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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical 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, (idemponent) 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 programming language designed to help with proofs in formalized mathematics.

There are two threads of current development in proof systems: foundational and coverage.

The foundational work tries to find the best meta-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 produce something larger.

The much larger work has happened for decades 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.)


Examples of proof assistant software:

Projects for formalization of mathematics wth proof assistants:

A historical projects that died out:

Specifically for higher category theory:


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

See also

Last revised on May 20, 2019 at 08:04:49. See the history of this page for a list of all contributions to it.