YES

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

  f(x,c(y)) -> f(x,s(f(y,y)))
  f(s(x),y) -> f(x,s(c(y)))

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

  f#(x,c(y)) -> f#(x,s(f(y,y)))
  f#(x,c(y)) -> f#(y,y)
  f#(s(x),y) -> f#(x,s(c(y)))

and Q consists of the rules:

  f(x,c(y)) -> f(x,s(f(y,y)))
  f(s(x),y) -> f(x,s(c(y)))
  rand(x) -> x
  rand(x) -> rand(s(x))

The weakly monotone algebra (N^2, >_lex) with

  f#_A((x1,y1),(x2,y2)) = (x2, 0)
  c_A((x1,y1)) = (x1 + 1, 1)
  s_A((x1,y1)) = (0, 2)
  f_A((x1,y1),(x2,y2)) = (1, 1)
  rand_A((x1,y1)) = (x1 + 1, 0)

strictly orients the following dependency pairs:

  f#(x,c(y)) -> f#(x,s(f(y,y)))
  f#(x,c(y)) -> f#(y,y)

We remove them from the set of dependency pairs.

The weakly monotone algebra (N^2, >_lex) with

  f#_A((x1,y1),(x2,y2)) = (x1, y1)
  s_A((x1,y1)) = (x1, y1 + 1)
  c_A((x1,y1)) = (2, 0)
  f_A((x1,y1),(x2,y2)) = (x1 + 1, 0)
  rand_A((x1,y1)) = (x1 + 1, 0)

strictly orients the following dependency pairs:

  f#(s(x),y) -> f#(x,s(c(y)))

We remove them from the set of dependency pairs.

No dependency pair remains.