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

TRS termination of:
[1] minus(x,0) -> x
[2] minus(s(x),s(y)) -> minus(x,y)
[3] quot(0,s(y)) -> 0
[4] quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
[5] plus(0,y) -> y
[6] plus(s(x),y) -> s(plus(x,y))
[7] plus(minus(x,s(0)),minus(y,s(s(z)))) ->
 plus(minus(y,s(s(z))),minus(x,s(0)))


Sub problem: 
guided: 

DP termination of:
<Marked_minus(s(x),s(y)) , Marked_minus(x,y)>
<Marked_quot(s(x),s(y)) , Marked_quot(minus(x,y),s(y))>
<Marked_quot(s(x),s(y)) , Marked_minus(x,y)>
<Marked_plus(s(x),y) , Marked_plus(x,y)>
<Marked_plus(minus(x,s(0)),minus(y,s(s(z)))) , Marked_plus(minus(y,s(s(z))),
                                                minus(x,s(0)))>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 3 components:

{  <Marked_quot(s(x),s(y)) , Marked_quot(minus(x,y),s(y))> 
      --> <Marked_quot(s(x),s(y)) , Marked_quot(minus(x,y),s(y))>
}

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

{  <Marked_plus(minus(x,s(0)),minus(y,s(s(z)))) , Marked_plus(minus(y,s(s(z))),
                                                   minus(x,s(0)))> 
      --> <Marked_plus(minus(x,s(0)),minus(y,s(s(z)))) , Marked_plus(
                                                          minus(y,s(s(z))),
                                                          minus(x,s(0)))>
  <Marked_plus(minus(x,s(0)),minus(y,s(s(z)))) , Marked_plus(minus(y,s(s(z))),
                                                  minus(x,s(0)))> 
     --> <Marked_plus(s(x),y) , Marked_plus(x,y)>
  <Marked_plus(s(x),y) , Marked_plus(x,y)> 
     --> <Marked_plus(minus(x,s(0)),minus(y,s(s(z)))) , Marked_plus(minus(
                                                                    y,
                                                                    s(s(z))),
                                                         minus(x,s(0)))>
  <Marked_plus(s(x),y) , Marked_plus(x,y)> 
     --> <Marked_plus(s(x),y) , Marked_plus(x,y)>
}
APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  minus(s(x),s(y)) >= minus(x,y) ;
  minus(x,0) >= x ;
  quot(0,s(y)) >= 0 ;
  quot(s(x),s(y)) >= s(quot(minus(x,y),s(y))) ;
  plus(minus(x,s(0)),minus(y,s(s(z)))) >= plus(minus(y,s(s(z))),minus(x,s(0))) ;
  plus(0,y) >= y ;
  plus(s(x),y) >= s(plus(x,y)) ;
  Marked_quot(s(x),s(y)) >= Marked_quot(minus(x,y),s(y)) ;
 }
 + Disjunctions:{ 
{
  Marked_quot(s(x),s(y)) > Marked_quot(minus(x,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
Sat solver result read
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
constraint: minus(s(x),s(y)) >= minus(x,y)
constraint: minus(x,0) >= x
constraint: quot(0,s(y)) >= 0
constraint: quot(s(x),s(y)) >= s(quot(minus(x,y),s(y)))
constraint: plus(minus(x,s(0)),minus(y,s(s(z)))) >= plus(minus(y,s(s(z))),
                                                     minus(x,s(0)))
constraint: plus(0,y) >= y
constraint: plus(s(x),y) >= s(plus(x,y))
constraint: Marked_quot(s(x),s(y)) >= Marked_quot(minus(x,y),s(y))
APPLY CRITERIA (Subterm criterion)

ST: Marked_minus -> 1

APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  minus(s(x),s(y)) >= minus(x,y) ;
  minus(x,0) >= x ;
  quot(0,s(y)) >= 0 ;
  quot(s(x),s(y)) >= s(quot(minus(x,y),s(y))) ;
  plus(minus(x,s(0)),minus(y,s(s(z)))) >= plus(minus(y,s(s(z))),minus(x,s(0))) ;
  plus(0,y) >= y ;
  plus(s(x),y) >= s(plus(x,y)) ;
  Marked_plus(minus(x,s(0)),minus(y,s(s(z)))) >= Marked_plus(minus(y,s(s(z))),
                                                  minus(x,s(0))) ;
  Marked_plus(s(x),y) >= Marked_plus(x,y) ;
 }
 + Disjunctions:{ 
{
  Marked_plus(minus(x,s(0)),minus(y,s(s(z)))) > Marked_plus(minus(y,s(s(z))),
                                                 minus(x,s(0))) ;
 }
{
  Marked_plus(s(x),y) > Marked_plus(x,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
Sat solver result read
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
constraint: minus(s(x),s(y)) >= minus(x,y)
constraint: minus(x,0) >= x
constraint: quot(0,s(y)) >= 0
constraint: quot(s(x),s(y)) >= s(quot(minus(x,y),s(y)))
constraint: plus(minus(x,s(0)),minus(y,s(s(z)))) >= plus(minus(y,s(s(z))),
                                                     minus(x,s(0)))
constraint: plus(0,y) >= y
constraint: plus(s(x),y) >= s(plus(x,y))
constraint: Marked_plus(minus(x,s(0)),minus(y,s(s(z)))) >= Marked_plus(
                                                            minus(y,s(s(z))),
                                                            minus(x,s(0)))
constraint: Marked_plus(s(x),y) >= Marked_plus(x,y)
APPLY CRITERIA (Graph splitting)
Found 0 components:
APPLY CRITERIA (Graph splitting)
Found 0 components:
APPLY CRITERIA (Graph splitting)
Found 1 components:

{  <Marked_plus(minus(x,s(0)),minus(y,s(s(z)))) , Marked_plus(minus(y,s(s(z))),
                                                   minus(x,s(0)))> 
      --> <Marked_plus(minus(x,s(0)),minus(y,s(s(z)))) , Marked_plus(
                                                          minus(y,s(s(z))),
                                                          minus(x,s(0)))>
}
APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  minus(s(x),s(y)) >= minus(x,y) ;
  minus(x,0) >= x ;
  quot(0,s(y)) >= 0 ;
  quot(s(x),s(y)) >= s(quot(minus(x,y),s(y))) ;
  plus(minus(x,s(0)),minus(y,s(s(z)))) >= plus(minus(y,s(s(z))),minus(x,s(0))) ;
  plus(0,y) >= y ;
  plus(s(x),y) >= s(plus(x,y)) ;
  Marked_plus(minus(x,s(0)),minus(y,s(s(z)))) >= Marked_plus(minus(y,s(s(z))),
                                                  minus(x,s(0))) ;
 }
 + Disjunctions:{ 
{
  Marked_plus(minus(x,s(0)),minus(y,s(s(z)))) > Marked_plus(minus(y,s(s(z))),
                                                 minus(x,s(0))) ;
 }
 } 
=== 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 : 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
Sat solver result read
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
constraint: minus(s(x),s(y)) >= minus(x,y)
constraint: minus(x,0) >= x
constraint: quot(0,s(y)) >= 0
constraint: quot(s(x),s(y)) >= s(quot(minus(x,y),s(y)))
constraint: plus(minus(x,s(0)),minus(y,s(s(z)))) >= plus(minus(y,s(s(z))),
                                                     minus(x,s(0)))
constraint: plus(0,y) >= y
constraint: plus(s(x),y) >= s(plus(x,y))
constraint: Marked_plus(minus(x,s(0)),minus(y,s(s(z)))) >= Marked_plus(
                                                            minus(y,s(s(z))),
                                                            minus(x,s(0)))
APPLY CRITERIA (Graph splitting)
Found 0 components:
SOLVED
{
 
TRS termination of:
[1] minus(x,0) -> x
[2] minus(s(x),s(y)) -> minus(x,y)
[3] quot(0,s(y)) -> 0
[4] quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
[5] plus(0,y) -> y
[6] plus(s(x),y) -> s(plus(x,y))
[7] plus(minus(x,s(0)),minus(y,s(s(z)))) ->
 plus(minus(y,s(s(z))),minus(x,s(0)))
 ,
 CRITERION: MDP
 [ {
      
     DP termination of:
     <Marked_minus(s(x),s(y)) , Marked_minus(x,y)>
<Marked_quot(s(x),s(y)) , Marked_quot(minus(x,y),s(y))>
<Marked_quot(s(x),s(y)) , Marked_minus(x,y)>
<Marked_plus(s(x),y) , Marked_plus(x,y)>
<Marked_plus(minus(x,s(0)),minus(y,s(s(z)))) , Marked_plus(minus(y,s(s(z))),
                                                minus(x,s(0)))>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_quot(s(x),s(y)) , Marked_quot(minus(x,y),s(y))>
 ,
 CRITERION: ORD
 [ 
     Solution found:
     polynomial interpretation = 
      [ minus ] (X0,X1) = 1 + 1*X0 + 0; 
 [ plus ] (X0,X1) = 2*X0 + 2*X1 + 0; 
 [ s ] (X0) = 2 + 1*X0 + 0; 
 [ Marked_quot ] (X0,X1) = 2*X0 + 0; 
 [ 0 ] () = 2 + 0; 
 [ quot ] (X0,X1) = 2*X0 + 2*X1 + 0; 
 ]} 
{
 
DP termination of:
<Marked_minus(s(x),s(y)) , Marked_minus(x,y)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
{
 
DP termination of:
<Marked_plus(s(x),y) , Marked_plus(x,y)>
<Marked_plus(minus(x,s(0)),minus(y,s(s(z)))) , Marked_plus(minus(y,s(s(z))),
                                                minus(x,s(0)))>
 ,
 CRITERION: CG using polynomial interpretation = 
 [ minus ] (X0,X1) = 1*X0 + 1; 
 [ plus ] (X0,X1) = 2*X1 + 2*X0; 
 [ s ] (X0) = 1*X0 + 2; 
 [ 0 ] () = 0; 
 [ Marked_plus ] (X0,X1) = 2*X1 + 2*X0; 
 [ quot ] (X0,X1) = 2*X0; 
 removing <Marked_plus(s(x),y),Marked_plus(x,y)>
 [ {
      
     DP termination of:
     <Marked_plus(minus(x,s(0)),minus(y,s(s(z)))) , Marked_plus(minus(
                                                                 y,s(s(z))),
                                                     minus(x,s(0)))>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_plus(minus(x,s(0)),minus(y,s(s(z)))) , Marked_plus(minus(
                                                                 y,s(s(z))),
                                                     minus(x,s(0)))>
 ,
 CRITERION: ORD
 [ 
     Solution found:
     Matrix polynomial interpretation (strict sub matrices size: 1x1) = 
      [ minus ] (X0,X1) = [ [ 0 , 1 , 0 ] [ 0 , 0 , 0 ] [ 0 , 0 , 0 ] ]*X1 + 
     [ [ 1 , 0 , 1 ] [ 0 , 1 , 0 ] [ 1 , 0 , 1 ] ]*X0 + [ [ 0 , 0 , 0 ] [ 0 , 0 , 0 ] [ 0 , 0 , 0 ] ]; 
 [ plus ] (X0,X1) = [ [ 1 , 0 , 1 ] [ 0 , 1 , 0 ] [ 1 , 0 , 1 ] ]*X1 + 
[ [ 1 , 0 , 1 ] [ 0 , 1 , 0 ] [ 1 , 0 , 1 ] ]*X0 + [ [ 0 , 0 , 0 ] [ 0 , 0 , 0 ] [ 0 , 0 , 0 ] ]; 
 [ s ] (X0) = [ [ 0 , 0 , 1 ] [ 0 , 1 , 0 ] [ 1 , 0 , 0 ] ]*X0 + [ [ 0 , 0 , 0 ] [ 1 , 0 , 0 ] [ 0 , 0 , 0 ] ]; 
 [ 0 ] () = [ [ 0 , 0 , 0 ] [ 0 , 0 , 0 ] [ 0 , 0 , 0 ] ]; 
 [ Marked_plus ] (X0,X1) = [ [ 1 , 0 , 0 ] [ 0 , 0 , 0 ] [ 0 , 0 , 0 ] ]*X1 + 
[ [ 0 , 0 , 1 ] [ 0 , 0 , 0 ] [ 0 , 0 , 0 ] ]*X0 + [ [ 0 , 0 , 0 ] [ 0 , 0 , 0 ] [ 0 , 0 , 0 ] ]; 
 [ quot ] (X0,X1) = [ [ 0 , 0 , 0 ] [ 0 , 0 , 0 ] [ 0 , 0 , 0 ] ]*X1 + 
[ [ 0 , 0 , 0 ] [ 0 , 1 , 0 ] [ 0 , 0 , 0 ] ]*X0 + [ [ 0 , 0 , 0 ] [ 0 , 0 , 0 ] [ 0 , 0 , 0 ] ]; 
 ]} 
 ]} 
 ]} 
 ]} 
 ]} 

Cime worked for 2.247232 seconds (real time)
Cime Exit Status: 0