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/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive 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, (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


There are good reasons why the theorems should all be easy and the definitions hard. (Michael Spivak, preface to “Calculus on Manifolds” )



In the traditional language of mathematics, a theorem is a statement which is of interest in its own right and which has been proven to be true, though the proof may not be immediately obvious. This contrasts with a lemma (which is usually of interest primarily because of its implications for other statements), a conjecture (which has not yet been proved), an axiom (which is obviously true or assumed to be true), a definition (which becomes true by virtue of its assigning meaning to a word or phrase), a proposition (which usually follows more easily from known facts than a theorem does), or a corollary (which follows immediately from facts recently proven).

The discipline of logic formalizes the notion of proof, but not the notions of “interest” or “immediacy”. Thus, to a logician, any proved statement is often called a theorem. (Mathematicians know this meaning too, but still usually reserve the term ‘theorem’ for important theorems in their published work.) The term ‘proposition’, to a logician, means any statement and does not imply the existence of a proof. The term ‘axiom’ is used in a way that somewhat matches its ordinary usage, but as a logician counts trivial proofs as proofs, an axiom is also a special case of a theorem. Logic rarely studies definitions explicitly, but in some theories they do play a role, similar to their informal usage. The other terms appear not to be used in logic.


In a given logic, in a given context, we have various propositions and various proofs of propositions. In that context, a theorem is a proposition with a proof.

Classically, a theorem is a proposition for which there exists a proof, but in some contexts (such as, perhaps, fully formalized constructive type theory), one may use “theorem” to mean a proposition together with a proof.

A theorem should be contrasted with a tautology: a proposition that is true in all models. If every theorem in a given logic is a tautology in a given class of models for that logic, then we say that the class of models is sound for that logic; if conversely every tautology is a theorem, then we say that the class of models is complete.


… we might list famous important theorems/lemmas/etc in the nnLab here …