YES

TRS:

  0(#()) -> #()
  +(#(),x) -> x
  +(x,#()) -> x
  +(0(x),0(y)) -> 0(+(x,y))
  +(0(x),1(y)) -> 1(+(x,y))
  +(1(x),0(y)) -> 1(+(x,y))
  +(1(x),1(y)) -> 0(+(+(x,y),1(#())))
  +(+(x,y),z) -> +(x,+(y,z))
  -(#(),x) -> #()
  -(x,#()) -> x
  -(0(x),0(y)) -> 0(-(x,y))
  -(0(x),1(y)) -> 1(-(-(x,y),1(#())))
  -(1(x),0(y)) -> 1(-(x,y))
  -(1(x),1(y)) -> 0(-(x,y))
  not(true()) -> false()
  not(false()) -> true()
  if(true(),x,y) -> x
  if(false(),x,y) -> y
  ge(0(x),0(y)) -> ge(x,y)
  ge(0(x),1(y)) -> not(ge(y,x))
  ge(1(x),0(y)) -> ge(x,y)
  ge(1(x),1(y)) -> ge(x,y)
  ge(x,#()) -> true()
  ge(#(),0(x)) -> ge(#(),x)
  ge(#(),1(x)) -> false()
  log(x) -> -(log'(x),1(#()))
  log'(#()) -> #()
  log'(1(x)) -> +(log'(x),1(#()))
  log'(0(x)) -> if(ge(x,1(#())),+(log'(x),1(#())),#())

linear polynomial interpretations on N:

  0_A(x1) = x1 + 4
  0#_A(x1) = 7
  #_A = 0
  ##_A = 5
  +_A(x1,x2) = x1 + x2
  +#_A(x1,x2) = x1 + x2
  1_A(x1) = x1 + 4
  1#_A(x1) = 6
  -_A(x1,x2) = x1
  -#_A(x1,x2) = x1 + x2 + 6
  not_A(x1) = 1
  not#_A(x1) = 8
  true_A = 1
  true#_A = 0
  false_A = 1
  false#_A = 7
  if_A(x1,x2,x3) = x2 + x3
  if#_A(x1,x2,x3) = 0
  ge_A(x1,x2) = x1 + 1
  ge#_A(x1,x2) = x1 + x2 + 8
  log_A(x1) = x1 + 1
  log#_A(x1) = x1 + 12
  log'_A(x1) = x1 + 1
  log'#_A(x1) = x1 + 9

precedence:

  - > 0 > if = ge > not = log' > 1 > + > # > false = log > true