natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
logic | category theory | type theory |
---|---|---|
true | terminal object/(-2)-truncated object | h-level 0-type/unit 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
</table>
basic constructions:
strong axioms
further
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
a pure type system, hence
a system of natural deduction with dependent types;
with the natural-deduction rule for dependent product types specified;
with a rule for how to introduce new natural-deduction rules for arbitrary (co)inductive types.
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
Thierry Coquand, Gérard Huet, The Calculus of Constructions (inria, pdf)
M. Bunder, Jonathan Seldin, Variants of the basic calculus of constructions (Cite Seer)
Surveys include
Frade, Calculus of inductive constructions (pdf)
Wikipedia, Calculus of constructions
A categorical semantics for CoC is discussed in
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.