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

TRS termination of:
[1] times(x,0) -> 0
[2] times(x,s(y)) -> plus(times(x,y),x)
[3] plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
[4] plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
[5] plus(zero,y) -> y
[6] plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
[7] id(x) -> x
[8] if(true,x,y) -> x
[9] if(false,x,y) -> y
[10] not(x) -> if(x,false,true)
[11] gt(s(x),zero) -> true
[12] gt(zero,y) -> false
[13] gt(s(x),s(y)) -> gt(x,y)


Sub problem: 
guided: 

DP termination of:
<Marked_times(x,s(y)) , Marked_plus(times(x,y),x)>
<Marked_times(x,s(y)) , Marked_times(x,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),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 3 components:

{  <Marked_times(x,s(y)) , Marked_times(x,y)> 
      --> <Marked_times(x,s(y)) , Marked_times(x,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(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 (Subterm criterion)

ST: Marked_times -> 2

APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  times(x,0) >= 0 ;
  times(x,s(y)) >= plus(times(x,y),x) ;
  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 ;
  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 (Simple graph)
Found the following constraints:
{
  times(x,0) >= 0 ;
  times(x,s(y)) >= plus(times(x,y),x) ;
  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 ;
  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))) ;
 }

APPLY CRITERIA (SOLVE_ORD)
Trying to solve the following constraints:
{
  times(x,0) >= 0 ;
  times(x,s(y)) >= plus(times(x,y),x) ;
  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 ;
  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:{ 
 } 
=== 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 ===
No solution found for these parameters.(1900 bt (1437) [831])
=== 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 143.204297 seconds (real time)
Cime Exit Status: 0