nLab The QED Project



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




The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms



The QED project was a project – or at least a manifesto for a project, see below – for the formalization of large parts of mathematics (formal proof) in formal language supported by proof assistants.

While as a project this seems to have died out around 1996, the idea of the manifesto itself is still pertinent and waiting for its implementation (see also Wiedijk 07).

Notice that since then the issues discussed in (Wiedijk 07) may appear in a different light in view of developments in homotopy type theory (“univalent foundations for mathematics”).

The QED manifesto

The following is taken literally from The QED Manifesto.

The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules. – Kurt Gödel

If civilization continues to advance, in the next two thousand years the overwhelming novelty in human thought will be the dominance of mathematical understanding. – Alfred North Whitehead

1 What Is the QED Project and Why Is It Important?

QED is the very tentative title of a project to build a computer system that effectively represents all important mathematical knowledge and techniques. The QED system will conform to the highest standards of mathematical rigor, including the use of strict formality in the internal representation of knowledge and the use of mechanical methods to check proofs of the correctness of all entries in the system.

The QED project will be a major scientific undertaking requiring the cooperation and effort of hundreds of deep mathematical minds, considerable ingenuity by many computer scientists, and broad support and leadership from research agencies. In the interest of enlisting a wide community of collaborators and supporters, we now offer reasons that the QED project should be undertaken.

First, the increase of mathematical knowledge during the last two hundred years has made the knowledge, let alone understanding, of all, or even of the most important, mathematical results something beyond the capacity of any human. For example, few mathematicians, if any, will ever understand the entirety of the recently settled structure of simple finite groups or the proof of the four color theorem. Remarkably, however, the creation of mathematical logic and the advance of computing technology have also provided the means for building a computing system that represents all important mathematical knowledge in an entirely rigorous and mechanically usable fashion. The QED system we imagine will provide a means by which mathematicians and scientists can scan the entirety of mathematical knowledge for relevant results and, using tools of the QED system, build upon such results with reliability and confidence but without the need for minute comprehension of the details or even the ultimate foundations of the parts of the system upon which they build. Note that the approach will almost surely be an incremental one: the most important and applicable results will likely become available before the more obscure and purely theoretical ones are tackled, thus leading to a useful system in the relatively near term.

Second, the development of high technology is an endeavor of fabulously increasing mathematical complexity. The internal documentation of the next generation of microprocessor chips may run, we have heard, to thousands of pages. The specification of a major new industrial system, such as a fly-by-wire airliner or an autonomous undersea mining operation, is likely to be even an order of magnitude greater in complexity, not the least reason being that such a system would perhaps include dozens of microprocessors. We believe that an industrial designer will be able to take parts of the QED system and use them to build reliable formal mathematical models of not only a new industrial system but even the interaction of that system with a formalization of the external world. We believe that such large mathematical models will provide a key principle for the construction of systems substantially more complex than those of today, with no loss but rather an increase in reliability. As such models become increasingly complex, it will be a major benefit to have them available in stable, rigorous, public form for use by many. The QED system will be a key component of systems for verifying and even synthesizing computing systems, both hardware and software.

The third motivation for the QED project is education. Nothing is more important than mathematics education to the creation of infrastructure for technology-based economic growth. The development of mathematical ability is notoriously dependent upon ‘doing’ rather than upon ‘being told’ or ‘remembering’. The QED system will provide, via such techniques as interactive proof checking algorithms and an endless variety of mathematical results at all levels, an opportunity for the one-on-one presenting, checking, and debugging of mathematical technique, which it is so expensive to provide by the method of one trained mathematician in dialogue with one student. QED can provide an engaging and non-threatening framework for the carrying out of proofs by students, in the same spirit as a long-standing program of Suppes at Stanford for example. Students will be able to get a deeper understanding of mathematics by seeing better the role that lemmas play in proofs and by seeing which kinds of manipulations are valid in which kinds of structures. Today few students get a grasp of mathematics at a detailed level, but via experimentation with a computerized laboratory, that number will increase. In fact, students can be used (eagerly, we think) to contribute to the development of the body of definitions and proved theorems in QED. Let also us make the observation that the relationship of QED to education may be seen in the following broad context: with increasing technology available, governments will look not only to cut costs of education but will increasingly turn to make education and its delivery more cost-effective and beneficial for the state and the individual.

Fourth, although it is not a practical motivation, nevertheless perhaps the foremost motivation for the QED project is cultural. Mathematics is arguably the foremost creation of the human mind. The QED system will be an object of significant cultural character, demonstrably and physically expressing the staggering depth and power of mathematics. Like the great pyramids, the effort required (especially early on) may be great; but the rewards can be even more staggering than this effort. Mathematics is one of the most basic things that unites all people, and helps illuminate some of the most fundamental truths of nature, even of being itself. In the last one hundred years, many traditional cultural values of our civilization have taken a severe beating, and the advance of science has received no small blame for this beating. The QED system will provide a beautiful and compelling monument to the fundamental reality of truth. It will thus provide some antidote to the degenerative effects of cultural relativism and nihilism. In providing motivations for things, one runs the danger of an infinite regression. In the end, we take some things as inherently valuable in themselves. We believe that the construction, use, and even contemplation of the QED system will be one of these, over and above the practical values of such a system. In support of this line of thought, let us cite Aristotle, the Philosopher, the Father of Logic, ‘That which is proper to each thing is by nature best and most pleasant for each thing; for man, therefore, the life according to reason is best and pleasantest, since reason more than anything else is man.’ We speculate that this cultural motivation may be the foremost motivation for the QED project. Sheer aesthetic beauty is a major, perhaps the major, force in the motivation of mathematicians, so it may be that such a cultural, aesthetic motivation will be the key motivation inciting mathematicians to participate.

Fifth, the QED system may help preserve mathematics from corruption. We must remember that mathematics essentially disappeared from Western civilization once, during the dark ages. Could it happen again? We must also remember how unprecedented in the history of mathematics is the clarity, even perfection, that developed in this century in regard to the idea of formal proof, and the foundation of essentially the entirety of known mathematics upon set theory. One can easily imagine corrupting forces that could undermine these achievements. For example, one might suspect that there is already a trend towards believing some recent “theorems” in physics because they offer some predictive power rather than that they have any meaning, much less rigorous proof, with a possible erosion in established standards of rigor. The QED system could offer an antidote to any such tendency. The standard, impartial answer to the question ‘Has it been proved?’ could become ‘Has it been checked by the QED system?’. Such a mechanical proof checker could provide answers immune to pressures of emotion, fashion, and politics.

Sixth, the ‘noise level’ of published mathematics is too high. It has been estimated that something between 50 and 100 thousand mathematical papers are published per year. Nobody knows for sure how many contain errors or how many are repetitions, but some pessimists claim the number of both is high. QED can help to reduce the level of noise, both by helping to find errors and by helping to support computer searches for duplication.

Seventh, QED can help to make mathematics more coherent. There are similar techniques used in various fields of mathematics, a fact that category theory has exploited very well. It is quite natural for formalizers to generalize definitions and propositions because it can make their work much easier.

Eighth, by its insistence upon formalization, the QED project will add to the body of explicitly formulated mathematics. There is mathematical knowledge that is neither taught in classes nor published in monographs. It is below what mathematicians call “folklore,” which is explicitly formulated. Let us call this lower level of unformulated knowledge “mathlore”. In formalization efforts, we must formalize everything, and that includes mathlore lemmas.

Ninth, the QED project will help improve the low level of self-consciousness in mathematics. Good mathematicians understand trends and connections in their field. The QED project will enable mathematicians to analyze, perhaps statistically, the whole structure of the mathematics, to discover new trends, to forecast developments and so on.

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:


  • The QED Manifesto in Automated Deduction - CADE 12, Springer-Verlag, Lecture Notes in Artificial Intelligence, Vol. 814, pp. 238-251, 1994. (web)

  • Wikipedia article

  • Freek Wiedijk, The QED Manifesto Revisited, 2007 (pdf)

Last revised on September 12, 2019 at 07:08:35. See the history of this page for a list of all contributions to it.