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

TRS termination of:
[1] f(0) -> 0
[2] f(s(0)) -> s(0)
[3] f(s(s(x))) -> p(h(g(x)))
[4] g(0) -> pair(s(0),s(0))
[5] g(s(x)) -> h(g(x))
[6] h(x) -> pair(+(p(x),q(x)),p(x))
[7] p(pair(x,y)) -> x
[8] q(pair(x,y)) -> y
[9] +(x,0) -> x
[10] +(x,s(y)) -> s(+(x,y))
[11] f(s(s(x))) -> +(p(g(x)),q(g(x)))
[12] g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x)))


Sub problem: 
guided: 

DP termination of:
<Marked_f(s(s(x))) , Marked_p(h(g(x)))>
<Marked_f(s(s(x))) , Marked_h(g(x))>
<Marked_f(s(s(x))) , Marked_g(x)>
<Marked_g(s(x)) , Marked_h(g(x))>
<Marked_g(s(x)) , Marked_g(x)>
<Marked_h(x) , Marked_+(p(x),q(x))>
<Marked_h(x) , Marked_p(x)>
<Marked_h(x) , Marked_q(x)>
<Marked_h(x) , Marked_p(x)>
<Marked_+(x,s(y)) , Marked_+(x,y)>
<Marked_f(s(s(x))) , Marked_+(p(g(x)),q(g(x)))>
<Marked_f(s(s(x))) , Marked_p(g(x))>
<Marked_f(s(s(x))) , Marked_g(x)>
<Marked_f(s(s(x))) , Marked_q(g(x))>
<Marked_f(s(s(x))) , Marked_g(x)>
<Marked_g(s(x)) , Marked_+(p(g(x)),q(g(x)))>
<Marked_g(s(x)) , Marked_p(g(x))>
<Marked_g(s(x)) , Marked_g(x)>
<Marked_g(s(x)) , Marked_q(g(x))>
<Marked_g(s(x)) , Marked_g(x)>
<Marked_g(s(x)) , Marked_p(g(x))>
<Marked_g(s(x)) , Marked_g(x)>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 2 components:

{  <Marked_g(s(x)) , Marked_g(x)> --> <Marked_g(s(x)) , Marked_g(x)>
  <Marked_g(s(x)) , Marked_g(x)> --> <Marked_g(s(x)) , Marked_g(x)>
  <Marked_g(s(x)) , Marked_g(x)> --> <Marked_g(s(x)) , Marked_g(x)>
  <Marked_g(s(x)) , Marked_g(x)> --> <Marked_g(s(x)) , Marked_g(x)>
  <Marked_g(s(x)) , Marked_g(x)> --> <Marked_g(s(x)) , Marked_g(x)>
  <Marked_g(s(x)) , Marked_g(x)> --> <Marked_g(s(x)) , Marked_g(x)>
  <Marked_g(s(x)) , Marked_g(x)> --> <Marked_g(s(x)) , Marked_g(x)>
  <Marked_g(s(x)) , Marked_g(x)> --> <Marked_g(s(x)) , Marked_g(x)>
  <Marked_g(s(x)) , Marked_g(x)> --> <Marked_g(s(x)) , Marked_g(x)>
  <Marked_g(s(x)) , Marked_g(x)> --> <Marked_g(s(x)) , Marked_g(x)>
  <Marked_g(s(x)) , Marked_g(x)> --> <Marked_g(s(x)) , Marked_g(x)>
  <Marked_g(s(x)) , Marked_g(x)> --> <Marked_g(s(x)) , Marked_g(x)>
  <Marked_g(s(x)) , Marked_g(x)> --> <Marked_g(s(x)) , Marked_g(x)>
  <Marked_g(s(x)) , Marked_g(x)> --> <Marked_g(s(x)) , Marked_g(x)>
  <Marked_g(s(x)) , Marked_g(x)> --> <Marked_g(s(x)) , Marked_g(x)>
  <Marked_g(s(x)) , Marked_g(x)> --> <Marked_g(s(x)) , Marked_g(x)>
}

{  <Marked_+(x,s(y)) , Marked_+(x,y)> --> <Marked_+(x,s(y)) , Marked_+(x,y)>
}
APPLY CRITERIA (Subterm criterion)

ST: Marked_g -> 1

APPLY CRITERIA (Subterm criterion)

ST: Marked_+ -> 2

APPLY CRITERIA (Graph splitting)
Found 0 components:
APPLY CRITERIA (Graph splitting)
Found 0 components:
SOLVED
{
 
TRS termination of:
[1] f(0) -> 0
[2] f(s(0)) -> s(0)
[3] f(s(s(x))) -> p(h(g(x)))
[4] g(0) -> pair(s(0),s(0))
[5] g(s(x)) -> h(g(x))
[6] h(x) -> pair(+(p(x),q(x)),p(x))
[7] p(pair(x,y)) -> x
[8] q(pair(x,y)) -> y
[9] +(x,0) -> x
[10] +(x,s(y)) -> s(+(x,y))
[11] f(s(s(x))) -> +(p(g(x)),q(g(x)))
[12] g(s(x)) -> pair(+(p(g(x)),q(g(x))),p(g(x)))
 ,
 CRITERION: MDP
 [ {
      
     DP termination of:
     <Marked_f(s(s(x))) , Marked_p(h(g(x)))>
<Marked_f(s(s(x))) , Marked_h(g(x))>
<Marked_f(s(s(x))) , Marked_g(x)>
<Marked_g(s(x)) , Marked_h(g(x))>
<Marked_g(s(x)) , Marked_g(x)>
<Marked_h(x) , Marked_+(p(x),q(x))>
<Marked_h(x) , Marked_p(x)>
<Marked_h(x) , Marked_q(x)>
<Marked_h(x) , Marked_p(x)>
<Marked_+(x,s(y)) , Marked_+(x,y)>
<Marked_f(s(s(x))) , Marked_+(p(g(x)),q(g(x)))>
<Marked_f(s(s(x))) , Marked_p(g(x))>
<Marked_f(s(s(x))) , Marked_g(x)>
<Marked_f(s(s(x))) , Marked_q(g(x))>
<Marked_f(s(s(x))) , Marked_g(x)>
<Marked_g(s(x)) , Marked_+(p(g(x)),q(g(x)))>
<Marked_g(s(x)) , Marked_p(g(x))>
<Marked_g(s(x)) , Marked_g(x)>
<Marked_g(s(x)) , Marked_q(g(x))>
<Marked_g(s(x)) , Marked_g(x)>
<Marked_g(s(x)) , Marked_p(g(x))>
<Marked_g(s(x)) , Marked_g(x)>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_g(s(x)) , Marked_g(x)>
<Marked_g(s(x)) , Marked_g(x)>
<Marked_g(s(x)) , Marked_g(x)>
<Marked_g(s(x)) , Marked_g(x)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
{
 
DP termination of:
<Marked_+(x,s(y)) , Marked_+(x,y)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
 ]} 
 ]} 

Cime worked for 0.008805 seconds (real time)
Cime Exit Status: 0