MAYBE

Problem:
 fact(X) -> if(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X))))
 add(0(),X) -> X
 add(s(X),Y) -> s(add(X,Y))
 prod(0(),X) -> 0()
 prod(s(X),Y) -> add(Y,prod(X,Y))
 if(true(),X,Y) -> activate(X)
 if(false(),X,Y) -> activate(Y)
 zero(0()) -> true()
 zero(s(X)) -> false()
 p(s(X)) -> X
 s(X) -> n__s(X)
 0() -> n__0()
 prod(X1,X2) -> n__prod(X1,X2)
 fact(X) -> n__fact(X)
 p(X) -> n__p(X)
 activate(n__s(X)) -> s(activate(X))
 activate(n__0()) -> 0()
 activate(n__prod(X1,X2)) -> prod(activate(X1),activate(X2))
 activate(n__fact(X)) -> fact(activate(X))
 activate(n__p(X)) -> p(activate(X))
 activate(X) -> X

Proof:
 DP Processor:
  DPs:
   fact#(X) -> zero#(X)
   fact#(X) -> if#(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X))))
   add#(s(X),Y) -> add#(X,Y)
   add#(s(X),Y) -> s#(add(X,Y))
   prod#(s(X),Y) -> prod#(X,Y)
   prod#(s(X),Y) -> add#(Y,prod(X,Y))
   if#(true(),X,Y) -> activate#(X)
   if#(false(),X,Y) -> activate#(Y)
   activate#(n__s(X)) -> activate#(X)
   activate#(n__s(X)) -> s#(activate(X))
   activate#(n__0()) -> 0#()
   activate#(n__prod(X1,X2)) -> activate#(X2)
   activate#(n__prod(X1,X2)) -> activate#(X1)
   activate#(n__prod(X1,X2)) -> prod#(activate(X1),activate(X2))
   activate#(n__fact(X)) -> activate#(X)
   activate#(n__fact(X)) -> fact#(activate(X))
   activate#(n__p(X)) -> activate#(X)
   activate#(n__p(X)) -> p#(activate(X))
  TRS:
   fact(X) -> if(zero(X),n__s(n__0()),n__prod(X,n__fact(n__p(X))))
   add(0(),X) -> X
   add(s(X),Y) -> s(add(X,Y))
   prod(0(),X) -> 0()
   prod(s(X),Y) -> add(Y,prod(X,Y))
   if(true(),X,Y) -> activate(X)
   if(false(),X,Y) -> activate(Y)
   zero(0()) -> true()
   zero(s(X)) -> false()
   p(s(X)) -> X
   s(X) -> n__s(X)
   0() -> n__0()
   prod(X1,X2) -> n__prod(X1,X2)
   fact(X) -> n__fact(X)
   p(X) -> n__p(X)
   activate(n__s(X)) -> s(activate(X))
   activate(n__0()) -> 0()
   activate(n__prod(X1,X2)) -> prod(activate(X1),activate(X2))
   activate(n__fact(X)) -> fact(activate(X))
   activate(n__p(X)) -> p(activate(X))
   activate(X) -> X
  Open