- : unit = ()
- : unit = ()
h : heuristic = <heuristic>
- : unit = ()
APPLY CRITERIA (Marked dependency pairs)

TRS termination of:
[1] a__fact(X) -> a__if(a__zero(mark(X)),s(0),prod(X,fact(p(X))))
[2] a__add(0,X) -> mark(X)
[3] a__add(s(X),Y) -> s(a__add(mark(X),mark(Y)))
[4] a__prod(0,X) -> 0
[5] a__prod(s(X),Y) -> a__add(mark(Y),a__prod(mark(X),mark(Y)))
[6] a__if(true,X,Y) -> mark(X)
[7] a__if(false,X,Y) -> mark(Y)
[8] a__zero(0) -> true
[9] a__zero(s(X)) -> false
[10] a__p(s(X)) -> mark(X)
[11] mark(fact(X)) -> a__fact(mark(X))
[12] mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3)
[13] mark(zero(X)) -> a__zero(mark(X))
[14] mark(prod(X1,X2)) -> a__prod(mark(X1),mark(X2))
[15] mark(p(X)) -> a__p(mark(X))
[16] mark(add(X1,X2)) -> a__add(mark(X1),mark(X2))
[17] mark(s(X)) -> s(mark(X))
[18] mark(0) -> 0
[19] mark(true) -> true
[20] mark(false) -> false
[21] a__fact(X) -> fact(X)
[22] a__if(X1,X2,X3) -> if(X1,X2,X3)
[23] a__zero(X) -> zero(X)
[24] a__prod(X1,X2) -> prod(X1,X2)
[25] a__p(X) -> p(X)
[26] a__add(X1,X2) -> add(X1,X2)


Sub problem: 
guided: 

DP termination of:
<Marked_a__fact(X) , Marked_a__if(a__zero(mark(X)),s(0),prod(X,fact(p(X))))>
<Marked_a__fact(X) , Marked_a__zero(mark(X))>
<Marked_a__fact(X) , Marked_mark(X)>
<Marked_a__add(0,X) , Marked_mark(X)>
<Marked_a__add(s(X),Y) , Marked_a__add(mark(X),mark(Y))>
<Marked_a__add(s(X),Y) , Marked_mark(X)>
<Marked_a__add(s(X),Y) , Marked_mark(Y)>
<Marked_a__prod(s(X),Y) , Marked_a__add(mark(Y),a__prod(mark(X),mark(Y)))>
<Marked_a__prod(s(X),Y) , Marked_mark(Y)>
<Marked_a__prod(s(X),Y) , Marked_a__prod(mark(X),mark(Y))>
<Marked_a__prod(s(X),Y) , Marked_mark(X)>
<Marked_a__prod(s(X),Y) , Marked_mark(Y)>
<Marked_a__if(true,X,Y) , Marked_mark(X)>
<Marked_a__if(false,X,Y) , Marked_mark(Y)>
<Marked_a__p(s(X)) , Marked_mark(X)>
<Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
<Marked_mark(fact(X)) , Marked_mark(X)>
<Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
<Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
<Marked_mark(zero(X)) , Marked_a__zero(mark(X))>
<Marked_mark(zero(X)) , Marked_mark(X)>
<Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
<Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
<Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
<Marked_mark(p(X)) , Marked_a__p(mark(X))>
<Marked_mark(p(X)) , Marked_mark(X)>
<Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
<Marked_mark(add(X1,X2)) , Marked_mark(X1)>
<Marked_mark(add(X1,X2)) , Marked_mark(X2)>
<Marked_mark(s(X)) , Marked_mark(X)>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 1 components:

{  <Marked_mark(s(X)) , Marked_mark(X)> 
      --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_mark(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_mark(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_mark(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_mark(s(X)) , Marked_mark(X)> --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_mark(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_mark(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_mark(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_mark(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_mark(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_mark(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_mark(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_mark(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_mark(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_mark(add(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_mark(add(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_mark(add(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_mark(add(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_mark(add(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_mark(add(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_mark(add(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_mark(add(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_mark(add(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))> 
     --> <Marked_a__add(s(X),Y) , Marked_mark(Y)>
  <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))> 
     --> <Marked_a__add(s(X),Y) , Marked_mark(X)>
  <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))> 
     --> <Marked_a__add(s(X),Y) , Marked_a__add(mark(X),mark(Y))>
  <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))> 
     --> <Marked_a__add(0,X) , Marked_mark(X)>
  <Marked_mark(p(X)) , Marked_mark(X)> --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_mark(p(X)) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_mark(p(X)) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_mark(p(X)) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_mark(p(X)) , Marked_mark(X)> --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_mark(p(X)) , Marked_mark(X)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_mark(p(X)) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_mark(p(X)) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_mark(p(X)) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_mark(p(X)) , Marked_mark(X)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_mark(p(X)) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_mark(p(X)) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_mark(p(X)) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_mark(p(X)) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_mark(p(X)) , Marked_a__p(mark(X))> 
     --> <Marked_a__p(s(X)) , Marked_mark(X)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X2)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_mark(prod(X1,X2)) , Marked_mark(X1)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))> 
     --> <Marked_a__prod(s(X),Y) , Marked_mark(Y)>
  <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))> 
     --> <Marked_a__prod(s(X),Y) , Marked_mark(X)>
  <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))> 
     --> <Marked_a__prod(s(X),Y) , Marked_a__prod(mark(X),mark(Y))>
  <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))> 
     --> <Marked_a__prod(s(X),Y) , Marked_mark(Y)>
  <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))> 
     --> <Marked_a__prod(s(X),Y) , Marked_a__add(mark(Y),
                                    a__prod(mark(X),mark(Y)))>
  <Marked_mark(zero(X)) , Marked_mark(X)> 
     --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_mark(zero(X)) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_mark(zero(X)) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_mark(zero(X)) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_mark(zero(X)) , Marked_mark(X)> 
     --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_mark(zero(X)) , Marked_mark(X)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_mark(zero(X)) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_mark(zero(X)) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_mark(zero(X)) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_mark(zero(X)) , Marked_mark(X)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_mark(zero(X)) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_mark(zero(X)) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_mark(zero(X)) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_mark(zero(X)) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)> 
     --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)> 
     --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)> 
     --> <Marked_a__if(false,X,Y) , Marked_mark(Y)>
  <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)> 
     --> <Marked_a__if(true,X,Y) , Marked_mark(X)>
  <Marked_mark(fact(X)) , Marked_mark(X)> 
     --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_mark(fact(X)) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_mark(fact(X)) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_mark(fact(X)) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_mark(fact(X)) , Marked_mark(X)> 
     --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_mark(fact(X)) , Marked_mark(X)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_mark(fact(X)) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_mark(fact(X)) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_mark(fact(X)) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_mark(fact(X)) , Marked_mark(X)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_mark(fact(X)) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_mark(fact(X)) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_mark(fact(X)) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_mark(fact(X)) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_mark(fact(X)) , Marked_a__fact(mark(X))> 
     --> <Marked_a__fact(X) , Marked_mark(X)>
  <Marked_mark(fact(X)) , Marked_a__fact(mark(X))> 
     --> <Marked_a__fact(X) , Marked_a__if(a__zero(mark(X)),s(0),
                               prod(X,fact(p(X))))>
  <Marked_a__p(s(X)) , Marked_mark(X)> --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_a__p(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_a__p(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_a__p(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_a__p(s(X)) , Marked_mark(X)> --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_a__p(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_a__p(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_a__p(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_a__p(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_a__p(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_a__p(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_a__p(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_a__p(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_a__p(s(X)) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_a__if(false,X,Y) , Marked_mark(Y)> 
     --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_a__if(false,X,Y) , Marked_mark(Y)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_a__if(false,X,Y) , Marked_mark(Y)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_a__if(false,X,Y) , Marked_mark(Y)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_a__if(false,X,Y) , Marked_mark(Y)> 
     --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_a__if(false,X,Y) , Marked_mark(Y)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_a__if(false,X,Y) , Marked_mark(Y)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_a__if(false,X,Y) , Marked_mark(Y)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_a__if(false,X,Y) , Marked_mark(Y)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_a__if(false,X,Y) , Marked_mark(Y)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_a__if(false,X,Y) , Marked_mark(Y)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_a__if(false,X,Y) , Marked_mark(Y)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_a__if(false,X,Y) , Marked_mark(Y)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_a__if(false,X,Y) , Marked_mark(Y)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_a__if(true,X,Y) , Marked_mark(X)> 
     --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_a__if(true,X,Y) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_a__if(true,X,Y) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_a__if(true,X,Y) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_a__if(true,X,Y) , Marked_mark(X)> 
     --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_a__if(true,X,Y) , Marked_mark(X)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_a__if(true,X,Y) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_a__if(true,X,Y) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_a__if(true,X,Y) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_a__if(true,X,Y) , Marked_mark(X)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_a__if(true,X,Y) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_a__if(true,X,Y) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_a__if(true,X,Y) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_a__if(true,X,Y) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_a__prod(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_a__prod(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_a__prod(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_a__prod(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_a__prod(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_a__prod(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_a__prod(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_a__prod(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_a__prod(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_a__prod(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_a__prod(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_a__prod(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_a__prod(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_a__prod(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_a__prod(s(X),Y) , Marked_a__prod(mark(X),mark(Y))> 
     --> <Marked_a__prod(s(X),Y) , Marked_mark(Y)>
  <Marked_a__prod(s(X),Y) , Marked_a__prod(mark(X),mark(Y))> 
     --> <Marked_a__prod(s(X),Y) , Marked_mark(X)>
  <Marked_a__prod(s(X),Y) , Marked_a__prod(mark(X),mark(Y))> 
     --> <Marked_a__prod(s(X),Y) , Marked_a__prod(mark(X),mark(Y))>
  <Marked_a__prod(s(X),Y) , Marked_a__prod(mark(X),mark(Y))> 
     --> <Marked_a__prod(s(X),Y) , Marked_mark(Y)>
  <Marked_a__prod(s(X),Y) , Marked_a__prod(mark(X),mark(Y))> 
     --> <Marked_a__prod(s(X),Y) , Marked_a__add(mark(Y),
                                    a__prod(mark(X),mark(Y)))>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_a__prod(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_a__prod(s(X),Y) , Marked_a__add(mark(Y),a__prod(mark(X),mark(Y)))> 
     --> <Marked_a__add(s(X),Y) , Marked_mark(Y)>
  <Marked_a__prod(s(X),Y) , Marked_a__add(mark(Y),a__prod(mark(X),mark(Y)))> 
     --> <Marked_a__add(s(X),Y) , Marked_mark(X)>
  <Marked_a__prod(s(X),Y) , Marked_a__add(mark(Y),a__prod(mark(X),mark(Y)))> 
     --> <Marked_a__add(s(X),Y) , Marked_a__add(mark(X),mark(Y))>
  <Marked_a__prod(s(X),Y) , Marked_a__add(mark(Y),a__prod(mark(X),mark(Y)))> 
     --> <Marked_a__add(0,X) , Marked_mark(X)>
  <Marked_a__add(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_a__add(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_a__add(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_a__add(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_a__add(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_a__add(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_a__add(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_a__add(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_a__add(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_a__add(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_a__add(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_a__add(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_a__add(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_a__add(s(X),Y) , Marked_mark(Y)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_a__add(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_a__add(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_a__add(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_a__add(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_a__add(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_a__add(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_a__add(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_a__add(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_a__add(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_a__add(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_a__add(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_a__add(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_a__add(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_a__add(s(X),Y) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_a__add(s(X),Y) , Marked_a__add(mark(X),mark(Y))> 
     --> <Marked_a__add(s(X),Y) , Marked_mark(Y)>
  <Marked_a__add(s(X),Y) , Marked_a__add(mark(X),mark(Y))> 
     --> <Marked_a__add(s(X),Y) , Marked_mark(X)>
  <Marked_a__add(s(X),Y) , Marked_a__add(mark(X),mark(Y))> 
     --> <Marked_a__add(s(X),Y) , Marked_a__add(mark(X),mark(Y))>
  <Marked_a__add(s(X),Y) , Marked_a__add(mark(X),mark(Y))> 
     --> <Marked_a__add(0,X) , Marked_mark(X)>
  <Marked_a__add(0,X) , Marked_mark(X)> 
     --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_a__add(0,X) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_a__add(0,X) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_a__add(0,X) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_a__add(0,X) , Marked_mark(X)> 
     --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_a__add(0,X) , Marked_mark(X)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_a__add(0,X) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_a__add(0,X) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_a__add(0,X) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_a__add(0,X) , Marked_mark(X)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_a__add(0,X) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_a__add(0,X) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_a__add(0,X) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_a__add(0,X) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_a__fact(X) , Marked_mark(X)> --> <Marked_mark(s(X)) , Marked_mark(X)>
  <Marked_a__fact(X) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X2)>
  <Marked_a__fact(X) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_mark(X1)>
  <Marked_a__fact(X) , Marked_mark(X)> 
     --> <Marked_mark(add(X1,X2)) , Marked_a__add(mark(X1),mark(X2))>
  <Marked_a__fact(X) , Marked_mark(X)> --> <Marked_mark(p(X)) , Marked_mark(X)>
  <Marked_a__fact(X) , Marked_mark(X)> 
     --> <Marked_mark(p(X)) , Marked_a__p(mark(X))>
  <Marked_a__fact(X) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X2)>
  <Marked_a__fact(X) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_mark(X1)>
  <Marked_a__fact(X) , Marked_mark(X)> 
     --> <Marked_mark(prod(X1,X2)) , Marked_a__prod(mark(X1),mark(X2))>
  <Marked_a__fact(X) , Marked_mark(X)> 
     --> <Marked_mark(zero(X)) , Marked_mark(X)>
  <Marked_a__fact(X) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_mark(X1)>
  <Marked_a__fact(X) , Marked_mark(X)> 
     --> <Marked_mark(if(X1,X2,X3)) , Marked_a__if(mark(X1),X2,X3)>
  <Marked_a__fact(X) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_mark(X)>
  <Marked_a__fact(X) , Marked_mark(X)> 
     --> <Marked_mark(fact(X)) , Marked_a__fact(mark(X))>
  <Marked_a__fact(X) , Marked_a__if(a__zero(mark(X)),s(0),prod(X,fact(p(X))))> 
     --> <Marked_a__if(false,X,Y) , Marked_mark(Y)>
  <Marked_a__fact(X) , Marked_a__if(a__zero(mark(X)),s(0),prod(X,fact(p(X))))> 
     --> <Marked_a__if(true,X,Y) , Marked_mark(X)>
}
APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  a__if(true,X,Y) >= mark(X) ;
  a__if(false,X,Y) >= mark(Y) ;
  a__if(X1,X2,X3) >= if(X1,X2,X3) ;
  a__zero(s(X)) >= false ;
  a__zero(0) >= true ;
  a__zero(X) >= zero(X) ;
  mark(s(X)) >= s(mark(X)) ;
  mark(0) >= 0 ;
  mark(prod(X1,X2)) >= a__prod(mark(X1),mark(X2)) ;
  mark(fact(X)) >= a__fact(mark(X)) ;
  mark(p(X)) >= a__p(mark(X)) ;
  mark(true) >= true ;
  mark(false) >= false ;
  mark(if(X1,X2,X3)) >= a__if(mark(X1),X2,X3) ;
  mark(zero(X)) >= a__zero(mark(X)) ;
  mark(add(X1,X2)) >= a__add(mark(X1),mark(X2)) ;
  a__fact(X) >= a__if(a__zero(mark(X)),s(0),prod(X,fact(p(X)))) ;
  a__fact(X) >= fact(X) ;
  a__add(s(X),Y) >= s(a__add(mark(X),mark(Y))) ;
  a__add(0,X) >= mark(X) ;
  a__add(X1,X2) >= add(X1,X2) ;
  a__prod(s(X),Y) >= a__add(mark(Y),a__prod(mark(X),mark(Y))) ;
  a__prod(0,X) >= 0 ;
  a__prod(X1,X2) >= prod(X1,X2) ;
  a__p(s(X)) >= mark(X) ;
  a__p(X) >= p(X) ;
  Marked_a__add(s(X),Y) >= Marked_a__add(mark(X),mark(Y)) ;
  Marked_a__add(s(X),Y) >= Marked_mark(X) ;
  Marked_a__add(s(X),Y) >= Marked_mark(Y) ;
  Marked_a__add(0,X) >= Marked_mark(X) ;
  Marked_a__p(s(X)) >= Marked_mark(X) ;
  Marked_a__prod(s(X),Y) >= Marked_a__add(mark(Y),a__prod(mark(X),mark(Y))) ;
  Marked_a__prod(s(X),Y) >= Marked_a__prod(mark(X),mark(Y)) ;
  Marked_a__prod(s(X),Y) >= Marked_mark(X) ;
  Marked_a__prod(s(X),Y) >= Marked_mark(Y) ;
  Marked_a__if(true,X,Y) >= Marked_mark(X) ;
  Marked_a__if(false,X,Y) >= Marked_mark(Y) ;
  Marked_a__fact(X) >= Marked_a__if(a__zero(mark(X)),s(0),prod(X,fact(p(X)))) ;
  Marked_a__fact(X) >= Marked_mark(X) ;
  Marked_mark(s(X)) >= Marked_mark(X) ;
  Marked_mark(prod(X1,X2)) >= Marked_a__prod(mark(X1),mark(X2)) ;
  Marked_mark(prod(X1,X2)) >= Marked_mark(X1) ;
  Marked_mark(prod(X1,X2)) >= Marked_mark(X2) ;
  Marked_mark(fact(X)) >= Marked_a__fact(mark(X)) ;
  Marked_mark(fact(X)) >= Marked_mark(X) ;
  Marked_mark(p(X)) >= Marked_a__p(mark(X)) ;
  Marked_mark(p(X)) >= Marked_mark(X) ;
  Marked_mark(if(X1,X2,X3)) >= Marked_a__if(mark(X1),X2,X3) ;
  Marked_mark(if(X1,X2,X3)) >= Marked_mark(X1) ;
  Marked_mark(zero(X)) >= Marked_mark(X) ;
  Marked_mark(add(X1,X2)) >= Marked_a__add(mark(X1),mark(X2)) ;
  Marked_mark(add(X1,X2)) >= Marked_mark(X1) ;
  Marked_mark(add(X1,X2)) >= Marked_mark(X2) ;
 }
 + Disjunctions:{ 
{
  Marked_a__add(s(X),Y) > Marked_a__add(mark(X),mark(Y)) ;
 }
{
  Marked_a__add(s(X),Y) > Marked_mark(X) ;
 }
{
  Marked_a__add(s(X),Y) > Marked_mark(Y) ;
 }
{
  Marked_a__add(0,X) > Marked_mark(X) ;
 }
{
  Marked_a__p(s(X)) > Marked_mark(X) ;
 }
{
  Marked_a__prod(s(X),Y) > Marked_a__add(mark(Y),a__prod(mark(X),mark(Y))) ;
 }
{
  Marked_a__prod(s(X),Y) > Marked_a__prod(mark(X),mark(Y)) ;
 }
{
  Marked_a__prod(s(X),Y) > Marked_mark(X) ;
 }
{
  Marked_a__prod(s(X),Y) > Marked_mark(Y) ;
 }
{
  Marked_a__if(true,X,Y) > Marked_mark(X) ;
 }
{
  Marked_a__if(false,X,Y) > Marked_mark(Y) ;
 }
{
  Marked_a__fact(X) > Marked_a__if(a__zero(mark(X)),s(0),prod(X,fact(p(X)))) ;
 }
{
  Marked_a__fact(X) > Marked_mark(X) ;
 }
{
  Marked_mark(s(X)) > Marked_mark(X) ;
 }
{
  Marked_mark(prod(X1,X2)) > Marked_a__prod(mark(X1),mark(X2)) ;
 }
{
  Marked_mark(prod(X1,X2)) > Marked_mark(X1) ;
 }
{
  Marked_mark(prod(X1,X2)) > Marked_mark(X2) ;
 }
{
  Marked_mark(fact(X)) > Marked_a__fact(mark(X)) ;
 }
{
  Marked_mark(fact(X)) > Marked_mark(X) ;
 }
{
  Marked_mark(p(X)) > Marked_a__p(mark(X)) ;
 }
{
  Marked_mark(p(X)) > Marked_mark(X) ;
 }
{
  Marked_mark(if(X1,X2,X3)) > Marked_a__if(mark(X1),X2,X3) ;
 }
{
  Marked_mark(if(X1,X2,X3)) > Marked_mark(X1) ;
 }
{
  Marked_mark(zero(X)) > Marked_mark(X) ;
 }
{
  Marked_mark(add(X1,X2)) > Marked_a__add(mark(X1),mark(X2)) ;
 }
{
  Marked_mark(add(X1,X2)) > Marked_mark(X1) ;
 }
{
  Marked_mark(add(X1,X2)) > Marked_mark(X2) ;
 }
 } 
=== TIMER virtual : 10.000000 ===
Entering poly_solver
Starting Sat solver initialization
Calling Sat solver...
=== STOPING TIMER virtual ===
=== TIMER real : 10.000000 ===
=== STOPING TIMER real ===
Sat solver returned
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
No solution found for these parameters.
Entering rpo_solver
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 15.000000 ===
Entering poly_solver
Starting Sat solver initialization
Calling Sat solver...
=== STOPING TIMER virtual ===
=== TIMER real : 15.000000 ===
=== STOPING TIMER real ===
Sat solver returned
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
No solution found for these parameters.
=== TIMER virtual : 50.000000 ===
trying sub matrices of size: 1
Matrix interpretation constraints generated.
Search parameters: LINEAR MATRIX 3x3 (strict=1x1) ; time limit: 50.. 
Termination constraints generated.
Starting Sat solver initialization
Calling Sat solver...
=== STOPING TIMER virtual ===
=== TIMER real : 50.000000 ===
=== STOPING TIMER real ===
Sat solver returned
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
No solution found for these parameters.
No solution found for these constraints.
APPLY CRITERIA (Simple graph)
Found the following constraints:
{
  a__if(true,X,Y) >= mark(X) ;
  a__if(false,X,Y) >= mark(Y) ;
  a__if(X1,X2,X3) >= if(X1,X2,X3) ;
  a__zero(s(X)) >= false ;
  a__zero(0) >= true ;
  a__zero(X) >= zero(X) ;
  mark(s(X)) >= s(mark(X)) ;
  mark(0) >= 0 ;
  mark(prod(X1,X2)) >= a__prod(mark(X1),mark(X2)) ;
  mark(fact(X)) >= a__fact(mark(X)) ;
  mark(p(X)) >= a__p(mark(X)) ;
  mark(true) >= true ;
  mark(false) >= false ;
  mark(if(X1,X2,X3)) >= a__if(mark(X1),X2,X3) ;
  mark(zero(X)) >= a__zero(mark(X)) ;
  mark(add(X1,X2)) >= a__add(mark(X1),mark(X2)) ;
  a__fact(X) >= a__if(a__zero(mark(X)),s(0),prod(X,fact(p(X)))) ;
  a__fact(X) >= fact(X) ;
  a__add(s(X),Y) >= s(a__add(mark(X),mark(Y))) ;
  a__add(0,X) >= mark(X) ;
  a__add(X1,X2) >= add(X1,X2) ;
  a__prod(s(X),Y) >= a__add(mark(Y),a__prod(mark(X),mark(Y))) ;
  a__prod(0,X) >= 0 ;
  a__prod(X1,X2) >= prod(X1,X2) ;
  a__p(s(X)) >= mark(X) ;
  a__p(X) >= p(X) ;
  Marked_a__add(s(X),Y) > Marked_a__add(mark(X),mark(Y)) ;
  Marked_a__add(s(X),Y) > Marked_mark(X) ;
  Marked_a__add(s(X),Y) > Marked_mark(Y) ;
  Marked_a__add(0,X) > Marked_mark(X) ;
  Marked_a__p(s(X)) > Marked_mark(X) ;
  Marked_a__prod(s(X),Y) > Marked_a__add(mark(Y),a__prod(mark(X),mark(Y))) ;
  Marked_a__prod(s(X),Y) > Marked_a__prod(mark(X),mark(Y)) ;
  Marked_a__prod(s(X),Y) > Marked_mark(X) ;
  Marked_a__prod(s(X),Y) > Marked_mark(Y) ;
  Marked_a__if(true,X,Y) > Marked_mark(X) ;
  Marked_a__if(false,X,Y) > Marked_mark(Y) ;
  Marked_a__fact(X) > Marked_a__if(a__zero(mark(X)),s(0),prod(X,fact(p(X)))) ;
  Marked_a__fact(X) > Marked_mark(X) ;
  Marked_mark(s(X)) > Marked_mark(X) ;
  Marked_mark(prod(X1,X2)) >= Marked_a__prod(mark(X1),mark(X2)) ;
  Marked_mark(prod(X1,X2)) > Marked_mark(X1) ;
  Marked_mark(prod(X1,X2)) > Marked_mark(X2) ;
  Marked_mark(fact(X)) >= Marked_a__fact(mark(X)) ;
  Marked_mark(fact(X)) > Marked_mark(X) ;
  Marked_mark(p(X)) >= Marked_a__p(mark(X)) ;
  Marked_mark(p(X)) > Marked_mark(X) ;
  Marked_mark(if(X1,X2,X3)) >= Marked_a__if(mark(X1),X2,X3) ;
  Marked_mark(if(X1,X2,X3)) > Marked_mark(X1) ;
  Marked_mark(zero(X)) > Marked_mark(X) ;
  Marked_mark(add(X1,X2)) >= Marked_a__add(mark(X1),mark(X2)) ;
  Marked_mark(add(X1,X2)) > Marked_mark(X1) ;
  Marked_mark(add(X1,X2)) > Marked_mark(X2) ;
 }

APPLY CRITERIA (SOLVE_ORD)
Trying to solve the following constraints:
{
  a__if(true,X,Y) >= mark(X) ;
  a__if(false,X,Y) >= mark(Y) ;
  a__if(X1,X2,X3) >= if(X1,X2,X3) ;
  a__zero(s(X)) >= false ;
  a__zero(0) >= true ;
  a__zero(X) >= zero(X) ;
  mark(s(X)) >= s(mark(X)) ;
  mark(0) >= 0 ;
  mark(prod(X1,X2)) >= a__prod(mark(X1),mark(X2)) ;
  mark(fact(X)) >= a__fact(mark(X)) ;
  mark(p(X)) >= a__p(mark(X)) ;
  mark(true) >= true ;
  mark(false) >= false ;
  mark(if(X1,X2,X3)) >= a__if(mark(X1),X2,X3) ;
  mark(zero(X)) >= a__zero(mark(X)) ;
  mark(add(X1,X2)) >= a__add(mark(X1),mark(X2)) ;
  a__fact(X) >= a__if(a__zero(mark(X)),s(0),prod(X,fact(p(X)))) ;
  a__fact(X) >= fact(X) ;
  a__add(s(X),Y) >= s(a__add(mark(X),mark(Y))) ;
  a__add(0,X) >= mark(X) ;
  a__add(X1,X2) >= add(X1,X2) ;
  a__prod(s(X),Y) >= a__add(mark(Y),a__prod(mark(X),mark(Y))) ;
  a__prod(0,X) >= 0 ;
  a__prod(X1,X2) >= prod(X1,X2) ;
  a__p(s(X)) >= mark(X) ;
  a__p(X) >= p(X) ;
  Marked_a__add(s(X),Y) > Marked_a__add(mark(X),mark(Y)) ;
  Marked_a__add(s(X),Y) > Marked_mark(X) ;
  Marked_a__add(s(X),Y) > Marked_mark(Y) ;
  Marked_a__add(0,X) > Marked_mark(X) ;
  Marked_a__p(s(X)) > Marked_mark(X) ;
  Marked_a__prod(s(X),Y) > Marked_a__add(mark(Y),a__prod(mark(X),mark(Y))) ;
  Marked_a__prod(s(X),Y) > Marked_a__prod(mark(X),mark(Y)) ;
  Marked_a__prod(s(X),Y) > Marked_mark(X) ;
  Marked_a__prod(s(X),Y) > Marked_mark(Y) ;
  Marked_a__if(true,X,Y) > Marked_mark(X) ;
  Marked_a__if(false,X,Y) > Marked_mark(Y) ;
  Marked_a__fact(X) > Marked_a__if(a__zero(mark(X)),s(0),prod(X,fact(p(X)))) ;
  Marked_a__fact(X) > Marked_mark(X) ;
  Marked_mark(s(X)) > Marked_mark(X) ;
  Marked_mark(prod(X1,X2)) >= Marked_a__prod(mark(X1),mark(X2)) ;
  Marked_mark(prod(X1,X2)) > Marked_mark(X1) ;
  Marked_mark(prod(X1,X2)) > Marked_mark(X2) ;
  Marked_mark(fact(X)) >= Marked_a__fact(mark(X)) ;
  Marked_mark(fact(X)) > Marked_mark(X) ;
  Marked_mark(p(X)) >= Marked_a__p(mark(X)) ;
  Marked_mark(p(X)) > Marked_mark(X) ;
  Marked_mark(if(X1,X2,X3)) >= Marked_a__if(mark(X1),X2,X3) ;
  Marked_mark(if(X1,X2,X3)) > Marked_mark(X1) ;
  Marked_mark(zero(X)) > Marked_mark(X) ;
  Marked_mark(add(X1,X2)) >= Marked_a__add(mark(X1),mark(X2)) ;
  Marked_mark(add(X1,X2)) > Marked_mark(X1) ;
  Marked_mark(add(X1,X2)) > Marked_mark(X2) ;
 }
 + Disjunctions:{ 
 } 
=== TIMER virtual : 10.000000 ===
Entering poly_solver
Starting Sat solver initialization
Calling Sat solver...
=== STOPING TIMER virtual ===
=== TIMER real : 10.000000 ===
=== STOPING TIMER real ===
Sat solver returned
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
No solution found for these parameters.
Entering rpo_solver
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
No solution found for these parameters.(9336 bt (9550) [4338])
=== TIMER virtual : 15.000000 ===
Entering poly_solver
Starting Sat solver initialization
Calling Sat solver...
=== STOPING TIMER virtual ===
=== TIMER real : 15.000000 ===
=== STOPING TIMER real ===
Sat solver returned
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
No solution found for these parameters.
=== TIMER virtual : 50.000000 ===
trying sub matrices of size: 1
Matrix interpretation constraints generated.
Search parameters: LINEAR MATRIX 3x3 (strict=1x1) ; time limit: 50.. 
Termination constraints generated.
Starting Sat solver initialization
Calling Sat solver...
=== STOPING TIMER virtual ===
=== TIMER real : 50.000000 ===
=== STOPING TIMER real ===
Sat solver returned
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
No solution found for these parameters.
No solution found for these constraints.
APPLY CRITERIA (ID_CRIT)
NOT SOLVED
No proof found
Cime worked for 287.190906 seconds (real time)
Cime Exit Status: 0