-- Functions from the previous post (../1/index.qmd)
module Previous where

import Data.Tree
import Data.List (nubBy)

-- Breadth-first search on a tree
bfs (Node root children) = bfs' root children where
  bfs' v []               = [v]
  bfs' v ((Node y ys):xs) = v:bfs' y (xs ++ ys)

-- Tree of all binary (dyadic) fractions
binFracTree = unfoldTree make $ (1,1) where
  make v@(vn, vd)
    = (v, [
      (2*vn - 1, 2*vd),
      (2*vn + 1, 2*vd)
    ])

-- Stern-Brocot Tree, as (numerator, denominator) pairs
sternBrocot = unfoldTree make $ ((1,1), (0,1), (1,0)) where
  make (v@(vn, vd), l@(ln, ld), r@(rn, rd))
    = (v, [
      (((ln + vn), (ld + vd)), l, v),
      (((vn + rn), (vd + rd)), v, r)
    ])

-- Build a list of pairs which sum to n
listPairs n = [ (k, n - k) | k <- [0..n] ]

-- "Triangular" enumeration of all pairs of positive integers
allPairs = concat $ map listPairs [0..]

-- Equality for rational numbers expressed as pairs of positive integers
ratEqual (a, b) (c, d) = a * d == b * c
-- All positive rational numbers in least terms, as pairs of positive integers
allRationals = nubBy ratEqual $ map (\(a,b) -> (a+1, b+1)) allPairs