YES

TRS:

  f_0(x) -> a()
  f_1(x) -> g_1(x,x)
  g_1(s(x),y) -> b(f_0(y),g_1(x,y))
  f_2(x) -> g_2(x,x)
  g_2(s(x),y) -> b(f_1(y),g_2(x,y))
  f_3(x) -> g_3(x,x)
  g_3(s(x),y) -> b(f_2(y),g_3(x,y))
  f_4(x) -> g_4(x,x)
  g_4(s(x),y) -> b(f_3(y),g_4(x,y))
  f_5(x) -> g_5(x,x)
  g_5(s(x),y) -> b(f_4(y),g_5(x,y))
  f_6(x) -> g_6(x,x)
  g_6(s(x),y) -> b(f_5(y),g_6(x,y))
  f_7(x) -> g_7(x,x)
  g_7(s(x),y) -> b(f_6(y),g_7(x,y))
  f_8(x) -> g_8(x,x)
  g_8(s(x),y) -> b(f_7(y),g_8(x,y))
  f_9(x) -> g_9(x,x)
  g_9(s(x),y) -> b(f_8(y),g_9(x,y))
  f_10(x) -> g_10(x,x)
  g_10(s(x),y) -> b(f_9(y),g_10(x,y))

linear polynomial interpretations on N:

  f_0_A(x1) = 1
  f_0#_A(x1) = 1
  a_A = 0
  a#_A = 0
  f_1_A(x1) = 1
  f_1#_A(x1) = 3
  g_1_A(x1,x2) = 1
  g_1#_A(x1,x2) = 2
  s_A(x1) = x1 + 1
  s#_A(x1) = 0
  b_A(x1,x2) = x1
  b#_A(x1,x2) = x2
  f_2_A(x1) = 1
  f_2#_A(x1) = 5
  g_2_A(x1,x2) = 1
  g_2#_A(x1,x2) = 4
  f_3_A(x1) = 1
  f_3#_A(x1) = 7
  g_3_A(x1,x2) = 1
  g_3#_A(x1,x2) = 6
  f_4_A(x1) = 1
  f_4#_A(x1) = x1 + 8
  g_4_A(x1,x2) = 1
  g_4#_A(x1,x2) = x1 + 7
  f_5_A(x1) = 1
  f_5#_A(x1) = x1 + 10
  g_5_A(x1,x2) = 1
  g_5#_A(x1,x2) = x2 + 9
  f_6_A(x1) = x1 + 1
  f_6#_A(x1) = x1 + 12
  g_6_A(x1,x2) = x2 + 1
  g_6#_A(x1,x2) = x2 + 11
  f_7_A(x1) = x1 + 1
  f_7#_A(x1) = x1 + 14
  g_7_A(x1,x2) = x2 + 1
  g_7#_A(x1,x2) = x2 + 13
  f_8_A(x1) = x1 + 1
  f_8#_A(x1) = x1 + 16
  g_8_A(x1,x2) = x2 + 1
  g_8#_A(x1,x2) = x2 + 15
  f_9_A(x1) = x1 + 1
  f_9#_A(x1) = x1 + 18
  g_9_A(x1,x2) = x2 + 1
  g_9#_A(x1,x2) = x2 + 17
  f_10_A(x1) = x1 + 1
  f_10#_A(x1) = x1 + 20
  g_10_A(x1,x2) = x2 + 1
  g_10#_A(x1,x2) = x2 + 19

precedence:

  f_10 = g_10 > f_9 > g_9 > f_8 > s = g_8 > f_7 > g_7 > f_6 > g_6 > f_5 > g_5 > f_4 > g_4 > f_3 > g_3 > f_2 > g_2 > f_1 > g_1 > f_0 = b > a