Hi Beginners. I was musing over weather Naturals were isomorphic to Natural pairs and wrote the following "proof". I don't have any formal mathematical training beyond calculus, and was wondering if this was an okay approach to take, if there were any better techniques available, if my terminology makes sense, and if my use of Haskell could be improved. Any pointers would be greatly appreciated! The code follows, and is also available as a gist: https://gist.github.com/804248 ---------- import Control.Monad.Logic import Data.Maybe import Data.List import Data.Numbers.Primes import Test.QuickCheck -- This program demonstrates a mapping between the pairs of natural numbers and a subset of co-primes. -- The property should hold for all sized lists, not just pairs. Ordering is preserved. -- Question: Are we able to compress the range to create a bijection? -- Answer: Yes! We can use the breadth-wise indces of the products, rather than the products themselves. -- Pair to Number x_p = (evens !!) y_p = (odds !!) xy_p x y = x_p x * y_p y xy_c x = ns_nc . xy_p x -- Number to Pair p_xy n = (x,y) where x = fromJust $ findIndex (`divides` n) evens y = fromJust $ findIndex (`divides` n) odds c_xy = p_xy . nc_ns -- Sparse Number to Compact Number numbers = odds >>- \x -> evens >>- \y -> return (x*y) -- I don't really understand LogicT... :-( ns_nc n = fromJust $ findIndex (==n) numbers -- Compact Numner to Sparse Number nc_ns = (numbers !!) -- Helpers evens = map (primes !!) [0,2..] odds = map (primes !!) [1,3..] x `divides` y = y `mod` x == 0 -- Id referencing properties prop_p1 = forAll (elements [(x,y) | x <- [1..6], y <- [1..7]]) f where f (x,y) = (x,y) == c_xy (xy_c x y) prop_p2 = forAll (elements [1..100]) f where f c = c == uncurry xy_c (c_xy c)