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

TRS termination of:
[1] le(0,y) -> true
[2] le(s(x),0) -> false
[3] le(s(x),s(y)) -> le(x,y)
[4] app(nil,y) -> y
[5] app(add(n,x),y) -> add(n,app(x,y))
[6] low(n,nil) -> nil
[7] low(n,add(m,x)) -> if_low(le(m,n),n,add(m,x))
[8] if_low(true,n,add(m,x)) -> add(m,low(n,x))
[9] if_low(false,n,add(m,x)) -> low(n,x)
[10] high(n,nil) -> nil
[11] high(n,add(m,x)) -> if_high(le(m,n),n,add(m,x))
[12] if_high(true,n,add(m,x)) -> high(n,x)
[13] if_high(false,n,add(m,x)) -> add(m,high(n,x))
[14] head(add(n,x)) -> n
[15] tail(add(n,x)) -> x
[16] isempty(nil) -> true
[17] isempty(add(n,x)) -> false
[18] quicksort(x) ->
 if_qs(isempty(x),low(head(x),tail(x)),head(x),high(head(x),tail(x)))
[19] if_qs(true,x,n,y) -> nil
[20] if_qs(false,x,n,y) -> app(quicksort(x),add(n,quicksort(y)))


Sub problem: 
guided: 

DP termination of:
<Marked_le(s(x),s(y)) , Marked_le(x,y)>
<Marked_app(add(n,x),y) , Marked_app(x,y)>
<Marked_low(n,add(m,x)) , Marked_if_low(le(m,n),n,add(m,x))>
<Marked_low(n,add(m,x)) , Marked_le(m,n)>
<Marked_if_low(true,n,add(m,x)) , Marked_low(n,x)>
<Marked_if_low(false,n,add(m,x)) , Marked_low(n,x)>
<Marked_high(n,add(m,x)) , Marked_if_high(le(m,n),n,add(m,x))>
<Marked_high(n,add(m,x)) , Marked_le(m,n)>
<Marked_if_high(true,n,add(m,x)) , Marked_high(n,x)>
<Marked_if_high(false,n,add(m,x)) , Marked_high(n,x)>
<Marked_quicksort(x) , Marked_if_qs(isempty(x),low(head(x),tail(x)),head(x),
                        high(head(x),tail(x)))>
<Marked_quicksort(x) , Marked_isempty(x)>
<Marked_quicksort(x) , Marked_low(head(x),tail(x))>
<Marked_quicksort(x) , Marked_head(x)>
<Marked_quicksort(x) , Marked_tail(x)>
<Marked_quicksort(x) , Marked_head(x)>
<Marked_quicksort(x) , Marked_high(head(x),tail(x))>
<Marked_quicksort(x) , Marked_head(x)>
<Marked_quicksort(x) , Marked_tail(x)>
<Marked_if_qs(false,x,n,y) , Marked_app(quicksort(x),add(n,quicksort(y)))>
<Marked_if_qs(false,x,n,y) , Marked_quicksort(x)>
<Marked_if_qs(false,x,n,y) , Marked_quicksort(y)>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 5 components:

{  <Marked_if_qs(false,x,n,y) , Marked_quicksort(y)> 
      --> <Marked_quicksort(x) , Marked_if_qs(isempty(x),low(head(x),tail(x)),
                                  head(x),high(head(x),tail(x)))>
  <Marked_if_qs(false,x,n,y) , Marked_quicksort(x)> 
     --> <Marked_quicksort(x) , Marked_if_qs(isempty(x),low(head(x),tail(x)),
                                 head(x),high(head(x),tail(x)))>
  <Marked_quicksort(x) , Marked_if_qs(isempty(x),low(head(x),tail(x)),
                          head(x),high(head(x),tail(x)))> 
     --> <Marked_if_qs(false,x,n,y) , Marked_quicksort(y)>
  <Marked_quicksort(x) , Marked_if_qs(isempty(x),low(head(x),tail(x)),
                          head(x),high(head(x),tail(x)))> 
     --> <Marked_if_qs(false,x,n,y) , Marked_quicksort(x)>
}

{  <Marked_app(add(n,x),y) , Marked_app(x,y)> 
      --> <Marked_app(add(n,x),y) , Marked_app(x,y)>
}

{  <Marked_if_low(false,n,add(m,x)) , Marked_low(n,x)> 
      --> <Marked_low(n,add(m,x)) , Marked_if_low(le(m,n),n,add(m,x))>
  <Marked_if_low(true,n,add(m,x)) , Marked_low(n,x)> 
     --> <Marked_low(n,add(m,x)) , Marked_if_low(le(m,n),n,add(m,x))>
  <Marked_low(n,add(m,x)) , Marked_if_low(le(m,n),n,add(m,x))> 
     --> <Marked_if_low(false,n,add(m,x)) , Marked_low(n,x)>
  <Marked_low(n,add(m,x)) , Marked_if_low(le(m,n),n,add(m,x))> 
     --> <Marked_if_low(true,n,add(m,x)) , Marked_low(n,x)>
}

{  <Marked_if_high(false,n,add(m,x)) , Marked_high(n,x)> 
      --> <Marked_high(n,add(m,x)) , Marked_if_high(le(m,n),n,add(m,x))>
  <Marked_if_high(true,n,add(m,x)) , Marked_high(n,x)> 
     --> <Marked_high(n,add(m,x)) , Marked_if_high(le(m,n),n,add(m,x))>
  <Marked_high(n,add(m,x)) , Marked_if_high(le(m,n),n,add(m,x))> 
     --> <Marked_if_high(false,n,add(m,x)) , Marked_high(n,x)>
  <Marked_high(n,add(m,x)) , Marked_if_high(le(m,n),n,add(m,x))> 
     --> <Marked_if_high(true,n,add(m,x)) , Marked_high(n,x)>
}

{  <Marked_le(s(x),s(y)) , Marked_le(x,y)> 
      --> <Marked_le(s(x),s(y)) , Marked_le(x,y)>
}
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  le(0,y) >= true ;
  le(s(x),0) >= false ;
  le(s(x),s(y)) >= le(x,y) ;
  app(nil,y) >= y ;
  app(add(n,x),y) >= add(n,app(x,y)) ;
  low(n,nil) >= nil ;
  low(n,add(m,x)) >= if_low(le(m,n),n,add(m,x)) ;
  if_low(true,n,add(m,x)) >= add(m,low(n,x)) ;
  if_low(false,n,add(m,x)) >= low(n,x) ;
  high(n,nil) >= nil ;
  high(n,add(m,x)) >= if_high(le(m,n),n,add(m,x)) ;
  if_high(true,n,add(m,x)) >= high(n,x) ;
  if_high(false,n,add(m,x)) >= add(m,high(n,x)) ;
  head(add(n,x)) >= n ;
  tail(add(n,x)) >= x ;
  isempty(nil) >= true ;
  isempty(add(n,x)) >= false ;
  if_qs(true,x,n,y) >= nil ;
  if_qs(false,x,n,y) >= app(quicksort(x),add(n,quicksort(y))) ;
  quicksort(x) >= if_qs(isempty(x),low(head(x),tail(x)),head(x),
                   high(head(x),tail(x))) ;
  Marked_if_qs(false,x,n,y) >= Marked_quicksort(x) ;
  Marked_if_qs(false,x,n,y) >= Marked_quicksort(y) ;
  Marked_quicksort(x) >= Marked_if_qs(isempty(x),low(head(x),tail(x)),
                          head(x),high(head(x),tail(x))) ;
 }
 + Disjunctions:{ 
{
  Marked_if_qs(false,x,n,y) > Marked_quicksort(x) ;
 }
{
  Marked_if_qs(false,x,n,y) > Marked_quicksort(y) ;
 }
{
  Marked_quicksort(x) > Marked_if_qs(isempty(x),low(head(x),tail(x)),
                         head(x),high(head(x),tail(x))) ;
 }
 } 
=== 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 ===
Time out for these parameters.
=== 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 timeout reached
=== STOPING TIMER virtual ===
Time out 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 51.824930 seconds (real time)
Cime Exit Status: 0