YES

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

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

  g_A(x1,x2) = 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) = 0
  j#_A(x1) = x1
  f2#_A(x1) = x1
  f7#_A(x1) = x1
  f8#_A(x1) = x1

and precedence:

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