YES

TRS:

  gcd(x,0()) -> x
  gcd(0(),y) -> y
  gcd(s(x),s(y)) -> if(<(x,y),gcd(s(x),-(y,x)),gcd(-(x,y),s(y)))

linear polynomial interpretations on N:

  gcd_A(x1,x2) = x1 + x2
  gcd#_A(x1,x2) = x1 + x2
  0_A = 0
  0#_A = 0
  s_A(x1) = x1 + 1
  s#_A(x1) = 0
  if_A(x1,x2,x3) = x2
  if#_A(x1,x2,x3) = 0
  <_A(x1,x2) = x2 + 1
  <#_A(x1,x2) = 0
  -_A(x1,x2) = 0
  -#_A(x1,x2) = 0

precedence:

  gcd = if > 0 = s = < = -