Curry–Howard correspondence

Proofs are programs, types are propositions

The Curry-Howard correspondence establishes a direct relationship between computer programs and mathematical proofs. This means that writing a computer program is akin to constructing a mathematical proof, and conversely, proving a theorem is like writing a computer program. This correspondence provides a framework for understanding how logical deductions and computational processes are fundamentally connected.

Example

Consider a simple function in Haskell, such as `id :: a -> a`, which returns its input. This function can be seen as a proof of the proposition that for any type `a`, applying `id` to `a` yields `a`. In this case, the Haskell function `id` serves as the "program" that corresponds to the "proof" of the proposition.

Remember this

Understanding the Curry-Howard correspondence is crucial for fields like programming language theory and proof theory, as it bridges the gap between logical reasoning and computational execution.

Text adapted from Wikipedia, licensed under CC BY-SA 4.0.

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