YES

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

  f(0) -> true
  f(1) -> false
  f(s(x)) -> f(x)
  if(true,s(x),s(y)) -> s(x)
  if(false,s(x),s(y)) -> s(y)
  g(x,c(y)) -> c(g(x,y))
  g(x,c(y)) -> g(x,if(f(x),c(g(s(x),y)),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#(s(x)) -> f#(x)
  g#(x,c(y)) -> g#(x,y)
  g#(x,c(y)) -> g#(x,if(f(x),c(g(s(x),y)),c(y)))
  g#(x,c(y)) -> if#(f(x),c(g(s(x),y)),c(y))
  g#(x,c(y)) -> f#(x)
  g#(x,c(y)) -> g#(s(x),y)

and Q consists of the rules:

  f(0) -> true
  f(1) -> false
  f(s(x)) -> f(x)
  if(true,s(x),s(y)) -> s(x)
  if(false,s(x),s(y)) -> s(y)
  g(x,c(y)) -> c(g(x,y))
  g(x,c(y)) -> g(x,if(f(x),c(g(s(x),y)),c(y)))
  rand(x) -> x
  rand(x) -> rand(s(x))

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

  f#_A((x1,y1,z1)) = (0, 0, 0)
  s_A((x1,y1,z1)) = (x1, y1 + 1, z1 + 1)
  g#_A((x1,y1,z1),(x2,y2,z2)) = (x1 + 2, 1, 1)
  c_A((x1,y1,z1)) = (1, 2, 2)
  if_A((x1,y1,z1),(x2,y2,z2),(x3,y3,z3)) = (x2 + x3 + 1, 3, 1)
  f_A((x1,y1,z1)) = (x1 + 1, y1 + 1, z1 + 1)
  g_A((x1,y1,z1),(x2,y2,z2)) = (x1 + 2, 1, 1)
  if#_A((x1,y1,z1),(x2,y2,z2),(x3,y3,z3)) = (x1, 2, 0)
  0_A = (1, 1, 1)
  true_A = (1, 3, 3)
  1_A = (1, 1, 1)
  false_A = (1, 3, 3)
  rand_A((x1,y1,z1)) = (x1 + 1, 1, 0)

strictly orients the following dependency pairs:

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

We remove them from the set of dependency pairs.

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

  f#_A((x1,y1,z1)) = (x1, y1, z1)
  s_A((x1,y1,z1)) = (x1, y1 + 1, z1 + 1)
  g#_A((x1,y1,z1),(x2,y2,z2)) = (0, 0, 0)
  c_A((x1,y1,z1)) = (1, 2, 2)
  if_A((x1,y1,z1),(x2,y2,z2),(x3,y3,z3)) = (x2 + x3 + 1, 0, 0)
  f_A((x1,y1,z1)) = (x1 + 1, y1 + 1, z1 + 1)
  g_A((x1,y1,z1),(x2,y2,z2)) = (x1 + 2, 1, 1)
  0_A = (1, 1, 1)
  true_A = (1, 3, 3)
  1_A = (1, 1, 1)
  false_A = (1, 3, 3)
  rand_A((x1,y1,z1)) = (x1 + 1, 1, 0)

strictly orients the following dependency pairs:

  f#(s(x)) -> f#(x)

We remove them from the set of dependency pairs.

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

  g#_A((x1,y1,z1),(x2,y2,z2)) = (x2, y2, 0)
  c_A((x1,y1,z1)) = (x1 + 2, 2, 1)
  if_A((x1,y1,z1),(x2,y2,z2),(x3,y3,z3)) = (1, 2, 0)
  f_A((x1,y1,z1)) = (2, 2, 1)
  g_A((x1,y1,z1),(x2,y2,z2)) = (x2 + 1, 3, 1)
  s_A((x1,y1,z1)) = (0, 1, 1)
  0_A = (1, 1, 1)
  true_A = (1, 1, 2)
  1_A = (1, 1, 1)
  false_A = (1, 1, 1)
  rand_A((x1,y1,z1)) = (x1 + 1, 1, 0)

strictly orients the following dependency pairs:

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

We remove them from the set of dependency pairs.

No dependency pair remains.