nLab proof assistant



Constructivism, Realizability, Computability

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

logicset theory (internal logic of)category theorytype theory
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




A proof assistant or proof management system is a kind of software designed to help with proofs in formalized mathematics. Many proof assistants resemble and/or include a programming language.

There are arguably two threads of current development in proof systems, which may be called “foundational” and “coverage”.

The “foundational” work tries to find the best foundational theory to formalize mathematics (see also at foundations of mathematics). Out of that work first came dependent types (Automath, in the late 60s), then the calculus of constructions (early Coq), and the calculus of inductive constructions (current Coq). More recently a new wave of such work is being done in homotopy type theory as another step in this direction. Coq’s library is not that large, except in the area of group theory where the results of the work on Feit-Thompson theorem has produced something larger.

The “coverage” work tries to formalize as much as possible of mathematics in existing theories. For instance, for decades people have been building Mizar‘s library (Mizar is based on Tarski–Grothendieck set theory rather than type theory). Its library is a couple of orders of magnitude larger than anyone else’s. On the other hand, despite this quantity, it remains an issue to attack problems of contemporary research interest in these systems, see also at Mizar – problem of pertinence.

Similar to Mizar is NuPRL, HOL light and Isabelle, which all have decently sized libraries. (Isabelle can be used with either material set theory, like Mizar, or higher-order type theory, like the others.)


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:


Gentle exposition in

Further exposition:

On computer assisted proofs in analysis:

See also:

List of web resources:

See also

Parts of the above text are taken from this MO comment by Jacques Carette.

Proof assistants specifically for homotopy type theory:

See also

Last revised on January 25, 2023 at 09:24:19. See the history of this page for a list of all contributions to it.