The 'unique' (nub) function removes duplicate elements from a list while preserving the order of first occurrence. It operates correctly on even infinite (lazy) lists.

Note that 'r' is a list and 'u' is a function and that they have mutually recursive definitions – r depends on u and v.v.. Bird called programs with self-referential data-structures circular programs.

unique = lambda L. {remove duplicates from L (may be infinite)}
let rec
  r = u L 0,   { result }

  u = lambda L. lambda n.                   { returns L-r }
    if null L then nil
    else if member  hd L  r n then u  tl L  n { duplicate }
    else hd L :: (u  tl L  (n+1)),            { new value }

  member = lambda e. lambda L. lambda n.    { is e in L ? }
    if n = 0 then false        { n is current length of r }
    else if e = hd L then true
    else member  e  tl L  (n-1)
in r

{ Circular Program Unique }

Reference:

[All89] L. Allison, Circular programs and self referential structures. Software Practice and Experience, 19(2), pp.99-109, Feb 1989.

[All93] L. Allison, Applications of Recursively Defined Data Structures, Australian Computer Journal, 25(1), pp.14-20, arxiv:2206.12795, 1993.

let unique = lambda L. {remove duplicates from L (may be infinite)} let rec r = u L 0, { result } u = lambda L. lambda n. { returns L-r } if null L then nil else if member hd L r n then u tl L n { duplicate } else hd L :: (u tl L (n+1)), { new value } member = lambda e. lambda L. lambda n. { is e in L ? } if n = 0 then false { n is current length of r } else if e = hd L then true else member e tl L (n-1) in r in unique (1::2::1::3::2::4::nil) {<B> Circular Program Unique </B>}

output area