- : 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] minus(minus(x,y),z) -> minus(x,plus(y,z))
[8] app(nil,k) -> k
[9] app(l,nil) -> l
[10] app(cons(x,l),k) -> cons(x,app(l,k))
[11] sum(cons(x,nil)) -> cons(x,nil)
[12] sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
[13] sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
[14] plus(0,y) -> y
[15] plus(s(x),y) -> s(plus(x,y))


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_minus(minus(x,y),z) , Marked_minus(x,plus(y,z))>
<Marked_minus(minus(x,y),z) , Marked_plus(y,z)>
<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_plus(s(x),y) , Marked_plus(x,y)>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 6 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(minus(x,y),z) , Marked_minus(x,plus(y,z))> 
      --> <Marked_minus(minus(x,y),z) , Marked_minus(x,plus(y,z))>
  <Marked_minus(minus(x,y),z) , Marked_minus(x,plus(y,z))> 
     --> <Marked_minus(s(x),s(y)) , Marked_minus(x,y)>
  <Marked_minus(s(x),s(y)) , Marked_minus(x,y)> 
     --> <Marked_minus(minus(x,y),z) , Marked_minus(x,plus(y,z))>
  <Marked_minus(s(x),s(y)) , Marked_minus(x,y)> 
     --> <Marked_minus(s(x),s(y)) , Marked_minus(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_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(s(x),y) , Marked_plus(x,y)> 
      --> <Marked_plus(s(x),y) , Marked_plus(x,y)>
  <Marked_plus(s(x),y) , Marked_plus(x,y)> 
     --> <Marked_plus(s(x),y) , Marked_plus(x,y)>
  <Marked_plus(s(x),y) , Marked_plus(x,y)> 
     --> <Marked_plus(s(x),y) , Marked_plus(x,y)>
  <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(minus(x,y),z) >= minus(x,plus(y,z)) ;
  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(0,y) >= y ;
  plus(s(x),y) >= s(plus(x,y)) ;
  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)) ;
  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(minus(x,y),z) >= minus(x,plus(y,z))
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(0,y) >= y
constraint: plus(s(x),y) >= s(plus(x,y))
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: 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(minus(x,y),z) >= minus(x,plus(y,z)) ;
  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(0,y) >= y ;
  plus(s(x),y) >= s(plus(x,y)) ;
  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)) ;
  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: minus(minus(x,y),z) >= minus(x,plus(y,z))
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(0,y) >= y
constraint: plus(s(x),y) >= s(plus(x,y))
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: Marked_sum(app(l,cons(x,cons(y,k)))) >= Marked_sum(app(l,
                                                                sum(cons(
                                                                    x,
                                                                    cons(y,k)))))
                                                     
APPLY CRITERIA (Subterm criterion)

ST: Marked_app -> 1

APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  minus(minus(x,y),z) >= minus(x,plus(y,z)) ;
  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(0,y) >= y ;
  plus(s(x),y) >= s(plus(x,y)) ;
  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)) ;
  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: minus(minus(x,y),z) >= minus(x,plus(y,z))
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(0,y) >= y
constraint: plus(s(x),y) >= s(plus(x,y))
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: Marked_sum(cons(x,cons(y,l))) >= Marked_sum(cons(plus(x,y),l))
APPLY CRITERIA (Subterm criterion)

ST: Marked_plus -> 1

APPLY CRITERIA (Graph splitting)
Found 0 components:
APPLY CRITERIA (Graph splitting)
Found 0 components:
APPLY CRITERIA (Graph splitting)
Found 0 components:
APPLY CRITERIA (Graph splitting)
Found 0 components:
APPLY CRITERIA (Graph splitting)
Found 0 components:
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] minus(minus(x,y),z) -> minus(x,plus(y,z))
[8] app(nil,k) -> k
[9] app(l,nil) -> l
[10] app(cons(x,l),k) -> cons(x,app(l,k))
[11] sum(cons(x,nil)) -> cons(x,nil)
[12] sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
[13] sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
[14] plus(0,y) -> y
[15] plus(s(x),y) -> s(plus(x,y))
 ,
 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_minus(minus(x,y),z) , Marked_minus(x,plus(y,z))>
<Marked_minus(minus(x,y),z) , Marked_plus(y,z)>
<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_plus(s(x),y) , Marked_plus(x,y)>
 ,
 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*X0 + 0; 
 [ sum ] (X0) = 2 + 0; 
 [ plus ] (X0,X1) = 2*X0 + 1*X1 + 0; 
 [ s ] (X0) = 2 + 1*X0 + 0; 
 [ nil ] () = 0; 
 [ 0 ] () = 0; 
 [ app ] (X0,X1) = 2*X0 + 1*X1 + 0; 
 [ Marked_quot ] (X0,X1) = 2*X0 + 0; 
 [ quot ] (X0,X1) = 1*X0 + 0; 
 [ cons ] (X0,X1) = 2 + 0; 
 ]} 
{
 
DP termination of:
<Marked_minus(s(x),s(y)) , Marked_minus(x,y)>
<Marked_minus(minus(x,y),z) , Marked_minus(x,plus(y,z))>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
{
 
DP termination of:
<Marked_sum(app(l,cons(x,cons(y,k)))) , Marked_sum(app(l,
                                                    sum(cons(x,cons(y,k)))))>
 ,
 CRITERION: ORD
 [ 
     Solution found:
     polynomial interpretation = 
      [ minus ] (X0,X1) = 2*X0 + 0; 
 [ sum ] (X0) = 2 + 0; 
 [ plus ] (X0,X1) = 2*X1 + 0; 
 [ s ] (X0) = 1*X0 + 0; 
 [ Marked_sum ] (X0) = 2*X0 + 0; 
 [ nil ] () = 0; 
 [ 0 ] () = 0; 
 [ app ] (X0,X1) = 2*X0 + 2*X1 + 0; 
 [ quot ] (X0,X1) = 0; 
 [ cons ] (X0,X1) = 2 + 1*X1 + 0; 
 ]} 
{
 
DP termination of:
<Marked_app(cons(x,l),k) , Marked_app(l,k)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
{
 
DP termination of:
<Marked_sum(cons(x,cons(y,l))) , Marked_sum(cons(plus(x,y),l))>
 ,
 CRITERION: ORD
 [ 
     Solution found:
     polynomial interpretation = 
      [ minus ] (X0,X1) = 1*X0 + 2*X1 + 0; 
 [ sum ] (X0) = 2 + 0; 
 [ plus ] (X0,X1) = 1*X1 + 0; 
 [ s ] (X0) = 1*X0 + 0; 
 [ Marked_sum ] (X0) = 1*X0 + 0; 
 [ nil ] () = 0; 
 [ 0 ] () = 0; 
 [ app ] (X0,X1) = 2*X0 + 2*X1 + 0; 
 [ quot ] (X0,X1) = 0; 
 [ cons ] (X0,X1) = 2 + 1*X1 + 0; 
 ]} 
{
 
DP termination of:
<Marked_plus(s(x),y) , Marked_plus(x,y)>
<Marked_plus(s(x),y) , Marked_plus(x,y)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
 ]} 
 ]} 

Cime worked for 0.352342 seconds (real time)
Cime Exit Status: 0