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

TRS termination of:
[1] app(nil,k) -> k
[2] app(l,nil) -> l
[3] app(cons(x,l),k) -> cons(x,app(l,k))
[4] sum(cons(x,nil)) -> cons(x,nil)
[5] sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h),l))
[6] a(h,h,x) -> s(x)
[7] a(x,s(y),h) -> a(x,y,s(h))
[8] a(x,s(y),s(z)) -> a(x,y,a(x,s(y),z))
[9] a(s(x),h,z) -> a(x,z,z)


Sub problem: 
guided: 

DP termination of:
<Marked_app(cons(x,l),k) , Marked_app(l,k)>
<Marked_sum(cons(x,cons(y,l))) , Marked_sum(cons(a(x,y,h),l))>
<Marked_sum(cons(x,cons(y,l))) , Marked_a(x,y,h)>
<Marked_a(x,s(y),h) , Marked_a(x,y,s(h))>
<Marked_a(x,s(y),s(z)) , Marked_a(x,y,a(x,s(y),z))>
<Marked_a(x,s(y),s(z)) , Marked_a(x,s(y),z)>
<Marked_a(s(x),h,z) , Marked_a(x,z,z)>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 3 components:

{  <Marked_app(cons(x,l),k) , Marked_app(l,k)> 
      --> <Marked_app(cons(x,l),k) , Marked_app(l,k)>
}

{  <Marked_sum(cons(x,cons(y,l))) , Marked_sum(cons(a(x,y,h),l))> 
      --> <Marked_sum(cons(x,cons(y,l))) , Marked_sum(cons(a(x,y,h),l))>
}

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

ST: Marked_app -> 1

APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  app(nil,k) >= k ;
  app(cons(x,l),k) >= cons(x,app(l,k)) ;
  app(l,nil) >= l ;
  sum(cons(x,nil)) >= cons(x,nil) ;
  sum(cons(x,cons(y,l))) >= sum(cons(a(x,y,h),l)) ;
  a(h,h,x) >= s(x) ;
  a(s(x),h,z) >= a(x,z,z) ;
  a(x,s(y),h) >= a(x,y,s(h)) ;
  a(x,s(y),s(z)) >= a(x,y,a(x,s(y),z)) ;
  Marked_sum(cons(x,cons(y,l))) >= Marked_sum(cons(a(x,y,h),l)) ;
 }
 + Disjunctions:{ 
{
  Marked_sum(cons(x,cons(y,l))) > Marked_sum(cons(a(x,y,h),l)) ;
 }
 } 
=== 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
Sat solver result read
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
constraint: app(nil,k) >= k
constraint: app(cons(x,l),k) >= cons(x,app(l,k))
constraint: app(l,nil) >= l
constraint: sum(cons(x,nil)) >= cons(x,nil)
constraint: sum(cons(x,cons(y,l))) >= sum(cons(a(x,y,h),l))
constraint: a(h,h,x) >= s(x)
constraint: a(s(x),h,z) >= a(x,z,z)
constraint: a(x,s(y),h) >= a(x,y,s(h))
constraint: a(x,s(y),s(z)) >= a(x,y,a(x,s(y),z))
constraint: Marked_sum(cons(x,cons(y,l))) >= Marked_sum(cons(a(x,y,h),l))
APPLY CRITERIA (Subterm criterion)

ST: Marked_a -> 1

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

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

ST: Marked_a -> 2

APPLY CRITERIA (Graph splitting)
Found 1 components:

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

ST: Marked_a -> 3

APPLY CRITERIA (Graph splitting)
Found 0 components:
SOLVED
{
 
TRS termination of:
[1] app(nil,k) -> k
[2] app(l,nil) -> l
[3] app(cons(x,l),k) -> cons(x,app(l,k))
[4] sum(cons(x,nil)) -> cons(x,nil)
[5] sum(cons(x,cons(y,l))) -> sum(cons(a(x,y,h),l))
[6] a(h,h,x) -> s(x)
[7] a(x,s(y),h) -> a(x,y,s(h))
[8] a(x,s(y),s(z)) -> a(x,y,a(x,s(y),z))
[9] a(s(x),h,z) -> a(x,z,z)
 ,
 CRITERION: MDP
 [ {
      
     DP termination of:
     <Marked_app(cons(x,l),k) , Marked_app(l,k)>
<Marked_sum(cons(x,cons(y,l))) , Marked_sum(cons(a(x,y,h),l))>
<Marked_sum(cons(x,cons(y,l))) , Marked_a(x,y,h)>
<Marked_a(x,s(y),h) , Marked_a(x,y,s(h))>
<Marked_a(x,s(y),s(z)) , Marked_a(x,y,a(x,s(y),z))>
<Marked_a(x,s(y),s(z)) , Marked_a(x,s(y),z)>
<Marked_a(s(x),h,z) , Marked_a(x,z,z)>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_app(cons(x,l),k) , Marked_app(l,k)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
{
 
DP termination of:
<Marked_sum(cons(x,cons(y,l))) , Marked_sum(cons(a(x,y,h),l))>
 ,
 CRITERION: ORD
 [ 
     Solution found:
     polynomial interpretation = 
      [ app ] (X0,X1) = 2*X0 + 3*X1 + 0; 
 [ Marked_sum ] (X0) = 3*X0 + 0; 
 [ a ] (X0,X1,X2) = 0; 
 [ cons ] (X0,X1) = 2 + 1*X1 + 0; 
 [ s ] (X0) = 0; 
 [ nil ] () = 0; 
 [ h ] () = 0; 
 [ sum ] (X0) = 2 + 0; 
 ]} 
{
 
DP termination of:
<Marked_a(x,s(y),h) , Marked_a(x,y,s(h))>
<Marked_a(x,s(y),s(z)) , Marked_a(x,y,a(x,s(y),z))>
<Marked_a(x,s(y),s(z)) , Marked_a(x,s(y),z)>
<Marked_a(s(x),h,z) , Marked_a(x,z,z)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
     <Marked_a(x,s(y),s(z)) , Marked_a(x,s(y),z)>
<Marked_a(x,s(y),s(z)) , Marked_a(x,y,a(x,s(y),z))>
<Marked_a(x,s(y),h) , Marked_a(x,y,s(h))>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_a(x,s(y),s(z)) , Marked_a(x,s(y),z)>
<Marked_a(x,s(y),s(z)) , Marked_a(x,y,a(x,s(y),z))>
<Marked_a(x,s(y),h) , Marked_a(x,y,s(h))>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
     <Marked_a(x,s(y),s(z)) , Marked_a(x,s(y),z)>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_a(x,s(y),s(z)) , Marked_a(x,s(y),z)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
 ]} 
 ]} 
 ]} 
 ]} 
 ]} 
 ]} 

Cime worked for 0.069095 seconds (real time)
Cime Exit Status: 0