The idea of using compile-time typechecking to prove that a number is even in Scala involves defining an inductive type class IsEven to perform the check.
The inductive definition starts with the base case, IsEven[Zero], asserting that Zero is even, and extends it by defining that if N is even, then Succ[Succ[N]] is too.
This Scala implementation showcases how types can carry rich information and facilitate mathematical proofs directly in code, highlighting induction's role in type systems.
Through this method, running the proveEven function confirms at compile-time whether a given Nat type is even, ensuring type safety in the Scala language.
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