natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
A dependently typed functional programming language with applications to certified programming. It is also used as a proof assistant.
Besides Coq, Agda is one of the languages in which homotopy type theory has been implemented (Brunerie). Agda can be compiled to Haskell, Epic or Javascript.
Cubical Agda is a version of Agda (turned on by the flag --cubical
) that implements a type theory similar to CCHM (De Morgan) cubical type theory.
Its main difference from CCHM is that instead of an exotype of “cofibrant propositions” it uses the interval itself, replacing cofibrant propositions by statements of the form $r \equiv 1$ for some dimension expression $r$. This change does not prevent the construction of a model for the theory in De Morgan cubical sets, although it doesn’t technically fall under the Orton-Pitts axioms since $I$ is not a subobject of $\Omega$, and no one has checked whether this model can be strengthened to a Quillen model category.
More problematically, to support identity types a la Swan (which are distinct from both cubical “path types” and Martin-Lof “identity types” – the latter sometimes called “jdentity types” to emphasize their definition relative to the J-eliminator) the type of cofibrant propositions must support a dominance. Cubical Agda thus assumes that $I$ supports a dominance, but this is not true in De Morgan cubical sets. So the semantics of the entirety of Cubical Agda, with Swan identity types, is unclear. (For this reason, the Cubical Agda library generally avoids using Swan identity types, although Cubical Agda supports them.)
Ordinary Martin-Lof jdentity types should, in principle, also be definable in Cubical Agda as an indexed inductive family, with computational behavior as usual for any inductive types in cubical type theory. As of March 2021, however, there is a bug in Cubical Agda that prevents jdentity types from computing correctly.
based on plain type theory/set theory:
based on dependent type theory/homotopy type theory:
For monoidal category theory:
projects for formalization of mathematics with proof assistants:
Archive of Formal Proofs (using Isabelle)
ForMath project (using Coq)
UniMath project (using Coq)
Xena project (using Lean)
Historical projects that died out:
General information on Agda is at
Ulf Norell, James Chapman, Dependently Typed Programming in Agda (pdf)
Dan Licata, Ian Voysey, Programming and proving in Agda
Ulf Norell, Towards a practical programming language based on dependent type theory, 2007 (pdf)
A tutorial for use of Agda as an implementation of homotopy type theory is at
Guillaume Brunerie, Agda for homotopy type theory (web)
Guillaume Brunerie, The Agda proof assistant, slides, pdf
and specifically of Cubical Agda as an implementation of cubical type theory:
The HoTT-Agda library is at