YES

weighted path orer with linear polynomial on natural numbers:

  min_A(x1) = x1 + 1
  cons_A(x1,x2) = x1 + x2 + 2
  nil_A = 0
  le_A(x1,x2) = x2 + 2
  s_A(x1) = x1 + 3
  0_A = 5
  false_A = 6
  true_A = 4


precedence:

  nil > true > false > 0 > le > s > min > cons


status:
min: [1]
cons: [1,2]
nil: []
le: []
s: [1]
0: []
false: []
true: []

the condition le(x1,min(nil())) ->> true() is unsatisfiable for all substitutions.