YES

(VAR x z y x0 x1)
(RULES
  f8(f8(f8(f8(x)))) -> f8(x)
  f8(j(z)) -> f8(f8(z))
  f2(y) -> f8(y)
  f3(y) -> f8(y)
  f4(y) -> f8(y)
  f5(y) -> f8(y)
  f6(y) -> f8(y)
  f7(y) -> f8(y)
  f1(y) -> f8(y)
  g(x0,x1) -> f8(x1)
)

(COMMENT
Termination is shown by EKBO with interpretations on natural numbers

  g_A(x1,x2) = x1 + x2 + 1
  j_A(x1) = x1 + 2
  f1_A(x1) = x1 + 1
  f2_A(x1) = x1 + 1
  f3_A(x1) = x1 + 1
  f4_A(x1) = x1 + 1
  f5_A(x1) = x1 + 1
  f6_A(x1) = x1 + 1
  f7_A(x1) = x1 + 1
  f8_A(x1) = x1 + 1
  g#_A(x1,x2) = x1
  f1#_A(x1) = x1
  f8#_A(x1) = x1

weights

  w0 = 2
  w(g) = 0
  w(j) = 1
  w(f1) = 1
  w(f2) = 1
  w(f3) = 1
  w(f4) = 1
  w(f5) = 1
  w(f6) = 1
  w(f7) = 1
  w(f8) = 1

and precedence:

g > f7 > f5 > f4 > f3 > f2 > j > f1 > f6 > f8
)