YES

weighted path orer with linear polynomial on natural numbers:

  add_A(x1,x2) = x1 + x2 + 2
  s_A(x1) = 1
  0_A = 0
  gcd_A(x1,x2) = x1 + x2 + 1
  leq_A(x1,x2) = 0
  false_A = 1


precedence:

  leq > false > gcd > 0 > s > add


status:
add: [1,2]
s: []
0: []
gcd: []
leq: []
false: []

the condition leq(x2,add(x,x2)) ->> false() is unsatisfiable for all substitutions.