NO

1 decompositions
 #1 -----------
   1: a(x1) -> b(x1)
   2: a(x2) -> c(c(x2))
   3: b(x3) -> g(b(x3))

unjoinable peak
    g(b(x9))
*<- a(x9) ->*
    c(c(x9))
NOTE: input TRS is reduced

original is
   1: a(x1) -> b(x1)
   2: a(x2) -> c(c(x2))
   3: b(x3) -> g(b(x3))
   4: c(x4) -> c(x4)

reduced to
   1: a(x1) -> b(x1)
   2: a(x2) -> c(c(x2))
   3: b(x3) -> g(b(x3))