- : 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(plus(x,y),l))
[6] sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
[7] sum(plus(cons(0,x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
[8] pred(cons(s(x),nil)) -> cons(x,nil)
[9] plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
[10] plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
[11] plus(zero,y) -> y
[12] plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
[13] id(x) -> x
[14] if(true,x,y) -> x
[15] if(false,x,y) -> y
[16] not(x) -> if(x,false,true)
[17] gt(s(x),zero) -> true
[18] gt(zero,y) -> false
[19] gt(s(x),s(y)) -> gt(x,y)


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(plus(x,y),l))>
<Marked_sum(cons(x,cons(y,l))) , Marked_plus(x,y)>
<Marked_sum(app(l,cons(x,cons(y,k)))) , Marked_sum(app(l,
                                                    sum(cons(x,cons(y,k)))))>
<Marked_sum(app(l,cons(x,cons(y,k)))) , Marked_app(l,sum(cons(x,cons(y,k))))>
<Marked_sum(app(l,cons(x,cons(y,k)))) , Marked_sum(cons(x,cons(y,k)))>
<Marked_sum(plus(cons(0,x),cons(y,l))) , Marked_pred(sum(cons(s(x),cons(y,l))))>
<Marked_sum(plus(cons(0,x),cons(y,l))) , Marked_sum(cons(s(x),cons(y,l)))>
<Marked_plus(s(x),s(y)) , Marked_plus(if(gt(x,y),x,y),
                           if(not(gt(x,y)),id(x),id(y)))>
<Marked_plus(s(x),s(y)) , Marked_if(gt(x,y),x,y)>
<Marked_plus(s(x),s(y)) , Marked_gt(x,y)>
<Marked_plus(s(x),s(y)) , Marked_if(not(gt(x,y)),id(x),id(y))>
<Marked_plus(s(x),s(y)) , Marked_not(gt(x,y))>
<Marked_plus(s(x),s(y)) , Marked_gt(x,y)>
<Marked_plus(s(x),s(y)) , Marked_id(x)>
<Marked_plus(s(x),s(y)) , Marked_id(y)>
<Marked_plus(s(x),x) , Marked_plus(if(gt(x,x),id(x),id(x)),s(x))>
<Marked_plus(s(x),x) , Marked_if(gt(x,x),id(x),id(x))>
<Marked_plus(s(x),x) , Marked_gt(x,x)>
<Marked_plus(s(x),x) , Marked_id(x)>
<Marked_plus(s(x),x) , Marked_id(x)>
<Marked_plus(id(x),s(y)) , Marked_plus(x,if(gt(s(y),y),y,s(y)))>
<Marked_plus(id(x),s(y)) , Marked_if(gt(s(y),y),y,s(y))>
<Marked_plus(id(x),s(y)) , Marked_gt(s(y),y)>
<Marked_not(x) , Marked_if(x,false,true)>
<Marked_gt(s(x),s(y)) , Marked_gt(x,y)>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 5 components:

{  <Marked_sum(app(l,cons(x,cons(y,k)))) , Marked_sum(app(l,
                                                       sum(cons(x,cons(y,k)))))> 
      --> <Marked_sum(app(l,cons(x,cons(y,k)))) , Marked_sum(app(l,
                                                              sum(cons(
                                                                   x,
                                                                   cons(y,k)))))>
}

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

{  <Marked_plus(id(x),s(y)) , Marked_plus(x,if(gt(s(y),y),y,s(y)))> 
      --> <Marked_plus(id(x),s(y)) , Marked_plus(x,if(gt(s(y),y),y,s(y)))>
  <Marked_plus(id(x),s(y)) , Marked_plus(x,if(gt(s(y),y),y,s(y)))> 
     --> <Marked_plus(s(x),x) , Marked_plus(if(gt(x,x),id(x),id(x)),s(x))>
  <Marked_plus(id(x),s(y)) , Marked_plus(x,if(gt(s(y),y),y,s(y)))> 
     --> <Marked_plus(s(x),s(y)) , Marked_plus(if(gt(x,y),x,y),
                                    if(not(gt(x,y)),id(x),id(y)))>
  <Marked_plus(s(x),x) , Marked_plus(if(gt(x,x),id(x),id(x)),s(x))> 
     --> <Marked_plus(id(x),s(y)) , Marked_plus(x,if(gt(s(y),y),y,s(y)))>
  <Marked_plus(s(x),x) , Marked_plus(if(gt(x,x),id(x),id(x)),s(x))> 
     --> <Marked_plus(s(x),x) , Marked_plus(if(gt(x,x),id(x),id(x)),s(x))>
  <Marked_plus(s(x),x) , Marked_plus(if(gt(x,x),id(x),id(x)),s(x))> 
     --> <Marked_plus(s(x),s(y)) , Marked_plus(if(gt(x,y),x,y),
                                    if(not(gt(x,y)),id(x),id(y)))>
  <Marked_plus(s(x),s(y)) , Marked_plus(if(gt(x,y),x,y),
                             if(not(gt(x,y)),id(x),id(y)))> 
     --> <Marked_plus(id(x),s(y)) , Marked_plus(x,if(gt(s(y),y),y,s(y)))>
  <Marked_plus(s(x),s(y)) , Marked_plus(if(gt(x,y),x,y),
                             if(not(gt(x,y)),id(x),id(y)))> 
     --> <Marked_plus(s(x),x) , Marked_plus(if(gt(x,x),id(x),id(x)),s(x))>
  <Marked_plus(s(x),s(y)) , Marked_plus(if(gt(x,y),x,y),
                             if(not(gt(x,y)),id(x),id(y)))> 
     --> <Marked_plus(s(x),s(y)) , Marked_plus(if(gt(x,y),x,y),
                                    if(not(gt(x,y)),id(x),id(y)))>
}

{  <Marked_gt(s(x),s(y)) , Marked_gt(x,y)> 
      --> <Marked_gt(s(x),s(y)) , Marked_gt(x,y)>
}
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(app(l,cons(x,cons(y,k)))) >= sum(app(l,sum(cons(x,cons(y,k))))) ;
  sum(cons(x,nil)) >= cons(x,nil) ;
  sum(cons(x,cons(y,l))) >= sum(cons(plus(x,y),l)) ;
  sum(plus(cons(0,x),cons(y,l))) >= pred(sum(cons(s(x),cons(y,l)))) ;
  plus(s(x),s(y)) >= s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))))) ;
  plus(s(x),x) >= plus(if(gt(x,x),id(x),id(x)),s(x)) ;
  plus(id(x),s(y)) >= s(plus(x,if(gt(s(y),y),y,s(y)))) ;
  plus(zero,y) >= y ;
  pred(cons(s(x),nil)) >= cons(x,nil) ;
  if(true,x,y) >= x ;
  if(false,x,y) >= y ;
  gt(s(x),s(y)) >= gt(x,y) ;
  gt(s(x),zero) >= true ;
  gt(zero,y) >= false ;
  not(x) >= if(x,false,true) ;
  id(x) >= x ;
  Marked_sum(app(l,cons(x,cons(y,k)))) >= Marked_sum(app(l,
                                                      sum(cons(x,cons(y,k))))) ;
 }
 + Disjunctions:{ 
{
  Marked_sum(app(l,cons(x,cons(y,k)))) > Marked_sum(app(l,
                                                     sum(cons(x,cons(y,k))))) ;
 }
 } 
=== 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(app(l,cons(x,cons(y,k)))) >= sum(app(l,sum(cons(x,cons(y,k)))))
constraint: sum(cons(x,nil)) >= cons(x,nil)
constraint: sum(cons(x,cons(y,l))) >= sum(cons(plus(x,y),l))
constraint: sum(plus(cons(0,x),cons(y,l))) >= pred(sum(cons(s(x),cons(y,l))))
constraint: plus(s(x),s(y)) >= s(s(plus(if(gt(x,y),x,y),
                                    if(not(gt(x,y)),id(x),id(y)))))
constraint: plus(s(x),x) >= plus(if(gt(x,x),id(x),id(x)),s(x))
constraint: plus(id(x),s(y)) >= s(plus(x,if(gt(s(y),y),y,s(y))))
constraint: plus(zero,y) >= y
constraint: pred(cons(s(x),nil)) >= cons(x,nil)
constraint: if(true,x,y) >= x
constraint: if(false,x,y) >= y
constraint: gt(s(x),s(y)) >= gt(x,y)
constraint: gt(s(x),zero) >= true
constraint: gt(zero,y) >= false
constraint: not(x) >= if(x,false,true)
constraint: id(x) >= x
constraint: Marked_sum(app(l,cons(x,cons(y,k)))) >= Marked_sum(app(l,
                                                                sum(cons(
                                                                    x,
                                                                    cons(y,k)))))
                                                     
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(app(l,cons(x,cons(y,k)))) >= sum(app(l,sum(cons(x,cons(y,k))))) ;
  sum(cons(x,nil)) >= cons(x,nil) ;
  sum(cons(x,cons(y,l))) >= sum(cons(plus(x,y),l)) ;
  sum(plus(cons(0,x),cons(y,l))) >= pred(sum(cons(s(x),cons(y,l)))) ;
  plus(s(x),s(y)) >= s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))))) ;
  plus(s(x),x) >= plus(if(gt(x,x),id(x),id(x)),s(x)) ;
  plus(id(x),s(y)) >= s(plus(x,if(gt(s(y),y),y,s(y)))) ;
  plus(zero,y) >= y ;
  pred(cons(s(x),nil)) >= cons(x,nil) ;
  if(true,x,y) >= x ;
  if(false,x,y) >= y ;
  gt(s(x),s(y)) >= gt(x,y) ;
  gt(s(x),zero) >= true ;
  gt(zero,y) >= false ;
  not(x) >= if(x,false,true) ;
  id(x) >= x ;
  Marked_app(cons(x,l),k) >= Marked_app(l,k) ;
 }
 + Disjunctions:{ 
{
  Marked_app(cons(x,l),k) > Marked_app(l,k) ;
 }
 } 
=== 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(app(l,cons(x,cons(y,k)))) >= sum(app(l,sum(cons(x,cons(y,k)))))
constraint: sum(cons(x,nil)) >= cons(x,nil)
constraint: sum(cons(x,cons(y,l))) >= sum(cons(plus(x,y),l))
constraint: sum(plus(cons(0,x),cons(y,l))) >= pred(sum(cons(s(x),cons(y,l))))
constraint: plus(s(x),s(y)) >= s(s(plus(if(gt(x,y),x,y),
                                    if(not(gt(x,y)),id(x),id(y)))))
constraint: plus(s(x),x) >= plus(if(gt(x,x),id(x),id(x)),s(x))
constraint: plus(id(x),s(y)) >= s(plus(x,if(gt(s(y),y),y,s(y))))
constraint: plus(zero,y) >= y
constraint: pred(cons(s(x),nil)) >= cons(x,nil)
constraint: if(true,x,y) >= x
constraint: if(false,x,y) >= y
constraint: gt(s(x),s(y)) >= gt(x,y)
constraint: gt(s(x),zero) >= true
constraint: gt(zero,y) >= false
constraint: not(x) >= if(x,false,true)
constraint: id(x) >= x
constraint: Marked_app(cons(x,l),k) >= Marked_app(l,k)
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(app(l,cons(x,cons(y,k)))) >= sum(app(l,sum(cons(x,cons(y,k))))) ;
  sum(cons(x,nil)) >= cons(x,nil) ;
  sum(cons(x,cons(y,l))) >= sum(cons(plus(x,y),l)) ;
  sum(plus(cons(0,x),cons(y,l))) >= pred(sum(cons(s(x),cons(y,l)))) ;
  plus(s(x),s(y)) >= s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))))) ;
  plus(s(x),x) >= plus(if(gt(x,x),id(x),id(x)),s(x)) ;
  plus(id(x),s(y)) >= s(plus(x,if(gt(s(y),y),y,s(y)))) ;
  plus(zero,y) >= y ;
  pred(cons(s(x),nil)) >= cons(x,nil) ;
  if(true,x,y) >= x ;
  if(false,x,y) >= y ;
  gt(s(x),s(y)) >= gt(x,y) ;
  gt(s(x),zero) >= true ;
  gt(zero,y) >= false ;
  not(x) >= if(x,false,true) ;
  id(x) >= x ;
  Marked_sum(cons(x,cons(y,l))) >= Marked_sum(cons(plus(x,y),l)) ;
 }
 + Disjunctions:{ 
{
  Marked_sum(cons(x,cons(y,l))) > Marked_sum(cons(plus(x,y),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(app(l,cons(x,cons(y,k)))) >= sum(app(l,sum(cons(x,cons(y,k)))))
constraint: sum(cons(x,nil)) >= cons(x,nil)
constraint: sum(cons(x,cons(y,l))) >= sum(cons(plus(x,y),l))
constraint: sum(plus(cons(0,x),cons(y,l))) >= pred(sum(cons(s(x),cons(y,l))))
constraint: plus(s(x),s(y)) >= s(s(plus(if(gt(x,y),x,y),
                                    if(not(gt(x,y)),id(x),id(y)))))
constraint: plus(s(x),x) >= plus(if(gt(x,x),id(x),id(x)),s(x))
constraint: plus(id(x),s(y)) >= s(plus(x,if(gt(s(y),y),y,s(y))))
constraint: plus(zero,y) >= y
constraint: pred(cons(s(x),nil)) >= cons(x,nil)
constraint: if(true,x,y) >= x
constraint: if(false,x,y) >= y
constraint: gt(s(x),s(y)) >= gt(x,y)
constraint: gt(s(x),zero) >= true
constraint: gt(zero,y) >= false
constraint: not(x) >= if(x,false,true)
constraint: id(x) >= x
constraint: Marked_sum(cons(x,cons(y,l))) >= Marked_sum(cons(plus(x,y),l))
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(app(l,cons(x,cons(y,k)))) >= sum(app(l,sum(cons(x,cons(y,k))))) ;
  sum(cons(x,nil)) >= cons(x,nil) ;
  sum(cons(x,cons(y,l))) >= sum(cons(plus(x,y),l)) ;
  sum(plus(cons(0,x),cons(y,l))) >= pred(sum(cons(s(x),cons(y,l)))) ;
  plus(s(x),s(y)) >= s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))))) ;
  plus(s(x),x) >= plus(if(gt(x,x),id(x),id(x)),s(x)) ;
  plus(id(x),s(y)) >= s(plus(x,if(gt(s(y),y),y,s(y)))) ;
  plus(zero,y) >= y ;
  pred(cons(s(x),nil)) >= cons(x,nil) ;
  if(true,x,y) >= x ;
  if(false,x,y) >= y ;
  gt(s(x),s(y)) >= gt(x,y) ;
  gt(s(x),zero) >= true ;
  gt(zero,y) >= false ;
  not(x) >= if(x,false,true) ;
  id(x) >= x ;
  Marked_plus(s(x),s(y)) >= Marked_plus(if(gt(x,y),x,y),
                             if(not(gt(x,y)),id(x),id(y))) ;
  Marked_plus(s(x),x) >= Marked_plus(if(gt(x,x),id(x),id(x)),s(x)) ;
  Marked_plus(id(x),s(y)) >= Marked_plus(x,if(gt(s(y),y),y,s(y))) ;
 }
 + Disjunctions:{ 
{
  Marked_plus(s(x),s(y)) > Marked_plus(if(gt(x,y),x,y),
                            if(not(gt(x,y)),id(x),id(y))) ;
 }
{
  Marked_plus(s(x),x) > Marked_plus(if(gt(x,x),id(x),id(x)),s(x)) ;
 }
{
  Marked_plus(id(x),s(y)) > Marked_plus(x,if(gt(s(y),y),y,s(y))) ;
 }
 } 
=== 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 ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== TIMER virtual : 25.000000 ===
Search parameters: AFS type: 2 ; time limit: 25.. 
=== STOPING TIMER virtual ===
=== 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 99.847814 seconds (real time)
Cime Exit Status: 0