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

TRS termination of:
[1] terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
[2] sqr(0) -> 0
[3] sqr(s(X)) -> s(add(sqr(X),dbl(X)))
[4] dbl(0) -> 0
[5] dbl(s(X)) -> s(s(dbl(X)))
[6] add(0,X) -> X
[7] add(s(X),Y) -> s(add(X,Y))
[8] first(0,X) -> nil
[9] first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
[10] terms(X) -> n__terms(X)
[11] s(X) -> n__s(X)
[12] first(X1,X2) -> n__first(X1,X2)
[13] activate(n__terms(X)) -> terms(activate(X))
[14] activate(n__s(X)) -> s(activate(X))
[15] activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
[16] activate(X) -> X


Sub problem: 
guided: 

DP termination of:
<Marked_terms(N) , Marked_sqr(N)>
<Marked_sqr(s(X)) , Marked_s(add(sqr(X),dbl(X)))>
<Marked_sqr(s(X)) , Marked_add(sqr(X),dbl(X))>
<Marked_sqr(s(X)) , Marked_sqr(X)>
<Marked_sqr(s(X)) , Marked_dbl(X)>
<Marked_dbl(s(X)) , Marked_s(s(dbl(X)))>
<Marked_dbl(s(X)) , Marked_s(dbl(X))>
<Marked_dbl(s(X)) , Marked_dbl(X)>
<Marked_add(s(X),Y) , Marked_s(add(X,Y))>
<Marked_add(s(X),Y) , Marked_add(X,Y)>
<Marked_first(s(X),cons(Y,Z)) , Marked_activate(Z)>
<Marked_activate(n__terms(X)) , Marked_terms(activate(X))>
<Marked_activate(n__terms(X)) , Marked_activate(X)>
<Marked_activate(n__s(X)) , Marked_s(activate(X))>
<Marked_activate(n__s(X)) , Marked_activate(X)>
<Marked_activate(n__first(X1,X2)) , Marked_first(activate(X1),activate(X2))>
<Marked_activate(n__first(X1,X2)) , Marked_activate(X1)>
<Marked_activate(n__first(X1,X2)) , Marked_activate(X2)>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 4 components:

{  <Marked_activate(n__first(X1,X2)) , Marked_activate(X2)> 
      --> <Marked_activate(n__first(X1,X2)) , Marked_activate(X2)>
  <Marked_activate(n__first(X1,X2)) , Marked_activate(X2)> 
     --> <Marked_activate(n__first(X1,X2)) , Marked_activate(X1)>
  <Marked_activate(n__first(X1,X2)) , Marked_activate(X2)> 
     --> <Marked_activate(n__first(X1,X2)) , Marked_first(activate(X1),
                                              activate(X2))>
  <Marked_activate(n__first(X1,X2)) , Marked_activate(X2)> 
     --> <Marked_activate(n__s(X)) , Marked_activate(X)>
  <Marked_activate(n__first(X1,X2)) , Marked_activate(X2)> 
     --> <Marked_activate(n__terms(X)) , Marked_activate(X)>
  <Marked_activate(n__first(X1,X2)) , Marked_activate(X1)> 
     --> <Marked_activate(n__first(X1,X2)) , Marked_activate(X2)>
  <Marked_activate(n__first(X1,X2)) , Marked_activate(X1)> 
     --> <Marked_activate(n__first(X1,X2)) , Marked_activate(X1)>
  <Marked_activate(n__first(X1,X2)) , Marked_activate(X1)> 
     --> <Marked_activate(n__first(X1,X2)) , Marked_first(activate(X1),
                                              activate(X2))>
  <Marked_activate(n__first(X1,X2)) , Marked_activate(X1)> 
     --> <Marked_activate(n__s(X)) , Marked_activate(X)>
  <Marked_activate(n__first(X1,X2)) , Marked_activate(X1)> 
     --> <Marked_activate(n__terms(X)) , Marked_activate(X)>
  <Marked_activate(n__first(X1,X2)) , Marked_first(activate(X1),activate(X2))> 
     --> <Marked_first(s(X),cons(Y,Z)) , Marked_activate(Z)>
  <Marked_activate(n__s(X)) , Marked_activate(X)> 
     --> <Marked_activate(n__first(X1,X2)) , Marked_activate(X2)>
  <Marked_activate(n__s(X)) , Marked_activate(X)> 
     --> <Marked_activate(n__first(X1,X2)) , Marked_activate(X1)>
  <Marked_activate(n__s(X)) , Marked_activate(X)> 
     --> <Marked_activate(n__first(X1,X2)) , Marked_first(activate(X1),
                                              activate(X2))>
  <Marked_activate(n__s(X)) , Marked_activate(X)> 
     --> <Marked_activate(n__s(X)) , Marked_activate(X)>
  <Marked_activate(n__s(X)) , Marked_activate(X)> 
     --> <Marked_activate(n__terms(X)) , Marked_activate(X)>
  <Marked_activate(n__terms(X)) , Marked_activate(X)> 
     --> <Marked_activate(n__first(X1,X2)) , Marked_activate(X2)>
  <Marked_activate(n__terms(X)) , Marked_activate(X)> 
     --> <Marked_activate(n__first(X1,X2)) , Marked_activate(X1)>
  <Marked_activate(n__terms(X)) , Marked_activate(X)> 
     --> <Marked_activate(n__first(X1,X2)) , Marked_first(activate(X1),
                                              activate(X2))>
  <Marked_activate(n__terms(X)) , Marked_activate(X)> 
     --> <Marked_activate(n__s(X)) , Marked_activate(X)>
  <Marked_activate(n__terms(X)) , Marked_activate(X)> 
     --> <Marked_activate(n__terms(X)) , Marked_activate(X)>
  <Marked_first(s(X),cons(Y,Z)) , Marked_activate(Z)> 
     --> <Marked_activate(n__first(X1,X2)) , Marked_activate(X2)>
  <Marked_first(s(X),cons(Y,Z)) , Marked_activate(Z)> 
     --> <Marked_activate(n__first(X1,X2)) , Marked_activate(X1)>
  <Marked_first(s(X),cons(Y,Z)) , Marked_activate(Z)> 
     --> <Marked_activate(n__first(X1,X2)) , Marked_first(activate(X1),
                                              activate(X2))>
  <Marked_first(s(X),cons(Y,Z)) , Marked_activate(Z)> 
     --> <Marked_activate(n__s(X)) , Marked_activate(X)>
  <Marked_first(s(X),cons(Y,Z)) , Marked_activate(Z)> 
     --> <Marked_activate(n__terms(X)) , Marked_activate(X)>
}

{  <Marked_sqr(s(X)) , Marked_sqr(X)> --> <Marked_sqr(s(X)) , Marked_sqr(X)>
}

{  <Marked_add(s(X),Y) , Marked_add(X,Y)> 
      --> <Marked_add(s(X),Y) , Marked_add(X,Y)>
}

{  <Marked_dbl(s(X)) , Marked_dbl(X)> --> <Marked_dbl(s(X)) , Marked_dbl(X)>
}
APPLY CRITERIA (Subterm criterion)
APPLY CRITERIA (Choosing graph)
Trying to solve the following constraints:
{
  sqr(0) >= 0 ;
  sqr(s(X)) >= s(add(sqr(X),dbl(X))) ;
  terms(N) >= cons(recip(sqr(N)),n__terms(n__s(N))) ;
  terms(X) >= n__terms(X) ;
  s(X) >= n__s(X) ;
  add(0,X) >= X ;
  add(s(X),Y) >= s(add(X,Y)) ;
  dbl(0) >= 0 ;
  dbl(s(X)) >= s(s(dbl(X))) ;
  first(0,X) >= nil ;
  first(s(X),cons(Y,Z)) >= cons(Y,n__first(X,activate(Z))) ;
  first(X1,X2) >= n__first(X1,X2) ;
  activate(n__terms(X)) >= terms(activate(X)) ;
  activate(n__s(X)) >= s(activate(X)) ;
  activate(n__first(X1,X2)) >= first(activate(X1),activate(X2)) ;
  activate(X) >= X ;
  Marked_activate(n__terms(X)) >= Marked_activate(X) ;
  Marked_activate(n__s(X)) >= Marked_activate(X) ;
  Marked_activate(n__first(X1,X2)) >= Marked_activate(X1) ;
  Marked_activate(n__first(X1,X2)) >= Marked_activate(X2) ;
  Marked_activate(n__first(X1,X2)) >= Marked_first(activate(X1),activate(X2)) ;
  Marked_first(s(X),cons(Y,Z)) >= Marked_activate(Z) ;
 }
 + Disjunctions:{ 
{
  Marked_activate(n__terms(X)) > Marked_activate(X) ;
 }
{
  Marked_activate(n__s(X)) > Marked_activate(X) ;
 }
{
  Marked_activate(n__first(X1,X2)) > Marked_activate(X1) ;
 }
{
  Marked_activate(n__first(X1,X2)) > Marked_activate(X2) ;
 }
{
  Marked_activate(n__first(X1,X2)) > Marked_first(activate(X1),activate(X2)) ;
 }
{
  Marked_first(s(X),cons(Y,Z)) > Marked_activate(Z) ;
 }
 } 
=== 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: sqr(0) >= 0
constraint: sqr(s(X)) >= s(add(sqr(X),dbl(X)))
constraint: terms(N) >= cons(recip(sqr(N)),n__terms(n__s(N)))
constraint: terms(X) >= n__terms(X)
constraint: s(X) >= n__s(X)
constraint: add(0,X) >= X
constraint: add(s(X),Y) >= s(add(X,Y))
constraint: dbl(0) >= 0
constraint: dbl(s(X)) >= s(s(dbl(X)))
constraint: first(0,X) >= nil
constraint: first(s(X),cons(Y,Z)) >= cons(Y,n__first(X,activate(Z)))
constraint: first(X1,X2) >= n__first(X1,X2)
constraint: activate(n__terms(X)) >= terms(activate(X))
constraint: activate(n__s(X)) >= s(activate(X))
constraint: activate(n__first(X1,X2)) >= first(activate(X1),activate(X2))
constraint: activate(X) >= X
constraint: Marked_activate(n__terms(X)) >= Marked_activate(X)
constraint: Marked_activate(n__s(X)) >= Marked_activate(X)
constraint: Marked_activate(n__first(X1,X2)) >= Marked_activate(X1)
constraint: Marked_activate(n__first(X1,X2)) >= Marked_activate(X2)
constraint: Marked_activate(n__first(X1,X2)) >= Marked_first(activate(X1),
                                                 activate(X2))
constraint: Marked_first(s(X),cons(Y,Z)) >= Marked_activate(Z)
APPLY CRITERIA (Subterm criterion)

ST: Marked_sqr -> 1

APPLY CRITERIA (Subterm criterion)

ST: Marked_add -> 1

APPLY CRITERIA (Subterm criterion)

ST: Marked_dbl -> 1

APPLY CRITERIA (Graph splitting)
Found 1 components:

{  <Marked_activate(n__s(X)) , Marked_activate(X)> 
      --> <Marked_activate(n__s(X)) , Marked_activate(X)>
  <Marked_activate(n__s(X)) , Marked_activate(X)> 
     --> <Marked_activate(n__terms(X)) , Marked_activate(X)>
  <Marked_activate(n__terms(X)) , Marked_activate(X)> 
     --> <Marked_activate(n__s(X)) , Marked_activate(X)>
  <Marked_activate(n__terms(X)) , Marked_activate(X)> 
     --> <Marked_activate(n__terms(X)) , Marked_activate(X)>
}
APPLY CRITERIA (Subterm criterion)

ST: Marked_activate -> 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:
SOLVED
{
 
TRS termination of:
[1] terms(N) -> cons(recip(sqr(N)),n__terms(n__s(N)))
[2] sqr(0) -> 0
[3] sqr(s(X)) -> s(add(sqr(X),dbl(X)))
[4] dbl(0) -> 0
[5] dbl(s(X)) -> s(s(dbl(X)))
[6] add(0,X) -> X
[7] add(s(X),Y) -> s(add(X,Y))
[8] first(0,X) -> nil
[9] first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
[10] terms(X) -> n__terms(X)
[11] s(X) -> n__s(X)
[12] first(X1,X2) -> n__first(X1,X2)
[13] activate(n__terms(X)) -> terms(activate(X))
[14] activate(n__s(X)) -> s(activate(X))
[15] activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
[16] activate(X) -> X
 ,
 CRITERION: MDP
 [ {
      
     DP termination of:
     <Marked_terms(N) , Marked_sqr(N)>
<Marked_sqr(s(X)) , Marked_s(add(sqr(X),dbl(X)))>
<Marked_sqr(s(X)) , Marked_add(sqr(X),dbl(X))>
<Marked_sqr(s(X)) , Marked_sqr(X)>
<Marked_sqr(s(X)) , Marked_dbl(X)>
<Marked_dbl(s(X)) , Marked_s(s(dbl(X)))>
<Marked_dbl(s(X)) , Marked_s(dbl(X))>
<Marked_dbl(s(X)) , Marked_dbl(X)>
<Marked_add(s(X),Y) , Marked_s(add(X,Y))>
<Marked_add(s(X),Y) , Marked_add(X,Y)>
<Marked_first(s(X),cons(Y,Z)) , Marked_activate(Z)>
<Marked_activate(n__terms(X)) , Marked_terms(activate(X))>
<Marked_activate(n__terms(X)) , Marked_activate(X)>
<Marked_activate(n__s(X)) , Marked_s(activate(X))>
<Marked_activate(n__s(X)) , Marked_activate(X)>
<Marked_activate(n__first(X1,X2)) , Marked_first(activate(X1),activate(X2))>
<Marked_activate(n__first(X1,X2)) , Marked_activate(X1)>
<Marked_activate(n__first(X1,X2)) , Marked_activate(X2)>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_first(s(X),cons(Y,Z)) , Marked_activate(Z)>
<Marked_activate(n__terms(X)) , Marked_activate(X)>
<Marked_activate(n__s(X)) , Marked_activate(X)>
<Marked_activate(n__first(X1,X2)) , Marked_first(activate(X1),activate(X2))>
<Marked_activate(n__first(X1,X2)) , Marked_activate(X1)>
<Marked_activate(n__first(X1,X2)) , Marked_activate(X2)>
 ,
 CRITERION: CG using polynomial interpretation = 
 [ cons ] (X0,X1) = 1*X1; 
 [ add ] (X0,X1) = 2*X1; 
 [ n__s ] (X0) = 1*X0; 
 [ n__first ] (X0,X1) = 1*X1 + 2*X0 + 2; 
 [ sqr ] (X0) = 0; 
 [ nil ] () = 2; 
 [ 0 ] () = 0; 
 [ Marked_activate ] (X0) = 2*X0 + 1; 
 [ recip ] (X0) = 0; 
 [ dbl ] (X0) = 0; 
 [ terms ] (X0) = 2*X0; 
 [ activate ] (X0) = 1*X0; 
 [ n__terms ] (X0) = 2*X0; 
 [ first ] (X0,X1) = 1*X1 + 2*X0 + 2; 
 [ s ] (X0) = 1*X0; 
 [ Marked_first ] (X0,X1) = 2*X1 + 2; 
 removing <Marked_first(s(X),cons(Y,Z)),Marked_activate(Z)><Marked_activate(
                                                             n__first(
                                                              X1,X2)),
Marked_first(activate(X1),activate(X2))><Marked_activate(n__first(X1,X2)),
Marked_activate(X1)><Marked_activate(n__first(X1,X2)),Marked_activate(X2)>
 [ {
      
     DP termination of:
     <Marked_activate(n__terms(X)) , Marked_activate(X)>
<Marked_activate(n__s(X)) , Marked_activate(X)>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_activate(n__terms(X)) , Marked_activate(X)>
<Marked_activate(n__s(X)) , Marked_activate(X)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
 ]} 
 ]} 
{
 
DP termination of:
<Marked_sqr(s(X)) , Marked_sqr(X)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
{
 
DP termination of:
<Marked_add(s(X),Y) , Marked_add(X,Y)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
{
 
DP termination of:
<Marked_dbl(s(X)) , Marked_dbl(X)>
 ,
 CRITERION: ST
 [ {
      
     DP termination of:
      ,
      CRITERION: SG
      [  ]} 
      ]} 
 ]} 
 ]} 

Cime worked for 0.108923 seconds (real time)
Cime Exit Status: 0