Sentential Combinators par Curry

BCKW.swift

// Playground - noun: a place where people can play

/// From Curry's doctoral thesis, Grundlagen der Kombinatorischen Logik, these combinators
/// have been largely abandoned due to the discovery of the SKI calculus (really just S and K).
/// Interestingly, Curry's system is derived from a number of popular axioms in propositional
/// logic.  Namely:
///
/// B ≡ (B → C) → ((A → B) → (A → C))
/// C ≡ (A → (B → C)) → (B → (A → C))
/// K ≡ A → (B → A)
/// W ≡ (A → (A → B)) → (A → B)
/// 
/// Where function application stands for Modus Ponens.


/// Compose | Composes two functions.
func B<A, B, C>(f: A -> C) -> (B -> A) -> B -> C {
	return { (g : (B -> A)) -> B -> C in
		return { (x : B) -> C in
			return f(g(x))
		}
	}
}

/// Swap | Swaps the arguments to a function.
func C<A, B, C>(f: (A -> B -> C)) -> B -> A -> C {
	return { (x: B) -> A -> C in
		return { (y: A) -> C in
			return f(y)(x)
		}
	}
}

/// Discard | "Forgets" its second argument.
func K<A, B>(x : A) -> B -> A {
	return { (y : B) -> A in
		return x
	}
}

/// Duplicate | Applies its second argument to a function twice.
func W<A, B>(f: (A -> A -> B)) -> A -> B {
	return { (x : A) in
		return f(x)(x)
	}
}