YES

TRS:

  din(der(plus(X,Y))) -> u21(din(der(X)),X,Y)
  u21(dout(DX),X,Y) -> u22(din(der(Y)),X,Y,DX)
  u22(dout(DY),X,Y,DX) -> dout(plus(DX,DY))
  din(der(times(X,Y))) -> u31(din(der(X)),X,Y)
  u31(dout(DX),X,Y) -> u32(din(der(Y)),X,Y,DX)
  u32(dout(DY),X,Y,DX) -> dout(plus(times(X,DY),times(Y,DX)))
  din(der(der(X))) -> u41(din(der(X)),X)
  u41(dout(DX),X) -> u42(din(der(DX)),X,DX)
  u42(dout(DDX),X,DX) -> dout(DDX)

max/plus interpretations on N:

  din_A(x1) = max{16, 1 + x1}
  din#_A(x1) = max{16, 1 + x1}
  der_A(x1) = max{65, 65 + x1}
  der#_A(x1) = max{65, 65 + x1}
  plus_A(x1,x2) = max{4, x1, 7 + x2}
  plus#_A(x1,x2) = max{4, x1, 7 + x2}
  u21_A(x1,x2,x3) = max{71, x1, 64 + x2, 73 + x3}
  u21#_A(x1,x2,x3) = max{71, x1, 64 + x2, 73 + x3}
  dout_A(x1) = max{52, 36 + x1}
  dout#_A(x1) = max{52, 36 + x1}
  u22_A(x1,x2,x3,x4) = max{53, 7 + x1, 53 + x2, 53 + x3, 36 + x4}
  u22#_A(x1,x2,x3,x4) = max{53, 7 + x1, 53 + x2, 53 + x3, 36 + x4}
  times_A(x1,x2) = max{15, 14 + x1, 3 + x2}
  times#_A(x1,x2) = max{15, 14 + x1, 3 + x2}
  u31_A(x1,x2,x3) = max{67, 12 + x1, 64 + x2, 69 + x3}
  u31#_A(x1,x2,x3) = max{67, 12 + x1, 64 + x2, 69 + x3}
  u32_A(x1,x2,x3,x4) = max{52, 3 + x1, 53 + x2, 58 + x3, 47 + x4}
  u32#_A(x1,x2,x3,x4) = max{52, 3 + x1, 53 + x2, 58 + x3, 47 + x4}
  u41_A(x1,x2) = max{84, 48 + x1, 84 + x2}
  u41#_A(x1,x2) = max{84, 48 + x1, 84 + x2}
  u42_A(x1,x2,x3) = max{53, 17 + x1, 53 + x2, 53 + x3}
  u42#_A(x1,x2,x3) = max{53, 17 + x1, 53 + x2, 53 + x3}

precedence:

  u41 > din = u42 > u31 > u32 > times > der > u21 > u22 > plus = dout