A Theory Independent Curry-De Bruijn-Howard Correspondence

Brouwer, Heyting, and Kolmogorov have proposed to define constructive proofs as algorithms, for instance, a proof of A ⇒ B as an algorithm taking proofs of A as input and returning proofs of B as output. Curry, De Bruijn, and Howard have developed this idea further. First, they have proposed to express these algorithms in the lambda-calculus, writing for instance λf A⇒A⇒B λx A (f x x) for the proof of the proposition (A ⇒ A ⇒ B) ⇒ A ⇒ B taking a proof f of A ⇒ A ⇒ B and a proof x of A as input and returning the proof of B obtained by applying f to x twice. Then, they have remarked that, as proofs of A ⇒ B map proofs of A to proofs of B, their type proof(A ⇒ B) is proof(A) → proof(B). Thus the function proof mapping propositions to the type of their proofs is a morphism transforming the operation ⇒ into the operation →. In the same way, this morphism transforms cut-reduction in proofs into beta-reduction in lambda-terms. This expression of proofs as lambda-terms has been extensively used in proof a more compact representation of proofs, than natural deduction or sequent calculus proof-trees. This representation is convenient, for instance to store proofs on a disk and to communicate them through a network. This has lead to the development of several typed lambda-calculi: Automath, the system F, the system Fω, the lambda-Pi-calculus, Martin-Löf intuitionistic type theory, the Calculus of Constructions, the Calculus of Inductive Constructions , etc. And we may wonder why so many different calculi are needed. In some cases, the differences in the lambda-calculi reflect differences in the logic where proofs are expressed: some calculi, for instance, express constructive proofs, others classical ones. In other cases, they reflect differences in the inductive rules used to define proofs: some calculi are based on natural deduction , others on sequent calculus. But most of the times, the differences reflect differences in the theory where the proofs are expressed: arithmetic, the theory of classes—a.k.a. second-order logic—, simple type theory—a.k.a. higher-order logic—, predicative type theory, etc. Instead of developing a customized typed lambda-calculus for each specific theory, we may attempt to design a general parametric calculus that permits to express the proofs of any theory. This way, the problem of expressing proofs in the lambda-calculus would be completely separated from that of choosing …