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

TRS termination of:
[1] p(s(x)) -> x
[2] plus(x,0) -> x
[3] plus(0,y) -> y
[4] plus(s(x),y) -> s(plus(x,y))
[5] plus(s(x),y) -> s(plus(p(s(x)),y))
[6] plus(x,s(y)) -> s(plus(x,p(s(y))))
[7] times(0,y) -> 0
[8] times(s(0),y) -> y
[9] times(s(x),y) -> plus(y,times(x,y))
[10] div(0,y) -> 0
[11] div(x,y) -> quot(x,y,y)
[12] quot(0,s(y),z) -> 0
[13] quot(s(x),s(y),z) -> quot(x,y,z)
[14] quot(x,0,s(z)) -> s(div(x,s(z)))
[15] div(div(x,y),z) -> div(x,times(y,z))
[16] eq(0,0) -> true
[17] eq(s(x),0) -> false
[18] eq(0,s(y)) -> false
[19] eq(s(x),s(y)) -> eq(x,y)
[20] divides(y,x) -> eq(x,times(div(x,y),y))
[21] prime(s(s(x))) -> pr(s(s(x)),s(x))
[22] pr(x,s(0)) -> true
[23] pr(x,s(s(y))) -> if(divides(s(s(y)),x),x,s(y))
[24] if(true,x,y) -> false
[25] if(false,x,y) -> pr(x,y)


Sub problem: 
guided: 

DP termination of:
<Marked_plus(s(x),y) , Marked_plus(x,y)>
<Marked_plus(s(x),y) , Marked_plus(p(s(x)),y)>
<Marked_plus(s(x),y) , Marked_p(s(x))>
<Marked_plus(x,s(y)) , Marked_plus(x,p(s(y)))>
<Marked_plus(x,s(y)) , Marked_p(s(y))>
<Marked_times(s(x),y) , Marked_plus(y,times(x,y))>
<Marked_times(s(x),y) , Marked_times(x,y)>
<Marked_div(x,y) , Marked_quot(x,y,y)>
<Marked_quot(s(x),s(y),z) , Marked_quot(x,y,z)>
<Marked_quot(x,0,s(z)) , Marked_div(x,s(z))>
<Marked_div(div(x,y),z) , Marked_div(x,times(y,z))>
<Marked_div(div(x,y),z) , Marked_times(y,z)>
<Marked_eq(s(x),s(y)) , Marked_eq(x,y)>
<Marked_divides(y,x) , Marked_eq(x,times(div(x,y),y))>
<Marked_divides(y,x) , Marked_times(div(x,y),y)>
<Marked_divides(y,x) , Marked_div(x,y)>
<Marked_prime(s(s(x))) , Marked_pr(s(s(x)),s(x))>
<Marked_pr(x,s(s(y))) , Marked_if(divides(s(s(y)),x),x,s(y))>
<Marked_pr(x,s(s(y))) , Marked_divides(s(s(y)),x)>
<Marked_if(false,x,y) , Marked_pr(x,y)>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 5 components:

{  <Marked_if(false,x,y) , Marked_pr(x,y)> 
      --> <Marked_pr(x,s(s(y))) , Marked_if(divides(s(s(y)),x),x,s(y))>
  <Marked_pr(x,s(s(y))) , Marked_if(divides(s(s(y)),x),x,s(y))> 
     --> <Marked_if(false,x,y) , Marked_pr(x,y)>
}

{  <Marked_eq(s(x),s(y)) , Marked_eq(x,y)> 
      --> <Marked_eq(s(x),s(y)) , Marked_eq(x,y)>
}

{  <Marked_div(div(x,y),z) , Marked_div(x,times(y,z))> 
      --> <Marked_div(div(x,y),z) , Marked_div(x,times(y,z))>
  <Marked_div(div(x,y),z) , Marked_div(x,times(y,z))> 
     --> <Marked_div(x,y) , Marked_quot(x,y,y)>
  <Marked_quot(x,0,s(z)) , Marked_div(x,s(z))> 
     --> <Marked_div(div(x,y),z) , Marked_div(x,times(y,z))>
  <Marked_quot(x,0,s(z)) , Marked_div(x,s(z))> 
     --> <Marked_div(x,y) , Marked_quot(x,y,y)>
  <Marked_quot(s(x),s(y),z) , Marked_quot(x,y,z)> 
     --> <Marked_quot(x,0,s(z)) , Marked_div(x,s(z))>
  <Marked_quot(s(x),s(y),z) , Marked_quot(x,y,z)> 
     --> <Marked_quot(s(x),s(y),z) , Marked_quot(x,y,z)>
  <Marked_div(x,y) , Marked_quot(x,y,y)> 
     --> <Marked_quot(x,0,s(z)) , Marked_div(x,s(z))>
  <Marked_div(x,y) , Marked_quot(x,y,y)> 
     --> <Marked_quot(s(x),s(y),z) , Marked_quot(x,y,z)>
}

{  <Marked_times(s(x),y) , Marked_times(x,y)> 
      --> <Marked_times(s(x),y) , Marked_times(x,y)>
}

{  <Marked_plus(x,s(y)) , Marked_plus(x,p(s(y)))> 
      --> <Marked_plus(x,s(y)) , Marked_plus(x,p(s(y)))>
  <Marked_plus(x,s(y)) , Marked_plus(x,p(s(y)))> 
     --> <Marked_plus(s(x),y) , Marked_plus(p(s(x)),y)>
  <Marked_plus(x,s(y)) , Marked_plus(x,p(s(y)))> 
     --> <Marked_plus(s(x),y) , Marked_plus(x,y)>
  <Marked_plus(s(x),y) , Marked_plus(p(s(x)),y)> 
     --> <Marked_plus(x,s(y)) , Marked_plus(x,p(s(y)))>
  <Marked_plus(s(x),y) , Marked_plus(p(s(x)),y)> 
     --> <Marked_plus(s(x),y) , Marked_plus(p(s(x)),y)>
  <Marked_plus(s(x),y) , Marked_plus(p(s(x)),y)> 
     --> <Marked_plus(s(x),y) , Marked_plus(x,y)>
  <Marked_plus(s(x),y) , Marked_plus(x,y)> 
     --> <Marked_plus(x,s(y)) , Marked_plus(x,p(s(y)))>
  <Marked_plus(s(x),y) , Marked_plus(x,y)> 
     --> <Marked_plus(s(x),y) , Marked_plus(p(s(x)),y)>
  <Marked_plus(s(x),y) , Marked_plus(x,y)> 
     --> <Marked_plus(s(x),y) , Marked_plus(x,y)>
}
APPLY CRITERIA (Subterm criterion)

ST: Marked_if -> 3
Marked_pr -> 2

APPLY CRITERIA (Subterm criterion)

ST: Marked_eq -> 1

APPLY CRITERIA (Subterm criterion)

ST: Marked_quot -> 1
Marked_div -> 1

APPLY CRITERIA (Subterm criterion)

ST: Marked_times -> 1

APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  p(s(x)) >= x ;
  plus(s(x),y) >= s(plus(p(s(x)),y)) ;
  plus(s(x),y) >= s(plus(x,y)) ;
  plus(0,y) >= y ;
  plus(x,s(y)) >= s(plus(x,p(s(y)))) ;
  plus(x,0) >= x ;
  times(s(0),y) >= y ;
  times(s(x),y) >= plus(y,times(x,y)) ;
  times(0,y) >= 0 ;
  div(0,y) >= 0 ;
  div(div(x,y),z) >= div(x,times(y,z)) ;
  div(x,y) >= quot(x,y,y) ;
  quot(s(x),s(y),z) >= quot(x,y,z) ;
  quot(0,s(y),z) >= 0 ;
  quot(x,0,s(z)) >= s(div(x,s(z))) ;
  eq(s(x),s(y)) >= eq(x,y) ;
  eq(s(x),0) >= false ;
  eq(0,s(y)) >= false ;
  eq(0,0) >= true ;
  divides(y,x) >= eq(x,times(div(x,y),y)) ;
  pr(x,s(s(y))) >= if(divides(s(s(y)),x),x,s(y)) ;
  pr(x,s(0)) >= true ;
  prime(s(s(x))) >= pr(s(s(x)),s(x)) ;
  if(true,x,y) >= false ;
  if(false,x,y) >= pr(x,y) ;
  Marked_plus(s(x),y) >= Marked_plus(p(s(x)),y) ;
  Marked_plus(s(x),y) >= Marked_plus(x,y) ;
  Marked_plus(x,s(y)) >= Marked_plus(x,p(s(y))) ;
 }
 + Disjunctions:{ 
{
  Marked_plus(s(x),y) > Marked_plus(p(s(x)),y) ;
 }
{
  Marked_plus(s(x),y) > Marked_plus(x,y) ;
 }
{
  Marked_plus(x,s(y)) > Marked_plus(x,p(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 solver returned
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
No solution found 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:
{
  p(s(x)) >= x ;
  plus(s(x),y) >= s(plus(p(s(x)),y)) ;
  plus(s(x),y) >= s(plus(x,y)) ;
  plus(0,y) >= y ;
  plus(x,s(y)) >= s(plus(x,p(s(y)))) ;
  plus(x,0) >= x ;
  times(s(0),y) >= y ;
  times(s(x),y) >= plus(y,times(x,y)) ;
  times(0,y) >= 0 ;
  div(0,y) >= 0 ;
  div(div(x,y),z) >= div(x,times(y,z)) ;
  div(x,y) >= quot(x,y,y) ;
  quot(s(x),s(y),z) >= quot(x,y,z) ;
  quot(0,s(y),z) >= 0 ;
  quot(x,0,s(z)) >= s(div(x,s(z))) ;
  eq(s(x),s(y)) >= eq(x,y) ;
  eq(s(x),0) >= false ;
  eq(0,s(y)) >= false ;
  eq(0,0) >= true ;
  divides(y,x) >= eq(x,times(div(x,y),y)) ;
  pr(x,s(s(y))) >= if(divides(s(s(y)),x),x,s(y)) ;
  pr(x,s(0)) >= true ;
  prime(s(s(x))) >= pr(s(s(x)),s(x)) ;
  if(true,x,y) >= false ;
  if(false,x,y) >= pr(x,y) ;
  Marked_plus(s(x),y) > Marked_plus(p(s(x)),y) ;
  Marked_plus(s(x),y) > Marked_plus(x,y) ;
  Marked_plus(x,s(y)) > Marked_plus(x,p(s(y))) ;
 }

APPLY CRITERIA (SOLVE_ORD)
Trying to solve the following constraints:
{
  p(s(x)) >= x ;
  plus(s(x),y) >= s(plus(p(s(x)),y)) ;
  plus(s(x),y) >= s(plus(x,y)) ;
  plus(0,y) >= y ;
  plus(x,s(y)) >= s(plus(x,p(s(y)))) ;
  plus(x,0) >= x ;
  times(s(0),y) >= y ;
  times(s(x),y) >= plus(y,times(x,y)) ;
  times(0,y) >= 0 ;
  div(0,y) >= 0 ;
  div(div(x,y),z) >= div(x,times(y,z)) ;
  div(x,y) >= quot(x,y,y) ;
  quot(s(x),s(y),z) >= quot(x,y,z) ;
  quot(0,s(y),z) >= 0 ;
  quot(x,0,s(z)) >= s(div(x,s(z))) ;
  eq(s(x),s(y)) >= eq(x,y) ;
  eq(s(x),0) >= false ;
  eq(0,s(y)) >= false ;
  eq(0,0) >= true ;
  divides(y,x) >= eq(x,times(div(x,y),y)) ;
  pr(x,s(s(y))) >= if(divides(s(s(y)),x),x,s(y)) ;
  pr(x,s(0)) >= true ;
  prime(s(s(x))) >= pr(s(s(x)),s(x)) ;
  if(true,x,y) >= false ;
  if(false,x,y) >= pr(x,y) ;
  Marked_plus(s(x),y) > Marked_plus(p(s(x)),y) ;
  Marked_plus(s(x),y) > Marked_plus(x,y) ;
  Marked_plus(x,s(y)) > Marked_plus(x,p(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.(611 bt (736) [112])
=== 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 solver returned
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
No solution found 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 4.792895 seconds (real time)
Cime Exit Status: 0