading material:
The main idea is explained in “Running circles around in your proof assistant”.
See the HoTT-Agda respository README on how they deal with HITs in agda