YES

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

  le(0,y) -> true
  le(s(x),0) -> false
  le(s(x),s(y)) -> le(x,y)
  minus(0,y) -> 0
  minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
  if_minus(true,s(x),y) -> 0
  if_minus(false,s(x),y) -> s(minus(x,y))
  gcd(0,y) -> y
  gcd(s(x),0) -> s(x)
  gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
  if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
  if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),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

  le#(s(x),s(y)) -> le#(x,y)
  minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
  minus#(s(x),y) -> le#(s(x),y)
  if_minus#(false,s(x),y) -> minus#(x,y)
  gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
  gcd#(s(x),s(y)) -> le#(y,x)
  if_gcd#(true,s(x),s(y)) -> gcd#(minus(x,y),s(y))
  if_gcd#(true,s(x),s(y)) -> minus#(x,y)
  if_gcd#(false,s(x),s(y)) -> gcd#(minus(y,x),s(x))
  if_gcd#(false,s(x),s(y)) -> minus#(y,x)

and Q consists of the rules:

  le(0,y) -> true
  le(s(x),0) -> false
  le(s(x),s(y)) -> le(x,y)
  minus(0,y) -> 0
  minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
  if_minus(true,s(x),y) -> 0
  if_minus(false,s(x),y) -> s(minus(x,y))
  gcd(0,y) -> y
  gcd(s(x),0) -> s(x)
  gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
  if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
  if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
  rand(x) -> x
  rand(x) -> rand(s(x))

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

  le#_A((x1,y1,z1),(x2,y2,z2)) = (0, 0, 0)
  s_A((x1,y1,z1)) = (0, 3, 1)
  minus#_A((x1,y1,z1),(x2,y2,z2)) = (1, 3, 1)
  if_minus#_A((x1,y1,z1),(x2,y2,z2),(x3,y3,z3)) = (x2 + 1, y2, 1)
  le_A((x1,y1,z1),(x2,y2,z2)) = (2, 2, 1)
  false_A = (1, 1, 2)
  gcd#_A((x1,y1,z1),(x2,y2,z2)) = (2, 1, 2)
  if_gcd#_A((x1,y1,z1),(x2,y2,z2),(x3,y3,z3)) = (2, 1, 2)
  true_A = (1, 0, 1)
  minus_A((x1,y1,z1),(x2,y2,z2)) = (3, 0, 1)
  0_A = (1, 1, 2)
  if_minus_A((x1,y1,z1),(x2,y2,z2),(x3,y3,z3)) = (2, 4, 2)
  gcd_A((x1,y1,z1),(x2,y2,z2)) = (x2 + 1, y2 + 1, 0)
  if_gcd_A((x1,y1,z1),(x2,y2,z2),(x3,y3,z3)) = (x2 + 1, y2 + 1, 0)
  rand_A((x1,y1,z1)) = (x1 + 1, 1, 0)

strictly orients the following dependency pairs:

  minus#(s(x),y) -> le#(s(x),y)
  gcd#(s(x),s(y)) -> le#(y,x)
  if_gcd#(true,s(x),s(y)) -> minus#(x,y)
  if_gcd#(false,s(x),s(y)) -> minus#(y,x)

We remove them from the set of dependency pairs.

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

  le#_A((x1,y1,z1),(x2,y2,z2)) = (x1, y1, z1)
  s_A((x1,y1,z1)) = (x1, y1 + 3, z1 + 2)
  minus#_A((x1,y1,z1),(x2,y2,z2)) = (x1, y1 + 1, z1 + 1)
  if_minus#_A((x1,y1,z1),(x2,y2,z2),(x3,y3,z3)) = (x2, y2, z2)
  le_A((x1,y1,z1),(x2,y2,z2)) = (x2 + 1, y2 + 1, z2 + 1)
  false_A = (0, 1, 3)
  gcd#_A((x1,y1,z1),(x2,y2,z2)) = (x1 + x2, y1 + y2 + 1, 0)
  if_gcd#_A((x1,y1,z1),(x2,y2,z2),(x3,y3,z3)) = (x2 + x3, y2 + y3, 1)
  true_A = (0, 2, 2)
  minus_A((x1,y1,z1),(x2,y2,z2)) = (x1, y1 + 1, z1 + 1)
  0_A = (0, 1, 1)
  if_minus_A((x1,y1,z1),(x2,y2,z2),(x3,y3,z3)) = (x2, y2 + 1, z2 + 1)
  gcd_A((x1,y1,z1),(x2,y2,z2)) = (x1 + x2 + 1, 1, 0)
  if_gcd_A((x1,y1,z1),(x2,y2,z2),(x3,y3,z3)) = (x2 + x3 + 1, 1, 0)
  rand_A((x1,y1,z1)) = (x1 + 1, 1, 0)

strictly orients the following dependency pairs:

  le#(s(x),s(y)) -> le#(x,y)
  minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
  if_minus#(false,s(x),y) -> minus#(x,y)
  gcd#(s(x),s(y)) -> if_gcd#(le(y,x),s(x),s(y))
  if_gcd#(true,s(x),s(y)) -> gcd#(minus(x,y),s(y))
  if_gcd#(false,s(x),s(y)) -> gcd#(minus(y,x),s(x))

We remove them from the set of dependency pairs.

No dependency pair remains.