calculus of constructions


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





The Calculus of Constructions (CoC) is a type theory formal system for constructive proof in natural deduction style. This calculus goes back to Thierry Coquand and Gérard Huet

When supplemented by inductive types, it become the Calculus of Inductive Constructions (CIC). Sometimes coinductive types are included as well and one speaks of the Calculus of (co)inductive Constructions. This is what the Coq software implements.

More in detail, the Calculus of (co)Inductive Constructions is

  1. a pure type system, hence

    1. a system of natural deduction with dependent types;

    2. with the natural-deduction rule for dependent product types specified;

  2. with a rule for how to introduce new natural-deduction rules for arbitrary (co)inductive types.

  3. and with universes:

    a cumulative hierarchy of predicative types of types

    and an impredicative type of propositions.

All of the other standard type formations are subsumed by the existence of arbitrary inductive types, notably the empty type, dependent sum types and identity types are special inductive types. Specifying these hence makes the calculus of constructions be an intensional dependent type theory.


Original articles include

Surveys include

A categorical semantics for CoC is discussed in

  • Martin Hyland, Andy Pitts, The Theory of Constructions: Categorical Semantics and Topos-theoretic Models , Cont. Math. 92 (1989) pp.137-199. (draft)

For specifics of the implementation in Coq see

Last revised on November 2, 2017 at 10:25:32. See the history of this page for a list of all contributions to it.