The key realization for me was that in lambda calculus, "A function can’t call itself by name, so we will have to find an alternative way."
The rest is smart acrobatics.
The rest is smart acrobatics.
let odd'(n, odd, even) = if n == 0 then False else even(n-1, odd, even)
let even'(n, odd, even) = if n == 0 then True else odd(n-1, odd, even)
let odd(n) = odd'(n, odd', even')
let even(n) = even'(n, odd', even')
The only reason to use Y combinator in practice is when you for some reason don't want to keep manually passing the function to itself like "func fact(self, n) { return (n < 1) ? 1 : n * self(self, n-1) }; print(fact(fact, 5))" — maybe because it's tedious and error-prone, — and don't have a sufficiently ergonomic term-rewrite system at hand that would do this for you.
even' _ odd n = if n == 0 then True else (odd (n - 1)))
odd' even _ n = if n == 0 then False else (even (n - 1))
even = head $ vfix [even', odd']
odd = tail $ vfix [even', odd']