axiom K (type theory)


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


Disambiguation: For axiom K as a principle of modal logic, see axiom K (modal logic)



In type theory, the axiom K is an axiom that when added to intensional type theory turns it into extensional type theory — or more precisely, what is called here “propositionally extensional type theory”. In the language of homotopy type theory, this means that all types are h-sets, accordingly axiom K is incompatible with the univalence axiom.

Heuristically, the axiom asserts that each term of each identity type Id A(x,x)Id_A(x,x) (of equivalences of a term x:Ax \colon A) is propositionally equal to the canonical reflexivity equality proof refl x:Id A(x,x)refl_x \colon Id_A(x,x).

See also at extensional type theory – Propositional extensionality.


K:A:Typex:AP:Id A(x,x)Type(P(refl Ax)h:Id A(x,x)P(h)) K \colon \underset{A \colon Type}{\prod} \underset{x \colon A}{\prod} \underset{P \colon Id_A(x,x) \to Type}{\prod} \left( P(refl_A x) \to \underset{h \colon Id_A(x,x)}{\prod} P(h) \right)


Unlike its logical equivalent axiom UIP, axiom K can be endowed with computational behavior: K(A,x,P,d,refl Ax)K(A,x,P,d,refl_A x) computes to dd. This gives a way to specify a computational propositionally extensional type theory.

This sort of computational axiom K can also be implemented with, and is sufficient to imply, a general scheme of function definition by pattern-matching. This is implemented in the proof assistant Agda. (The flag --without-K alters Agda’s pattern-matching scheme to a weaker version appropriate for intensional type theory, including homotopy type theory.)


The axiom K was introduced in

For a review and discussion of the implementation in Coq, see

Discussion in the context of homotopy type theory is in

around theorem 7.2.1