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

TRS termination of:
[1] f(x,0,0) -> s(x)
[2] f(0,y,0) -> s(y)
[3] f(0,0,z) -> s(z)
[4] f(s(0),y,z) -> f(0,s(y),s(z))
[5] f(s(x),s(y),0) -> f(x,y,s(0))
[6] f(s(x),0,s(z)) -> f(x,s(0),z)
[7] f(0,s(0),s(0)) -> s(s(0))
[8] f(s(x),s(y),s(z)) -> f(x,y,f(s(x),s(y),z))
[9] f(0,s(s(y)),s(0)) -> f(0,y,s(0))
[10] f(0,s(0),s(s(z))) -> f(0,s(0),z)
[11] f(0,s(s(y)),s(s(z))) -> f(0,y,f(0,s(s(y)),s(z)))


Sub problem: 
guided: 

DP termination of:
<Marked_f(s(0),y,z) , Marked_f(0,s(y),s(z))>
<Marked_f(s(x),s(y),0) , Marked_f(x,y,s(0))>
<Marked_f(s(x),0,s(z)) , Marked_f(x,s(0),z)>
<Marked_f(s(x),s(y),s(z)) , Marked_f(x,y,f(s(x),s(y),z))>
<Marked_f(s(x),s(y),s(z)) , Marked_f(s(x),s(y),z)>
<Marked_f(0,s(s(y)),s(0)) , Marked_f(0,y,s(0))>
<Marked_f(0,s(0),s(s(z))) , Marked_f(0,s(0),z)>
<Marked_f(0,s(s(y)),s(s(z))) , Marked_f(0,y,f(0,s(s(y)),s(z)))>
<Marked_f(0,s(s(y)),s(s(z))) , Marked_f(0,s(s(y)),s(z))>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 4 components:

{  <Marked_f(s(x),s(y),s(z)) , Marked_f(s(x),s(y),z)> 
      --> <Marked_f(s(x),s(y),s(z)) , Marked_f(s(x),s(y),z)>
  <Marked_f(s(x),s(y),s(z)) , Marked_f(s(x),s(y),z)> 
     --> <Marked_f(s(x),s(y),s(z)) , Marked_f(x,y,f(s(x),s(y),z))>
  <Marked_f(s(x),s(y),s(z)) , Marked_f(s(x),s(y),z)> 
     --> <Marked_f(s(x),s(y),0) , Marked_f(x,y,s(0))>
  <Marked_f(s(x),s(y),s(z)) , Marked_f(x,y,f(s(x),s(y),z))> 
     --> <Marked_f(s(x),s(y),s(z)) , Marked_f(s(x),s(y),z)>
  <Marked_f(s(x),s(y),s(z)) , Marked_f(x,y,f(s(x),s(y),z))> 
     --> <Marked_f(s(x),s(y),s(z)) , Marked_f(x,y,f(s(x),s(y),z))>
  <Marked_f(s(x),s(y),s(z)) , Marked_f(x,y,f(s(x),s(y),z))> 
     --> <Marked_f(s(x),0,s(z)) , Marked_f(x,s(0),z)>
  <Marked_f(s(x),s(y),s(z)) , Marked_f(x,y,f(s(x),s(y),z))> 
     --> <Marked_f(s(x),s(y),0) , Marked_f(x,y,s(0))>
  <Marked_f(s(x),0,s(z)) , Marked_f(x,s(0),z)> 
     --> <Marked_f(s(x),s(y),s(z)) , Marked_f(s(x),s(y),z)>
  <Marked_f(s(x),0,s(z)) , Marked_f(x,s(0),z)> 
     --> <Marked_f(s(x),s(y),s(z)) , Marked_f(x,y,f(s(x),s(y),z))>
  <Marked_f(s(x),0,s(z)) , Marked_f(x,s(0),z)> 
     --> <Marked_f(s(x),s(y),0) , Marked_f(x,y,s(0))>
  <Marked_f(s(x),s(y),0) , Marked_f(x,y,s(0))> 
     --> <Marked_f(s(x),s(y),s(z)) , Marked_f(s(x),s(y),z)>
  <Marked_f(s(x),s(y),0) , Marked_f(x,y,s(0))> 
     --> <Marked_f(s(x),s(y),s(z)) , Marked_f(x,y,f(s(x),s(y),z))>
  <Marked_f(s(x),s(y),0) , Marked_f(x,y,s(0))> 
     --> <Marked_f(s(x),0,s(z)) , Marked_f(x,s(0),z)>
}

{  <Marked_f(0,s(s(y)),s(s(z))) , Marked_f(0,s(s(y)),s(z))> 
      --> <Marked_f(0,s(s(y)),s(s(z))) , Marked_f(0,s(s(y)),s(z))>
  <Marked_f(0,s(s(y)),s(s(z))) , Marked_f(0,s(s(y)),s(z))> 
     --> <Marked_f(0,s(s(y)),s(s(z))) , Marked_f(0,y,f(0,s(s(y)),s(z)))>
  <Marked_f(0,s(s(y)),s(s(z))) , Marked_f(0,y,f(0,s(s(y)),s(z)))> 
     --> <Marked_f(0,s(s(y)),s(s(z))) , Marked_f(0,s(s(y)),s(z))>
  <Marked_f(0,s(s(y)),s(s(z))) , Marked_f(0,y,f(0,s(s(y)),s(z)))> 
     --> <Marked_f(0,s(s(y)),s(s(z))) , Marked_f(0,y,f(0,s(s(y)),s(z)))>
}

{  <Marked_f(0,s(s(y)),s(0)) , Marked_f(0,y,s(0))> 
      --> <Marked_f(0,s(s(y)),s(0)) , Marked_f(0,y,s(0))>
}

{  <Marked_f(0,s(0),s(s(z))) , Marked_f(0,s(0),z)> 
      --> <Marked_f(0,s(0),s(s(z))) , Marked_f(0,s(0),z)>
}
APPLY CRITERIA (Subterm criterion)

ST: Marked_f -> 1

APPLY CRITERIA (Subterm criterion)

ST: Marked_f -> 2

APPLY CRITERIA (Subterm criterion)

ST: Marked_f -> 2

APPLY CRITERIA (Subterm criterion)

ST: Marked_f -> 3

APPLY CRITERIA (Graph splitting)
Found 1 components:

{  <Marked_f(s(x),s(y),s(z)) , Marked_f(s(x),s(y),z)> 
      --> <Marked_f(s(x),s(y),s(z)) , Marked_f(s(x),s(y),z)>
}
APPLY CRITERIA (Subterm criterion)

ST: Marked_f -> 3

APPLY CRITERIA (Graph splitting)
Found 0 components:
APPLY CRITERIA (Graph splitting)
Found 1 components:

{  <Marked_f(0,s(s(y)),s(s(z))) , Marked_f(0,s(s(y)),s(z))> 
      --> <Marked_f(0,s(s(y)),s(s(z))) , Marked_f(0,s(s(y)),s(z))>
}
APPLY CRITERIA (Subterm criterion)

ST: Marked_f -> 3

APPLY CRITERIA (Graph splitting)
Found 0 components:
APPLY CRITERIA (Graph splitting)
Found 0 components:
APPLY CRITERIA (Graph splitting)
Found 0 components:
SOLVED
{
 
TRS termination of:
[1] f(x,0,0) -> s(x)
[2] f(0,y,0) -> s(y)
[3] f(0,0,z) -> s(z)
[4] f(s(0),y,z) -> f(0,s(y),s(z))
[5] f(s(x),s(y),0) -> f(x,y,s(0))
[6] f(s(x),0,s(z)) -> f(x,s(0),z)
[7] f(0,s(0),s(0)) -> s(s(0))
[8] f(s(x),s(y),s(z)) -> f(x,y,f(s(x),s(y),z))
[9] f(0,s(s(y)),s(0)) -> f(0,y,s(0))
[10] f(0,s(0),s(s(z))) -> f(0,s(0),z)
[11] f(0,s(s(y)),s(s(z))) -> f(0,y,f(0,s(s(y)),s(z)))
 ,
 CRITERION: MDP
 [ {
      
     DP termination of:
     <Marked_f(s(0),y,z) , Marked_f(0,s(y),s(z))>
<Marked_f(s(x),s(y),0) , Marked_f(x,y,s(0))>
<Marked_f(s(x),0,s(z)) , Marked_f(x,s(0),z)>
<Marked_f(s(x),s(y),s(z)) , Marked_f(x,y,f(s(x),s(y),z))>
<Marked_f(s(x),s(y),s(z)) , Marked_f(s(x),s(y),z)>
<Marked_f(0,s(s(y)),s(0)) , Marked_f(0,y,s(0))>
<Marked_f(0,s(0),s(s(z))) , Marked_f(0,s(0),z)>
<Marked_f(0,s(s(y)),s(s(z))) , Marked_f(0,y,f(0,s(s(y)),s(z)))>
<Marked_f(0,s(s(y)),s(s(z))) , Marked_f(0,s(s(y)),s(z))>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_f(s(x),s(y),0) , Marked_f(x,y,s(0))>
<Marked_f(s(x),0,s(z)) , Marked_f(x,s(0),z)>
<Marked_f(s(x),s(y),s(z)) , Marked_f(x,y,f(s(x),s(y),z))>
<Marked_f(s(x),s(y),s(z)) , Marked_f(s(x),s(y),z)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
     <Marked_f(s(x),s(y),s(z)) , Marked_f(s(x),s(y),z)>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_f(s(x),s(y),s(z)) , Marked_f(s(x),s(y),z)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
 ]} 
 ]} 
{
 
DP termination of:
<Marked_f(0,s(s(y)),s(s(z))) , Marked_f(0,y,f(0,s(s(y)),s(z)))>
<Marked_f(0,s(s(y)),s(s(z))) , Marked_f(0,s(s(y)),s(z))>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
     <Marked_f(0,s(s(y)),s(s(z))) , Marked_f(0,s(s(y)),s(z))>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_f(0,s(s(y)),s(s(z))) , Marked_f(0,s(s(y)),s(z))>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
 ]} 
 ]} 
{
 
DP termination of:
<Marked_f(0,s(s(y)),s(0)) , Marked_f(0,y,s(0))>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
{
 
DP termination of:
<Marked_f(0,s(0),s(s(z))) , Marked_f(0,s(0),z)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
 ]} 
 ]} 

Cime worked for 0.022669 seconds (real time)
Cime Exit Status: 0