NO

3 decompositions
 #1 -----------
   1: b() -> a()
   2: b() -> c()

 #2 -----------
   4: d() -> c()
   5: d() -> e()

 #3 -----------
   6: f(x1,a()) -> A()
   7: f(x2,e()) -> A()
   8: f(x3,A()) -> A()
   9: f(c(),x4) -> A()

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

original is
   1: b() -> a()
   2: b() -> c()
   3: c() -> c()
   4: d() -> c()
   5: d() -> e()
   6: f(x1,a()) -> A()
   7: f(x2,e()) -> A()
   8: f(x3,A()) -> A()
   9: f(c(),x4) -> A()

reduced to
   1: b() -> a()
   2: b() -> c()
   4: d() -> c()
   5: d() -> e()
   6: f(x1,a()) -> A()
   7: f(x2,e()) -> A()
   8: f(x3,A()) -> A()
   9: f(c(),x4) -> A()