Curry-Howard correspondence

The Curry-Howard correspondence uses this to map proofs and computer programs to each other.

In the Curry-Howard correspondence, the bottom type corresponds to falsity.

Automath was also the first practical system that exploited the Curry-Howard correspondence.

These three steps can be stated succinctly using the Curry-Howard correspondence:

This became known as the Curry-Howard correspondence.

Curry is also known for Curry's paradox and the Curry-Howard correspondence.

The Curry-Howard correspondence relates logical conjunction to product types.

Includes a discussion of the Curry-Howard correspondence from a Computer Science perspective.

The sum type corresponds to intuitionistic logical disjunction under the Curry-Howard correspondence.

The Curry-Howard correspondence relates a constructivist form of disjunction to tagged union types.

However, the Curry-Howard correspondence can turn proofs into algorithms, and differences between algorithms are often important.

The Curry-Howard correspondence relates function application to the logical rule of modus ponens.

Exportation is associated with Currying via the Curry-Howard correspondence.

See also Curry-Howard correspondence.

Type theories are often the mathematical foundation used by computer proof assistants because of the Curry-Howard correspondence which equates programs with proofs.

The Curry-Howard correspondence implies that types can be constructed that express arbitrarily complex mathematical properties.

Speculatively, the Curry-Howard correspondence might be expected to lead to a substantial unification between mathematical logic and foundational computer science:

In its more general formulation, the Curry-Howard correspondence is a correspondence between formal proof calculi and type systems for models of computation.

Thanks to the Curry-Howard correspondence, a typed expression whose type corresponds to a logical formula is analogous to a proof of that formula.

In the Curry-Howard correspondence, product types are associated with logical conjunction (AND) in logic.

Using right-associative notation for these operations can be motivated by the Curry-Howard correspondence and by the currying isomorphism.

Some researchers tend to use the term Curry-Howard-de Bruijn correspondence in place of Curry-Howard correspondence.

In type theory, some analogous notions are used as in mathematical logic (giving rise to connections between the two fields, e.g. Curry-Howard correspondence).

The Curry-Howard correspondence between proofs and programs relates ML-style pattern matching to case analysis and proof by exhaustion.

The Curry-Howard correspondence also raised new questions regarding the computational content of proof concepts which were not covered by the original works of Curry and Howard.