YES

We show the termination of R/S, where R is

  half(0) -> 0
  half(s(0)) -> 0
  half(s(s(x))) -> s(half(x))
  lastbit(0) -> 0
  lastbit(s(0)) -> s(0)
  lastbit(s(s(x))) -> lastbit(x)
  conv(0) -> cons(nil,0)
  conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))

and S is:

  rand(x) -> x
  rand(x) -> rand(s(x))

Since R almost dominates S and S is non-duplicating,
it is enough to show finiteness of (P, Q).
Here P consists of the dependency pairs

  half#(s(s(x))) -> half#(x)
  lastbit#(s(s(x))) -> lastbit#(x)
  conv#(s(x)) -> conv#(half(s(x)))
  conv#(s(x)) -> half#(s(x))
  conv#(s(x)) -> lastbit#(s(x))

and Q consists of the rules:

  half(0) -> 0
  half(s(0)) -> 0
  half(s(s(x))) -> s(half(x))
  lastbit(0) -> 0
  lastbit(s(0)) -> s(0)
  lastbit(s(s(x))) -> lastbit(x)
  conv(0) -> cons(nil,0)
  conv(s(x)) -> cons(conv(half(s(x))),lastbit(s(x)))
  rand(x) -> x
  rand(x) -> rand(s(x))

The weakly monotone algebra (N^3, >_lex) with

  half#_A((x1,y1,z1)) = (x1, 2, 3)
  s_A((x1,y1,z1)) = (x1, y1 + 1, z1 + 1)
  lastbit#_A((x1,y1,z1)) = (x1, y1 + 1, z1)
  conv#_A((x1,y1,z1)) = (x1 + 1, y1, z1 + 1)
  half_A((x1,y1,z1)) = (x1, y1, z1)
  0_A = (1, 2, 2)
  lastbit_A((x1,y1,z1)) = (2, 1, 1)
  conv_A((x1,y1,z1)) = (1, 1, 1)
  cons_A((x1,y1,z1),(x2,y2,z2)) = (0, 2, 2)
  nil_A = (1, 1, 1)
  rand_A((x1,y1,z1)) = (x1 + 1, 1, 0)

strictly orients the following dependency pairs:

  lastbit#(s(s(x))) -> lastbit#(x)
  conv#(s(x)) -> half#(s(x))
  conv#(s(x)) -> lastbit#(s(x))

We remove them from the set of dependency pairs.

The weakly monotone algebra (N^3, >_lex) with

  half#_A((x1,y1,z1)) = (x1, 0, 0)
  s_A((x1,y1,z1)) = (x1, y1 + 1, 2)
  conv#_A((x1,y1,z1)) = (x1, y1, z1)
  half_A((x1,y1,z1)) = (x1, y1, 1)
  0_A = (1, 1, 1)
  lastbit_A((x1,y1,z1)) = (x1 + 1, y1 + 1, z1)
  conv_A((x1,y1,z1)) = (x1 + 2, 1, 1)
  cons_A((x1,y1,z1),(x2,y2,z2)) = (x2, 2, 2)
  nil_A = (1, 1, 1)
  rand_A((x1,y1,z1)) = (x1 + 1, 1, 0)

strictly orients the following dependency pairs:

  conv#(s(x)) -> conv#(half(s(x)))

We remove them from the set of dependency pairs.

The weakly monotone algebra (N^3, >_lex) with

  half#_A((x1,y1,z1)) = (x1, y1, z1)
  s_A((x1,y1,z1)) = (x1, y1 + 1, 3)
  half_A((x1,y1,z1)) = (x1 + 1, y1 + 1, 2)
  0_A = (1, 2, 2)
  lastbit_A((x1,y1,z1)) = (2, 1, 1)
  conv_A((x1,y1,z1)) = (2, 1, 1)
  cons_A((x1,y1,z1),(x2,y2,z2)) = (x1, y1, 1)
  nil_A = (1, 2, 1)
  rand_A((x1,y1,z1)) = (x1 + 1, 1, 0)

strictly orients the following dependency pairs:

  half#(s(s(x))) -> half#(x)

We remove them from the set of dependency pairs.

No dependency pair remains.