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

TRS termination of:
[1] +(x,0) -> x
[2] +(0,x) -> x
[3] +(s(x),s(y)) -> s(s(+(x,y)))
[4] +(+(x,y),z) -> +(x,+(y,z))
[5]  *(x,0) -> 0
[6]  *(0,x) -> 0
[7]  *(s(x),s(y)) -> s(+( *(x,y),+(x,y)))
[8]  *( *(x,y),z) ->  *(x, *(y,z))
[9] app(nil,l) -> l
[10] app(cons(x,l1),l2) -> cons(x,app(l1,l2))
[11] sum(nil) -> 0
[12] sum(cons(x,l)) -> +(x,sum(l))
[13] sum(app(l1,l2)) -> +(sum(l1),sum(l2))
[14] prod(nil) -> s(0)
[15] prod(cons(x,l)) ->  *(x,prod(l))
[16] prod(app(l1,l2)) ->  *(prod(l1),prod(l2))


Sub problem: 
guided: 

DP termination of:
<Marked_+(s(x),s(y)) , Marked_+(x,y)>
<Marked_+(+(x,y),z) , Marked_+(x,+(y,z))>
<Marked_+(+(x,y),z) , Marked_+(y,z)>
<Marked_*(s(x),s(y)) , Marked_+( *(x,y),+(x,y))>
<Marked_*(s(x),s(y)) , Marked_*(x,y)>
<Marked_*(s(x),s(y)) , Marked_+(x,y)>
<Marked_*( *(x,y),z) , Marked_*(x, *(y,z))>
<Marked_*( *(x,y),z) , Marked_*(y,z)>
<Marked_app(cons(x,l1),l2) , Marked_app(l1,l2)>
<Marked_sum(cons(x,l)) , Marked_+(x,sum(l))>
<Marked_sum(cons(x,l)) , Marked_sum(l)>
<Marked_sum(app(l1,l2)) , Marked_+(sum(l1),sum(l2))>
<Marked_sum(app(l1,l2)) , Marked_sum(l1)>
<Marked_sum(app(l1,l2)) , Marked_sum(l2)>
<Marked_prod(cons(x,l)) , Marked_*(x,prod(l))>
<Marked_prod(cons(x,l)) , Marked_prod(l)>
<Marked_prod(app(l1,l2)) , Marked_*(prod(l1),prod(l2))>
<Marked_prod(app(l1,l2)) , Marked_prod(l1)>
<Marked_prod(app(l1,l2)) , Marked_prod(l2)>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 5 components:

{  <Marked_app(cons(x,l1),l2) , Marked_app(l1,l2)> 
      --> <Marked_app(cons(x,l1),l2) , Marked_app(l1,l2)>
}

{  <Marked_sum(app(l1,l2)) , Marked_sum(l2)> 
      --> <Marked_sum(app(l1,l2)) , Marked_sum(l2)>
  <Marked_sum(app(l1,l2)) , Marked_sum(l2)> 
     --> <Marked_sum(app(l1,l2)) , Marked_sum(l1)>
  <Marked_sum(app(l1,l2)) , Marked_sum(l2)> 
     --> <Marked_sum(cons(x,l)) , Marked_sum(l)>
  <Marked_sum(app(l1,l2)) , Marked_sum(l1)> 
     --> <Marked_sum(app(l1,l2)) , Marked_sum(l2)>
  <Marked_sum(app(l1,l2)) , Marked_sum(l1)> 
     --> <Marked_sum(app(l1,l2)) , Marked_sum(l1)>
  <Marked_sum(app(l1,l2)) , Marked_sum(l1)> 
     --> <Marked_sum(cons(x,l)) , Marked_sum(l)>
  <Marked_sum(cons(x,l)) , Marked_sum(l)> 
     --> <Marked_sum(app(l1,l2)) , Marked_sum(l2)>
  <Marked_sum(cons(x,l)) , Marked_sum(l)> 
     --> <Marked_sum(app(l1,l2)) , Marked_sum(l1)>
  <Marked_sum(cons(x,l)) , Marked_sum(l)> 
     --> <Marked_sum(cons(x,l)) , Marked_sum(l)>
}

{  <Marked_prod(app(l1,l2)) , Marked_prod(l2)> 
      --> <Marked_prod(app(l1,l2)) , Marked_prod(l2)>
  <Marked_prod(app(l1,l2)) , Marked_prod(l2)> 
     --> <Marked_prod(app(l1,l2)) , Marked_prod(l1)>
  <Marked_prod(app(l1,l2)) , Marked_prod(l2)> 
     --> <Marked_prod(cons(x,l)) , Marked_prod(l)>
  <Marked_prod(app(l1,l2)) , Marked_prod(l1)> 
     --> <Marked_prod(app(l1,l2)) , Marked_prod(l2)>
  <Marked_prod(app(l1,l2)) , Marked_prod(l1)> 
     --> <Marked_prod(app(l1,l2)) , Marked_prod(l1)>
  <Marked_prod(app(l1,l2)) , Marked_prod(l1)> 
     --> <Marked_prod(cons(x,l)) , Marked_prod(l)>
  <Marked_prod(cons(x,l)) , Marked_prod(l)> 
     --> <Marked_prod(app(l1,l2)) , Marked_prod(l2)>
  <Marked_prod(cons(x,l)) , Marked_prod(l)> 
     --> <Marked_prod(app(l1,l2)) , Marked_prod(l1)>
  <Marked_prod(cons(x,l)) , Marked_prod(l)> 
     --> <Marked_prod(cons(x,l)) , Marked_prod(l)>
}

{  <Marked_*( *(x,y),z) , Marked_*(y,z)> 
      --> <Marked_*( *(x,y),z) , Marked_*(y,z)>
  <Marked_*( *(x,y),z) , Marked_*(y,z)> 
     --> <Marked_*( *(x,y),z) , Marked_*(x, *(y,z))>
  <Marked_*( *(x,y),z) , Marked_*(y,z)> 
     --> <Marked_*(s(x),s(y)) , Marked_*(x,y)>
  <Marked_*( *(x,y),z) , Marked_*(x, *(y,z))> 
     --> <Marked_*( *(x,y),z) , Marked_*(y,z)>
  <Marked_*( *(x,y),z) , Marked_*(x, *(y,z))> 
     --> <Marked_*( *(x,y),z) , Marked_*(x, *(y,z))>
  <Marked_*( *(x,y),z) , Marked_*(x, *(y,z))> 
     --> <Marked_*(s(x),s(y)) , Marked_*(x,y)>
  <Marked_*(s(x),s(y)) , Marked_*(x,y)> 
     --> <Marked_*( *(x,y),z) , Marked_*(y,z)>
  <Marked_*(s(x),s(y)) , Marked_*(x,y)> 
     --> <Marked_*( *(x,y),z) , Marked_*(x, *(y,z))>
  <Marked_*(s(x),s(y)) , Marked_*(x,y)> 
     --> <Marked_*(s(x),s(y)) , Marked_*(x,y)>
}

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

ST: Marked_app -> 1

APPLY CRITERIA (Subterm criterion)

ST: Marked_sum -> 1

APPLY CRITERIA (Subterm criterion)

ST: Marked_prod -> 1

APPLY CRITERIA (Subterm criterion)

ST: Marked_* -> 1

APPLY CRITERIA (Subterm criterion)

ST: Marked_+ -> 1

APPLY CRITERIA (Graph splitting)
Found 0 components:
APPLY CRITERIA (Graph splitting)
Found 0 components:
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] +(x,0) -> x
[2] +(0,x) -> x
[3] +(s(x),s(y)) -> s(s(+(x,y)))
[4] +(+(x,y),z) -> +(x,+(y,z))
[5]  *(x,0) -> 0
[6]  *(0,x) -> 0
[7]  *(s(x),s(y)) -> s(+( *(x,y),+(x,y)))
[8]  *( *(x,y),z) ->  *(x, *(y,z))
[9] app(nil,l) -> l
[10] app(cons(x,l1),l2) -> cons(x,app(l1,l2))
[11] sum(nil) -> 0
[12] sum(cons(x,l)) -> +(x,sum(l))
[13] sum(app(l1,l2)) -> +(sum(l1),sum(l2))
[14] prod(nil) -> s(0)
[15] prod(cons(x,l)) ->  *(x,prod(l))
[16] prod(app(l1,l2)) ->  *(prod(l1),prod(l2))
 ,
 CRITERION: MDP
 [ {
      
     DP termination of:
     <Marked_+(s(x),s(y)) , Marked_+(x,y)>
<Marked_+(+(x,y),z) , Marked_+(x,+(y,z))>
<Marked_+(+(x,y),z) , Marked_+(y,z)>
<Marked_*(s(x),s(y)) , Marked_+( *(x,y),+(x,y))>
<Marked_*(s(x),s(y)) , Marked_*(x,y)>
<Marked_*(s(x),s(y)) , Marked_+(x,y)>
<Marked_*( *(x,y),z) , Marked_*(x, *(y,z))>
<Marked_*( *(x,y),z) , Marked_*(y,z)>
<Marked_app(cons(x,l1),l2) , Marked_app(l1,l2)>
<Marked_sum(cons(x,l)) , Marked_+(x,sum(l))>
<Marked_sum(cons(x,l)) , Marked_sum(l)>
<Marked_sum(app(l1,l2)) , Marked_+(sum(l1),sum(l2))>
<Marked_sum(app(l1,l2)) , Marked_sum(l1)>
<Marked_sum(app(l1,l2)) , Marked_sum(l2)>
<Marked_prod(cons(x,l)) , Marked_*(x,prod(l))>
<Marked_prod(cons(x,l)) , Marked_prod(l)>
<Marked_prod(app(l1,l2)) , Marked_*(prod(l1),prod(l2))>
<Marked_prod(app(l1,l2)) , Marked_prod(l1)>
<Marked_prod(app(l1,l2)) , Marked_prod(l2)>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_app(cons(x,l1),l2) , Marked_app(l1,l2)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
{
 
DP termination of:
<Marked_sum(cons(x,l)) , Marked_sum(l)>
<Marked_sum(app(l1,l2)) , Marked_sum(l1)>
<Marked_sum(app(l1,l2)) , Marked_sum(l2)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
{
 
DP termination of:
<Marked_prod(cons(x,l)) , Marked_prod(l)>
<Marked_prod(app(l1,l2)) , Marked_prod(l1)>
<Marked_prod(app(l1,l2)) , Marked_prod(l2)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
{
 
DP termination of:
<Marked_*(s(x),s(y)) , Marked_*(x,y)>
<Marked_*( *(x,y),z) , Marked_*(x, *(y,z))>
<Marked_*( *(x,y),z) , Marked_*(y,z)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
{
 
DP termination of:
<Marked_+(s(x),s(y)) , Marked_+(x,y)>
<Marked_+(+(x,y),z) , Marked_+(x,+(y,z))>
<Marked_+(+(x,y),z) , Marked_+(y,z)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
 ]} 
 ]} 

Cime worked for 0.022493 seconds (real time)
Cime Exit Status: 0