nLab
Agda

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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 = propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-level 1-type/h-prop
proofgeneralized elementprogram
conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator monadmodal type theory, monad (in computer science)

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 implementsed (Brunerie). Agda can be compiled to Haskell, Epic or Javascript.

References

General information on Agda is at

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

  • Guillaume Brunerie, The Agda proof assistant, slides, pdf

category: software

Revised on March 20, 2013 16:56:42 by Zoran Škoda (161.53.130.104)
Edit | Back in time (3 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: type theory, Freudenthal suspension theorem, Whitehead theorem, identity type, homotopy type theory, type of types, bracket type, Coq, dependent type theory, extensional type theory, polymorphism, certified programming, Haskell, monad (in computer science), programming language, list of mathematics software, Mathematica