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

TRS termination of:
[1] terms(N) -> cons(recip(sqr(N)),n__terms(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] first(X1,X2) -> n__first(X1,X2)
[12] activate(n__terms(X)) -> terms(X)
[13] activate(n__first(X1,X2)) -> first(X1,X2)
[14] activate(X) -> X


Sub problem: 
guided: 

DP termination of:
<Marked_terms(N) , Marked_sqr(N)>
<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_dbl(X)>
<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(X)>
<Marked_activate(n__first(X1,X2)) , Marked_first(X1,X2)>
 
END GUIDED
APPLY CRITERIA (Graph splitting)
Found 4 components:

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

{  <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)

ST: Marked_activate -> 1
Marked_first -> 2

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 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(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] first(X1,X2) -> n__first(X1,X2)
[12] activate(n__terms(X)) -> terms(X)
[13] activate(n__first(X1,X2)) -> first(X1,X2)
[14] activate(X) -> X
 ,
 CRITERION: MDP
 [ {
      
     DP termination of:
     <Marked_terms(N) , Marked_sqr(N)>
<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_dbl(X)>
<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(X)>
<Marked_activate(n__first(X1,X2)) , Marked_first(X1,X2)>
 ,
 CRITERION: SG
 [ {
      
     DP termination of:
     <Marked_first(s(X),cons(Y,Z)) , Marked_activate(Z)>
<Marked_activate(n__first(X1,X2)) , Marked_first(X1,X2)>
 ,
 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 0.013056 seconds (real time)
Cime Exit Status: 0