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Concept
Calculus of constructions
Summary
In mathematical logic and computer science, the calculus of constructions (CoC) is a type theory created by Thierry Coquand. It can serve as both a typed programming language and as constructive foundation for mathematics. For this second reason, the CoC and its variants have been the basis for Coq and other proof assistants.
Some of its variants include the calculus of inductive constructions (which adds inductive types), the calculus of (co)inductive constructions (which adds coinduction), and the predicative calculus of inductive constructions (which removes some impredicativity).
The CoC is a higher-order typed lambda calculus, initially developed by Thierry Coquand. It is well known for being at the top of Barendregt's lambda cube. It is possible within CoC to define functions from terms to terms, as well as terms to types, types to types, and types to terms.
The CoC is strongly normalizing, and hence consistent.
The CoC has been developed alongside the Coq proof assistant. As features were added (or possible liabilities removed) to the theory, they became available in Coq.
Variants of the CoC are used in other proof assistants, such as Matita and Lean.
The calculus of constructions can be considered an extension of the Curry–Howard isomorphism. The Curry–Howard isomorphism associates a term in the simply typed lambda calculus with each natural-deduction proof in intuitionistic propositional logic. The calculus of constructions extends this isomorphism to proofs in the full intuitionistic predicate calculus, which includes proofs of quantified statements (which we will also call "propositions").
A term in the calculus of constructions is constructed using the following rules:
is a term (also called type);
is a term (also called prop, the type of all propositions);
Variables () are terms;
If and are terms, then so is ;
If and are terms and is a variable, then the following are also terms:
In other words, the term syntax, in BNF, is then:
The calculus of constructions has five kinds of objects:
proofs, which are terms whose types are propositions;
propositions, which are also known as small types;
predicates, which are functions that return propositions;
large types, which are the types of predicates ( is an example of a large type);
itself, which is the type of large types.
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