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Agda (Rev #16, changes) in nLab

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Context

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism = \linebreak propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
propositionsetobjecttype
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels

semantics

Contents

Overview

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

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≡1r \equiv 1 for some dimension expression rr. 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 II is not a subobject of Ω\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 “jdentity “identity types”) the type of cofibrant propositions must support adominance. Cubical Agda thus assumes that II 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. (However, ordinary Martin-Lof jdentity types can also be defined in Cubical Agda as an indexed inductive family, with computational behavior as usual for any inductive types in cubical type theory.)

proof assistants:

based on plain type theory/set theory:

based on dependent type theory/homotopy type theory:

based on cubical type theory:

based on modal type theory:

based on simplicial type theory:

For monoidal category theory:

For higher category theory:

projects for formalization of mathematics with proof assistants:

Other proof assistants

Historical projects that died out:

References

General information on Agda is at

A tutorial for use of Agda as an implementation of homotopy type theory is at

and specifically of Cubical Agda as an implementation of cubical type theory:

The HoTT-Agda library is at

Revision on March 16, 2021 at 04:17:40 by Urs Schreiber See the history of this page for a list of all contributions to it.