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

TRS termination of:
[1] lt(0,s(X)) -> true
[2] lt(s(X),0) -> false
[3] lt(s(X),s(Y)) -> lt(X,Y)
[4] append(nil,Y) -> Y
[5] append(add(N,X),Y) -> add(N,append(X,Y))
[6] split(N,nil) -> pair(nil,nil)
[7] split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
[8] f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
[9] f_2(true,N,M,Y,X,Z) -> pair(X,add(M,Z))
[10] f_2(false,N,M,Y,X,Z) -> pair(add(M,X),Z)
[11] qsort(nil) -> nil
[12] qsort(add(N,X)) -> f_3(split(N,X),N,X)
[13] f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))


Sub problem: 
guided: 

DP termination of:
<Marked_lt(s(X),s(Y)) , Marked_lt(X,Y)>
<Marked_append(add(N,X),Y) , Marked_append(X,Y)>
<Marked_split(N,add(M,Y)) , Marked_f_1(split(N,Y),N,M,Y)>
<Marked_split(N,add(M,Y)) , Marked_split(N,Y)>
<Marked_f_1(pair(X,Z),N,M,Y) , Marked_f_2(lt(N,M),N,M,Y,X,Z)>
<Marked_f_1(pair(X,Z),N,M,Y) , Marked_lt(N,M)>
<Marked_qsort(add(N,X)) , Marked_f_3(split(N,X),N,X)>
<Marked_qsort(add(N,X)) , Marked_split(N,X)>
<Marked_f_3(pair(Y,Z),N,X) , Marked_append(qsort(Y),add(X,qsort(Z)))>
<Marked_f_3(pair(Y,Z),N,X) , Marked_qsort(Y)>
<Marked_f_3(pair(Y,Z),N,X) , Marked_qsort(Z)>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 4 components:

{  <Marked_f_3(pair(Y,Z),N,X) , Marked_qsort(Z)> 
      --> <Marked_qsort(add(N,X)) , Marked_f_3(split(N,X),N,X)>
  <Marked_f_3(pair(Y,Z),N,X) , Marked_qsort(Y)> 
     --> <Marked_qsort(add(N,X)) , Marked_f_3(split(N,X),N,X)>
  <Marked_qsort(add(N,X)) , Marked_f_3(split(N,X),N,X)> 
     --> <Marked_f_3(pair(Y,Z),N,X) , Marked_qsort(Z)>
  <Marked_qsort(add(N,X)) , Marked_f_3(split(N,X),N,X)> 
     --> <Marked_f_3(pair(Y,Z),N,X) , Marked_qsort(Y)>
}

{  <Marked_append(add(N,X),Y) , Marked_append(X,Y)> 
      --> <Marked_append(add(N,X),Y) , Marked_append(X,Y)>
}

{  <Marked_split(N,add(M,Y)) , Marked_split(N,Y)> 
      --> <Marked_split(N,add(M,Y)) , Marked_split(N,Y)>
}

{  <Marked_lt(s(X),s(Y)) , Marked_lt(X,Y)> 
      --> <Marked_lt(s(X),s(Y)) , Marked_lt(X,Y)>
}
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  lt(0,s(X)) >= true ;
  lt(s(X),0) >= false ;
  lt(s(X),s(Y)) >= lt(X,Y) ;
  append(nil,Y) >= Y ;
  append(add(N,X),Y) >= add(N,append(X,Y)) ;
  split(N,nil) >= pair(nil,nil) ;
  split(N,add(M,Y)) >= f_1(split(N,Y),N,M,Y) ;
  f_1(pair(X,Z),N,M,Y) >= f_2(lt(N,M),N,M,Y,X,Z) ;
  f_2(true,N,M,Y,X,Z) >= pair(X,add(M,Z)) ;
  f_2(false,N,M,Y,X,Z) >= pair(add(M,X),Z) ;
  qsort(nil) >= nil ;
  qsort(add(N,X)) >= f_3(split(N,X),N,X) ;
  f_3(pair(Y,Z),N,X) >= append(qsort(Y),add(X,qsort(Z))) ;
  Marked_f_3(pair(Y,Z),N,X) >= Marked_qsort(Y) ;
  Marked_f_3(pair(Y,Z),N,X) >= Marked_qsort(Z) ;
  Marked_qsort(add(N,X)) >= Marked_f_3(split(N,X),N,X) ;
 }
 + Disjunctions:{ 
{
  Marked_f_3(pair(Y,Z),N,X) > Marked_qsort(Y) ;
 }
{
  Marked_f_3(pair(Y,Z),N,X) > Marked_qsort(Z) ;
 }
{
  Marked_qsort(add(N,X)) > Marked_f_3(split(N,X),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
Sat solver result read
=== STOPING TIMER real ===
=== STOPING TIMER virtual ===
constraint: lt(0,s(X)) >= true
constraint: lt(s(X),0) >= false
constraint: lt(s(X),s(Y)) >= lt(X,Y)
constraint: append(nil,Y) >= Y
constraint: append(add(N,X),Y) >= add(N,append(X,Y))
constraint: split(N,nil) >= pair(nil,nil)
constraint: split(N,add(M,Y)) >= f_1(split(N,Y),N,M,Y)
constraint: f_1(pair(X,Z),N,M,Y) >= f_2(lt(N,M),N,M,Y,X,Z)
constraint: f_2(true,N,M,Y,X,Z) >= pair(X,add(M,Z))
constraint: f_2(false,N,M,Y,X,Z) >= pair(add(M,X),Z)
constraint: qsort(nil) >= nil
constraint: qsort(add(N,X)) >= f_3(split(N,X),N,X)
constraint: f_3(pair(Y,Z),N,X) >= append(qsort(Y),add(X,qsort(Z)))
constraint: Marked_f_3(pair(Y,Z),N,X) >= Marked_qsort(Y)
constraint: Marked_f_3(pair(Y,Z),N,X) >= Marked_qsort(Z)
constraint: Marked_qsort(add(N,X)) >= Marked_f_3(split(N,X),N,X)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  lt(0,s(X)) >= true ;
  lt(s(X),0) >= false ;
  lt(s(X),s(Y)) >= lt(X,Y) ;
  append(nil,Y) >= Y ;
  append(add(N,X),Y) >= add(N,append(X,Y)) ;
  split(N,nil) >= pair(nil,nil) ;
  split(N,add(M,Y)) >= f_1(split(N,Y),N,M,Y) ;
  f_1(pair(X,Z),N,M,Y) >= f_2(lt(N,M),N,M,Y,X,Z) ;
  f_2(true,N,M,Y,X,Z) >= pair(X,add(M,Z)) ;
  f_2(false,N,M,Y,X,Z) >= pair(add(M,X),Z) ;
  qsort(nil) >= nil ;
  qsort(add(N,X)) >= f_3(split(N,X),N,X) ;
  f_3(pair(Y,Z),N,X) >= append(qsort(Y),add(X,qsort(Z))) ;
  Marked_append(add(N,X),Y) >= Marked_append(X,Y) ;
 }
 + Disjunctions:{ 
{
  Marked_append(add(N,X),Y) > Marked_append(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: lt(0,s(X)) >= true
constraint: lt(s(X),0) >= false
constraint: lt(s(X),s(Y)) >= lt(X,Y)
constraint: append(nil,Y) >= Y
constraint: append(add(N,X),Y) >= add(N,append(X,Y))
constraint: split(N,nil) >= pair(nil,nil)
constraint: split(N,add(M,Y)) >= f_1(split(N,Y),N,M,Y)
constraint: f_1(pair(X,Z),N,M,Y) >= f_2(lt(N,M),N,M,Y,X,Z)
constraint: f_2(true,N,M,Y,X,Z) >= pair(X,add(M,Z))
constraint: f_2(false,N,M,Y,X,Z) >= pair(add(M,X),Z)
constraint: qsort(nil) >= nil
constraint: qsort(add(N,X)) >= f_3(split(N,X),N,X)
constraint: f_3(pair(Y,Z),N,X) >= append(qsort(Y),add(X,qsort(Z)))
constraint: Marked_append(add(N,X),Y) >= Marked_append(X,Y)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  lt(0,s(X)) >= true ;
  lt(s(X),0) >= false ;
  lt(s(X),s(Y)) >= lt(X,Y) ;
  append(nil,Y) >= Y ;
  append(add(N,X),Y) >= add(N,append(X,Y)) ;
  split(N,nil) >= pair(nil,nil) ;
  split(N,add(M,Y)) >= f_1(split(N,Y),N,M,Y) ;
  f_1(pair(X,Z),N,M,Y) >= f_2(lt(N,M),N,M,Y,X,Z) ;
  f_2(true,N,M,Y,X,Z) >= pair(X,add(M,Z)) ;
  f_2(false,N,M,Y,X,Z) >= pair(add(M,X),Z) ;
  qsort(nil) >= nil ;
  qsort(add(N,X)) >= f_3(split(N,X),N,X) ;
  f_3(pair(Y,Z),N,X) >= append(qsort(Y),add(X,qsort(Z))) ;
  Marked_split(N,add(M,Y)) >= Marked_split(N,Y) ;
 }
 + Disjunctions:{ 
{
  Marked_split(N,add(M,Y)) > Marked_split(N,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: lt(0,s(X)) >= true
constraint: lt(s(X),0) >= false
constraint: lt(s(X),s(Y)) >= lt(X,Y)
constraint: append(nil,Y) >= Y
constraint: append(add(N,X),Y) >= add(N,append(X,Y))
constraint: split(N,nil) >= pair(nil,nil)
constraint: split(N,add(M,Y)) >= f_1(split(N,Y),N,M,Y)
constraint: f_1(pair(X,Z),N,M,Y) >= f_2(lt(N,M),N,M,Y,X,Z)
constraint: f_2(true,N,M,Y,X,Z) >= pair(X,add(M,Z))
constraint: f_2(false,N,M,Y,X,Z) >= pair(add(M,X),Z)
constraint: qsort(nil) >= nil
constraint: qsort(add(N,X)) >= f_3(split(N,X),N,X)
constraint: f_3(pair(Y,Z),N,X) >= append(qsort(Y),add(X,qsort(Z)))
constraint: Marked_split(N,add(M,Y)) >= Marked_split(N,Y)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  lt(0,s(X)) >= true ;
  lt(s(X),0) >= false ;
  lt(s(X),s(Y)) >= lt(X,Y) ;
  append(nil,Y) >= Y ;
  append(add(N,X),Y) >= add(N,append(X,Y)) ;
  split(N,nil) >= pair(nil,nil) ;
  split(N,add(M,Y)) >= f_1(split(N,Y),N,M,Y) ;
  f_1(pair(X,Z),N,M,Y) >= f_2(lt(N,M),N,M,Y,X,Z) ;
  f_2(true,N,M,Y,X,Z) >= pair(X,add(M,Z)) ;
  f_2(false,N,M,Y,X,Z) >= pair(add(M,X),Z) ;
  qsort(nil) >= nil ;
  qsort(add(N,X)) >= f_3(split(N,X),N,X) ;
  f_3(pair(Y,Z),N,X) >= append(qsort(Y),add(X,qsort(Z))) ;
  Marked_lt(s(X),s(Y)) >= Marked_lt(X,Y) ;
 }
 + Disjunctions:{ 
{
  Marked_lt(s(X),s(Y)) > Marked_lt(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: lt(0,s(X)) >= true
constraint: lt(s(X),0) >= false
constraint: lt(s(X),s(Y)) >= lt(X,Y)
constraint: append(nil,Y) >= Y
constraint: append(add(N,X),Y) >= add(N,append(X,Y))
constraint: split(N,nil) >= pair(nil,nil)
constraint: split(N,add(M,Y)) >= f_1(split(N,Y),N,M,Y)
constraint: f_1(pair(X,Z),N,M,Y) >= f_2(lt(N,M),N,M,Y,X,Z)
constraint: f_2(true,N,M,Y,X,Z) >= pair(X,add(M,Z))
constraint: f_2(false,N,M,Y,X,Z) >= pair(add(M,X),Z)
constraint: qsort(nil) >= nil
constraint: qsort(add(N,X)) >= f_3(split(N,X),N,X)
constraint: f_3(pair(Y,Z),N,X) >= append(qsort(Y),add(X,qsort(Z)))
constraint: Marked_lt(s(X),s(Y)) >= Marked_lt(X,Y)
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] lt(0,s(X)) -> true
[2] lt(s(X),0) -> false
[3] lt(s(X),s(Y)) -> lt(X,Y)
[4] append(nil,Y) -> Y
[5] append(add(N,X),Y) -> add(N,append(X,Y))
[6] split(N,nil) -> pair(nil,nil)
[7] split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
[8] f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
[9] f_2(true,N,M,Y,X,Z) -> pair(X,add(M,Z))
[10] f_2(false,N,M,Y,X,Z) -> pair(add(M,X),Z)
[11] qsort(nil) -> nil
[12] qsort(add(N,X)) -> f_3(split(N,X),N,X)
[13] f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))
 ,
 CRITERION: MDP
 [ {
      
     DP termination of:
     <Marked_lt(s(X),s(Y)) , Marked_lt(X,Y)>
<Marked_append(add(N,X),Y) , Marked_append(X,Y)>
<Marked_split(N,add(M,Y)) , Marked_f_1(split(N,Y),N,M,Y)>
<Marked_split(N,add(M,Y)) , Marked_split(N,Y)>
<Marked_f_1(pair(X,Z),N,M,Y) , Marked_f_2(lt(N,M),N,M,Y,X,Z)>
<Marked_f_1(pair(X,Z),N,M,Y) , Marked_lt(N,M)>
<Marked_qsort(add(N,X)) , Marked_f_3(split(N,X),N,X)>
<Marked_qsort(add(N,X)) , Marked_split(N,X)>
<Marked_f_3(pair(Y,Z),N,X) , Marked_append(qsort(Y),add(X,qsort(Z)))>
<Marked_f_3(pair(Y,Z),N,X) , Marked_qsort(Y)>
<Marked_f_3(pair(Y,Z),N,X) , Marked_qsort(Z)>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_qsort(add(N,X)) , Marked_f_3(split(N,X),N,X)>
<Marked_f_3(pair(Y,Z),N,X) , Marked_qsort(Y)>
<Marked_f_3(pair(Y,Z),N,X) , Marked_qsort(Z)>
 ,
 CRITERION: CG using polynomial interpretation = 
 [ true ] () = 1; 
 [ Marked_qsort ] (X0) = 2*X0; 
 [ pair ] (X0,X1) = 1*X1 + 1*X0; 
 [ false ] () = 1; 
 [ qsort ] (X0) = 2*X0; 
 [ 0 ] () = 0; 
 [ f_1 ] (X0,X1,X2,X3) = 1*X0 + 2; 
 [ nil ] () = 0; 
 [ Marked_f_3 ] (X0,X1,X2) = 2*X0; 
 [ lt ] (X0,X1) = 1; 
 [ split ] (X0,X1) = 1*X1; 
 [ append ] (X0,X1) = 1*X1 + 1*X0; 
 [ f_3 ] (X0,X1,X2) = 2*X0 + 2; 
 [ s ] (X0) = 3*X0; 
 [ f_2 ] (X0,X1,X2,X3,X4,X5) = 1*X5 + 1*X4 + 2*X0; 
 [ add ] (X0,X1) = 1*X1 + 2; 
 removing <Marked_qsort(add(N,X)),Marked_f_3(split(N,X),N,X)>
 [ {
      
     DP termination of:
     <Marked_f_3(pair(Y,Z),N,X) , Marked_qsort(Y)>
<Marked_f_3(pair(Y,Z),N,X) , Marked_qsort(Z)>
 ,
 CRITERION: SG
 [  ]} 
 ]} 
{
 
DP termination of:
<Marked_append(add(N,X),Y) , Marked_append(X,Y)>
 ,
 CRITERION: ORD
 [ 
     Solution found:
     polynomial interpretation = 
      [ true ] () = 0; 
 [ pair ] (X0,X1) = 1*X0 + 1*X1 + 0; 
 [ false ] () = 0; 
 [ qsort ] (X0) = 1*X0 + 0; 
 [ 0 ] () = 0; 
 [ f_1 ] (X0,X1,X2,X3) = 2 + 1*X0 + 0; 
 [ nil ] () = 0; 
 [ lt ] (X0,X1) = 0; 
 [ split ] (X0,X1) = 1*X1 + 0; 
 [ append ] (X0,X1) = 1*X0 + 1*X1 + 0; 
 [ f_3 ] (X0,X1,X2) = 2 + 1*X0 + 0; 
 [ s ] (X0) = 3*X0 + 0; 
 [ f_2 ] (X0,X1,X2,X3,X4,X5) = 2 + 1*X4 + 1*X5 + 0; 
 [ add ] (X0,X1) = 2 + 1*X1 + 0; 
 [ Marked_append ] (X0,X1) = 3*X0 + 0; 
 ]} 
{
 
DP termination of:
<Marked_split(N,add(M,Y)) , Marked_split(N,Y)>
 ,
 CRITERION: ORD
 [ 
     Solution found:
     polynomial interpretation = 
      [ true ] () = 0; 
 [ pair ] (X0,X1) = 1*X0 + 1*X1 + 0; 
 [ false ] () = 0; 
 [ qsort ] (X0) = 2*X0 + 0; 
 [ 0 ] () = 0; 
 [ f_1 ] (X0,X1,X2,X3) = 2 + 1*X0 + 0; 
 [ nil ] () = 0; 
 [ lt ] (X0,X1) = 0; 
 [ Marked_split ] (X0,X1) = 3*X1 + 0; 
 [ split ] (X0,X1) = 1 + 1*X1 + 0; 
 [ append ] (X0,X1) = 1*X0 + 1*X1 + 0; 
 [ f_3 ] (X0,X1,X2) = 2 + 2*X0 + 0; 
 [ s ] (X0) = 3*X0 + 0; 
 [ f_2 ] (X0,X1,X2,X3,X4,X5) = 2 + 1*X4 + 1*X5 + 0; 
 [ add ] (X0,X1) = 2 + 1*X1 + 0; 
 ]} 
{
 
DP termination of:
<Marked_lt(s(X),s(Y)) , Marked_lt(X,Y)>
 ,
 CRITERION: ORD
 [ 
     Solution found:
     polynomial interpretation = 
      [ true ] () = 0; 
 [ pair ] (X0,X1) = 0; 
 [ false ] () = 0; 
 [ Marked_lt ] (X0,X1) = 1*X1 + 0; 
 [ qsort ] (X0) = 0; 
 [ 0 ] () = 0; 
 [ f_1 ] (X0,X1,X2,X3) = 0; 
 [ nil ] () = 0; 
 [ lt ] (X0,X1) = 0; 
 [ split ] (X0,X1) = 0; 
 [ append ] (X0,X1) = 2*X1 + 0; 
 [ f_3 ] (X0,X1,X2) = 0; 
 [ s ] (X0) = 1 + 3*X0 + 0; 
 [ f_2 ] (X0,X1,X2,X3,X4,X5) = 0; 
 [ add ] (X0,X1) = 0; 
 ]} 
 ]} 
 ]} 

Cime worked for 0.276149 seconds (real time)
Cime Exit Status: 0