Curry–Howard correspondence

In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relationship between computer programs and mathematical proofs.

A proof written as a functional program
plus_comm = fun n m : nat => nat_ind (fun n0 : nat => n0 + m = m + n0) (plus_n_0 m) (fun (y : nat) (H : y + m = m + y) => eq_ind (S (m + y)) (fun n0 : nat => S (y + m) = n0) (f_equal S H) (m + S y) (plus_n_Sm m y)) n : forall n m : nat, n + m = m + n
A proof of commutativity of addition on natural numbers in the proof assistant Coq. nat_ind stands for mathematical induction, eq_ind for substitution of equals, and f_equal for taking the same function on both sides of the equality. Earlier theorems are referenced showing ${\displaystyle m=m+0}$ and ${\displaystyle S(m+y)=m+Sy}$.

It is a generalization of a syntactic analogy between systems of formal logic and computational calculi that was first discovered by the American mathematician Haskell Curry and the logician William Alvin Howard. It is the link between logic and computation that is usually attributed to Curry and Howard, although the idea is related to the operational interpretation of intuitionistic logic given in various formulations by L. E. J. Brouwer, Arend Heyting and Andrey Kolmogorov (see Brouwer–Heyting–Kolmogorov interpretation) and Stephen Kleene (see Realizability). The relationship has been extended to include category theory as the three-way Curry–Howard–Lambek correspondence.