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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.