Curry Howard Correspondence

+The Curry Howard Correspondence, also known as the proofs-as-programs correspondence, reveals a profound equivalence: programs are proofs, and types are propositions. It elegantly bridges the worlds of [Logic](/wiki/logic) and [Computation](/wiki/computation), demonstrating how formal proofs directly correspond to functional programs. This deep insight underpins much of modern [Type Theory](/wiki/type_theory).

+In this correspondence, if a type `T` can be interpreted as a logical proposition, then a program of type `T` acts as a constructive proof for that proposition. For instance, a function that accepts an argument of type `A` and returns a value of type `B` can be seen as a proof of the implication `A -> B`. This mapping extends to connect various logical connectives (like conjunction, disjunction, implication) with type constructors (like product types, sum types, function types).

+Originating from independent observations by Haskell Curry and William Alvin Howard, this principle is particularly strong in the context of [Intuitionistic Logic](/wiki/intuitionistic_logic), where proofs must be constructive. It has profoundly influenced the design of [Programming Languages](/wiki/programming_languages) with strong [Type Systems](/wiki/type_systems), such as those supporting [Dependent Types](/wiki/dependent_types), enabling the verification of program correctness directly through the type system.

+- [Intuitionistic Logic](/wiki/intuitionistic_logic)

+The Curry Howard Correspondence reveals a profound equivalence: programs are proofs, and types are propositions. It elegantly bridges the worlds of [Logic](/wiki/logic) and [Computation](/wiki/computation), demonstrating how formal proofs directly correspond to functional programs. This deep insight underpins much of modern type theory.

+## See also

+- [Type Theory](/wiki/type_theory)

+- [Lambda Calculus](/wiki/lambda_calculus)

+- [Proof Assistant](/wiki/proof_assistant)