I'm new to functional programming.
So the terms cons appends an element to the front of the list. Where
cons ≜ λx:λl:λc:λn: c x (l c n)
How should I go about proving that cons works correctly using beta reduction for a sample function call? For example reducing cons 3 [2,1] to [3,2,1]?
Is there a formula like for the arithmetic operations in lambda calculus? I'm a bit confused on how to approach this compared to an arithmetic operation (i.e. addition or multiplication).
Thanks.
cons ≜ λx:λl:λc:λn: c x (l c n) means that
cons x l c n =
c x (l c n)
(in functional / applicative / combinatory notation). So
cons 3 [2,1] c n =
= c 3 ([2,1] c n)
and what is [2,1] if not just shortcut notation for cons 2 [1] so that we continue
= c 3 (cons 2 [1] c n)
= c 3 (c 2 ([1] c n))
= c 3 (c 2 (cons 1 [] c n))
= c 3 (c 2 (c 1 ([] c n)))
So there's no reduction from cons 3 [2,1] to [3,2,1]; [3,2,1] is cons 3 [2,1]. And [2,1] is cons 2 [1], and [1] is cons 1 [].
The list cons x xs, when supplied with c and n arguments, will turn into c x (xs c n), and so will xs, in its turn; so any list's elements are used in the chain of applications of c on top one another.
And what should [] c n turn into? It has nothing in it to put through the c applications -- those are to be applied to a list's elements, and [] has none. So the only reasonable thing to do (and I'm sure you're already given that definition) is to turn [] c n into just n:
cons 3 [2,1] c n =
= c 3 (c 2 (c 1 ([] c n)))
= c 3 (c 2 (c 1 n ))
whatever c and n are.
And that's that.