- : 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] le(0,y) -> true
[6] le(s(x),0) -> false
[7] le(s(x),s(y)) -> le(x,y)
[8] app(nil,y) -> y
[9] app(add(n,x),y) -> add(n,app(x,y))
[10] low(n,nil) -> nil
[11] low(n,add(m,x)) -> if_low(le(m,n),n,add(m,x))
[12] if_low(true,n,add(m,x)) -> add(m,low(n,x))
[13] if_low(false,n,add(m,x)) -> low(n,x)
[14] high(n,nil) -> nil
[15] high(n,add(m,x)) -> if_high(le(m,n),n,add(m,x))
[16] if_high(true,n,add(m,x)) -> high(n,x)
[17] if_high(false,n,add(m,x)) -> add(m,high(n,x))
[18] quicksort(nil) -> nil
[19] quicksort(add(n,x)) ->
 app(quicksort(low(n,x)),add(n,quicksort(high(n,x))))


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_le(s(x),s(y)) , Marked_le(x,y)>
<Marked_app(add(n,x),y) , Marked_app(x,y)>
<Marked_low(n,add(m,x)) , Marked_if_low(le(m,n),n,add(m,x))>
<Marked_low(n,add(m,x)) , Marked_le(m,n)>
<Marked_if_low(true,n,add(m,x)) , Marked_low(n,x)>
<Marked_if_low(false,n,add(m,x)) , Marked_low(n,x)>
<Marked_high(n,add(m,x)) , Marked_if_high(le(m,n),n,add(m,x))>
<Marked_high(n,add(m,x)) , Marked_le(m,n)>
<Marked_if_high(true,n,add(m,x)) , Marked_high(n,x)>
<Marked_if_high(false,n,add(m,x)) , Marked_high(n,x)>
<Marked_quicksort(add(n,x)) , Marked_app(quicksort(low(n,x)),
                               add(n,quicksort(high(n,x))))>
<Marked_quicksort(add(n,x)) , Marked_quicksort(low(n,x))>
<Marked_quicksort(add(n,x)) , Marked_low(n,x)>
<Marked_quicksort(add(n,x)) , Marked_quicksort(high(n,x))>
<Marked_quicksort(add(n,x)) , Marked_high(n,x)>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 7 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_quicksort(add(n,x)) , Marked_quicksort(high(n,x))> 
      --> <Marked_quicksort(add(n,x)) , Marked_quicksort(high(n,x))>
  <Marked_quicksort(add(n,x)) , Marked_quicksort(high(n,x))> 
     --> <Marked_quicksort(add(n,x)) , Marked_quicksort(low(n,x))>
  <Marked_quicksort(add(n,x)) , Marked_quicksort(low(n,x))> 
     --> <Marked_quicksort(add(n,x)) , Marked_quicksort(high(n,x))>
  <Marked_quicksort(add(n,x)) , Marked_quicksort(low(n,x))> 
     --> <Marked_quicksort(add(n,x)) , Marked_quicksort(low(n,x))>
}

{  <Marked_app(add(n,x),y) , Marked_app(x,y)> 
      --> <Marked_app(add(n,x),y) , Marked_app(x,y)>
}

{  <Marked_if_low(false,n,add(m,x)) , Marked_low(n,x)> 
      --> <Marked_low(n,add(m,x)) , Marked_if_low(le(m,n),n,add(m,x))>
  <Marked_if_low(true,n,add(m,x)) , Marked_low(n,x)> 
     --> <Marked_low(n,add(m,x)) , Marked_if_low(le(m,n),n,add(m,x))>
  <Marked_low(n,add(m,x)) , Marked_if_low(le(m,n),n,add(m,x))> 
     --> <Marked_if_low(false,n,add(m,x)) , Marked_low(n,x)>
  <Marked_low(n,add(m,x)) , Marked_if_low(le(m,n),n,add(m,x))> 
     --> <Marked_if_low(true,n,add(m,x)) , Marked_low(n,x)>
}

{  <Marked_if_high(false,n,add(m,x)) , Marked_high(n,x)> 
      --> <Marked_high(n,add(m,x)) , Marked_if_high(le(m,n),n,add(m,x))>
  <Marked_if_high(true,n,add(m,x)) , Marked_high(n,x)> 
     --> <Marked_high(n,add(m,x)) , Marked_if_high(le(m,n),n,add(m,x))>
  <Marked_high(n,add(m,x)) , Marked_if_high(le(m,n),n,add(m,x))> 
     --> <Marked_if_high(false,n,add(m,x)) , Marked_high(n,x)>
  <Marked_high(n,add(m,x)) , Marked_if_high(le(m,n),n,add(m,x))> 
     --> <Marked_if_high(true,n,add(m,x)) , Marked_high(n,x)>
}

{  <Marked_le(s(x),s(y)) , Marked_le(x,y)> 
      --> <Marked_le(s(x),s(y)) , Marked_le(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))) ;
  le(0,y) >= true ;
  le(s(x),0) >= false ;
  le(s(x),s(y)) >= le(x,y) ;
  app(nil,y) >= y ;
  app(add(n,x),y) >= add(n,app(x,y)) ;
  low(n,nil) >= nil ;
  low(n,add(m,x)) >= if_low(le(m,n),n,add(m,x)) ;
  if_low(true,n,add(m,x)) >= add(m,low(n,x)) ;
  if_low(false,n,add(m,x)) >= low(n,x) ;
  high(n,nil) >= nil ;
  high(n,add(m,x)) >= if_high(le(m,n),n,add(m,x)) ;
  if_high(true,n,add(m,x)) >= high(n,x) ;
  if_high(false,n,add(m,x)) >= add(m,high(n,x)) ;
  quicksort(nil) >= nil ;
  quicksort(add(n,x)) >= app(quicksort(low(n,x)),add(n,quicksort(high(n,x)))) ;
  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: le(0,y) >= true
constraint: le(s(x),0) >= false
constraint: le(s(x),s(y)) >= le(x,y)
constraint: app(nil,y) >= y
constraint: app(add(n,x),y) >= add(n,app(x,y))
constraint: low(n,nil) >= nil
constraint: low(n,add(m,x)) >= if_low(le(m,n),n,add(m,x))
constraint: if_low(true,n,add(m,x)) >= add(m,low(n,x))
constraint: if_low(false,n,add(m,x)) >= low(n,x)
constraint: high(n,nil) >= nil
constraint: high(n,add(m,x)) >= if_high(le(m,n),n,add(m,x))
constraint: if_high(true,n,add(m,x)) >= high(n,x)
constraint: if_high(false,n,add(m,x)) >= add(m,high(n,x))
constraint: quicksort(nil) >= nil
constraint: quicksort(add(n,x)) >= app(quicksort(low(n,x)),
                                    add(n,quicksort(high(n,x))))
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))) ;
  le(0,y) >= true ;
  le(s(x),0) >= false ;
  le(s(x),s(y)) >= le(x,y) ;
  app(nil,y) >= y ;
  app(add(n,x),y) >= add(n,app(x,y)) ;
  low(n,nil) >= nil ;
  low(n,add(m,x)) >= if_low(le(m,n),n,add(m,x)) ;
  if_low(true,n,add(m,x)) >= add(m,low(n,x)) ;
  if_low(false,n,add(m,x)) >= low(n,x) ;
  high(n,nil) >= nil ;
  high(n,add(m,x)) >= if_high(le(m,n),n,add(m,x)) ;
  if_high(true,n,add(m,x)) >= high(n,x) ;
  if_high(false,n,add(m,x)) >= add(m,high(n,x)) ;
  quicksort(nil) >= nil ;
  quicksort(add(n,x)) >= app(quicksort(low(n,x)),add(n,quicksort(high(n,x)))) ;
  Marked_quicksort(add(n,x)) >= Marked_quicksort(low(n,x)) ;
  Marked_quicksort(add(n,x)) >= Marked_quicksort(high(n,x)) ;
 }
 + Disjunctions:{ 
{
  Marked_quicksort(add(n,x)) > Marked_quicksort(low(n,x)) ;
 }
{
  Marked_quicksort(add(n,x)) > Marked_quicksort(high(n,x)) ;
 }
 } 
=== 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 ===
=== 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: le(0,y) >= true
constraint: le(s(x),0) >= false
constraint: le(s(x),s(y)) >= le(x,y)
constraint: app(nil,y) >= y
constraint: app(add(n,x),y) >= add(n,app(x,y))
constraint: low(n,nil) >= nil
constraint: low(n,add(m,x)) >= if_low(le(m,n),n,add(m,x))
constraint: if_low(true,n,add(m,x)) >= add(m,low(n,x))
constraint: if_low(false,n,add(m,x)) >= low(n,x)
constraint: high(n,nil) >= nil
constraint: high(n,add(m,x)) >= if_high(le(m,n),n,add(m,x))
constraint: if_high(true,n,add(m,x)) >= high(n,x)
constraint: if_high(false,n,add(m,x)) >= add(m,high(n,x))
constraint: quicksort(nil) >= nil
constraint: quicksort(add(n,x)) >= app(quicksort(low(n,x)),
                                    add(n,quicksort(high(n,x))))
constraint: Marked_quicksort(add(n,x)) >= Marked_quicksort(low(n,x))
constraint: Marked_quicksort(add(n,x)) >= Marked_quicksort(high(n,x))
APPLY CRITERIA (Subterm criterion)

ST: Marked_app -> 1

APPLY CRITERIA (Subterm criterion)

ST: Marked_if_low -> 3
Marked_low -> 2

APPLY CRITERIA (Subterm criterion)

ST: Marked_if_high -> 3
Marked_high -> 2

APPLY CRITERIA (Subterm criterion)

ST: Marked_le -> 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:
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] le(0,y) -> true
[6] le(s(x),0) -> false
[7] le(s(x),s(y)) -> le(x,y)
[8] app(nil,y) -> y
[9] app(add(n,x),y) -> add(n,app(x,y))
[10] low(n,nil) -> nil
[11] low(n,add(m,x)) -> if_low(le(m,n),n,add(m,x))
[12] if_low(true,n,add(m,x)) -> add(m,low(n,x))
[13] if_low(false,n,add(m,x)) -> low(n,x)
[14] high(n,nil) -> nil
[15] high(n,add(m,x)) -> if_high(le(m,n),n,add(m,x))
[16] if_high(true,n,add(m,x)) -> high(n,x)
[17] if_high(false,n,add(m,x)) -> add(m,high(n,x))
[18] quicksort(nil) -> nil
[19] quicksort(add(n,x)) ->
 app(quicksort(low(n,x)),add(n,quicksort(high(n,x))))
 ,
 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_le(s(x),s(y)) , Marked_le(x,y)>
<Marked_app(add(n,x),y) , Marked_app(x,y)>
<Marked_low(n,add(m,x)) , Marked_if_low(le(m,n),n,add(m,x))>
<Marked_low(n,add(m,x)) , Marked_le(m,n)>
<Marked_if_low(true,n,add(m,x)) , Marked_low(n,x)>
<Marked_if_low(false,n,add(m,x)) , Marked_low(n,x)>
<Marked_high(n,add(m,x)) , Marked_if_high(le(m,n),n,add(m,x))>
<Marked_high(n,add(m,x)) , Marked_le(m,n)>
<Marked_if_high(true,n,add(m,x)) , Marked_high(n,x)>
<Marked_if_high(false,n,add(m,x)) , Marked_high(n,x)>
<Marked_quicksort(add(n,x)) , Marked_app(quicksort(low(n,x)),
                               add(n,quicksort(high(n,x))))>
<Marked_quicksort(add(n,x)) , Marked_quicksort(low(n,x))>
<Marked_quicksort(add(n,x)) , Marked_low(n,x)>
<Marked_quicksort(add(n,x)) , Marked_quicksort(high(n,x))>
<Marked_quicksort(add(n,x)) , Marked_high(n,x)>
 ,
 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; 
 [ nil ] () = 0; 
 [ true ] () = 0; 
 [ high ] (X0,X1) = 0; 
 [ s ] (X0) = 1 + 2*X0 + 0; 
 [ low ] (X0,X1) = 0; 
 [ false ] () = 0; 
 [ Marked_quot ] (X0,X1) = 2*X0 + 0; 
 [ quicksort ] (X0) = 1 + 0; 
 [ 0 ] () = 0; 
 [ add ] (X0,X1) = 0; 
 [ le ] (X0,X1) = 0; 
 [ if_high ] (X0,X1,X2) = 0; 
 [ quot ] (X0,X1) = 1*X0 + 0; 
 [ if_low ] (X0,X1,X2) = 0; 
 [ app ] (X0,X1) = 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_quicksort(add(n,x)) , Marked_quicksort(low(n,x))>
<Marked_quicksort(add(n,x)) , Marked_quicksort(high(n,x))>
 ,
 CRITERION: ORD
 [ 
     Solution found:
     RPO with AFS = AFS: minus -> 0high -> 1low -> 1if_high -> 2if_low -> 2
     and precedence:
     prec (All symbols are Lex.): { s  <  quot ; quot  >  s ; true  <  le ;
                                   le  >  true ; le  >  false ; false  <  le ;
                                   app  >  add ; app  <  quicksort ;
                                   add  <  app ; add  <  quicksort ;
                                   quicksort  >  app ; quicksort  >  add ; }

 ]} 
{
 
DP termination of:
<Marked_app(add(n,x),y) , Marked_app(x,y)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
{
 
DP termination of:
<Marked_low(n,add(m,x)) , Marked_if_low(le(m,n),n,add(m,x))>
<Marked_if_low(true,n,add(m,x)) , Marked_low(n,x)>
<Marked_if_low(false,n,add(m,x)) , Marked_low(n,x)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
     <Marked_low(n,add(m,x)) , Marked_if_low(le(m,n),n,add(m,x))>
 ,
 CRITERION: SG
 [  ]} 
 ]} 
{
 
DP termination of:
<Marked_high(n,add(m,x)) , Marked_if_high(le(m,n),n,add(m,x))>
<Marked_if_high(true,n,add(m,x)) , Marked_high(n,x)>
<Marked_if_high(false,n,add(m,x)) , Marked_high(n,x)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
     <Marked_high(n,add(m,x)) , Marked_if_high(le(m,n),n,add(m,x))>
 ,
 CRITERION: SG
 [  ]} 
 ]} 
{
 
DP termination of:
<Marked_le(s(x),s(y)) , Marked_le(x,y)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
 ]} 
 ]} 

Cime worked for 0.614759 seconds (real time)
Cime Exit Status: 0