Term Rewriting System R:

   [N, X, Y, Z, X1, X2]
   terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
   terms(X) -> n__terms(X)
   sqr(0) -> 0
   sqr(s(X)) -> s(add(sqr(X), dbl(X)))
   dbl(0) -> 0
   dbl(s(X)) -> s(s(dbl(X)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   first(0, X) -> nil
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   first(X1, X2) -> n__first(X1, X2)
   s(X) -> n__s(X)
   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

Termination of R to be shown.


R contains the following Dependency Pairs: 


   ACTIVATE(n__terms(X)) -> TERMS(activate(X))
   ACTIVATE(n__terms(X)) -> ACTIVATE(X)
   ACTIVATE(n__s(X)) -> S(activate(X))
   ACTIVATE(n__s(X)) -> ACTIVATE(X)
   ACTIVATE(n__first(X1, X2)) -> FIRST(activate(X1), activate(X2))
   ACTIVATE(n__first(X1, X2)) -> ACTIVATE(X1)
   ACTIVATE(n__first(X1, X2)) -> ACTIVATE(X2)
   TERMS(N) -> SQR(N)
   ADD(s(X), Y) -> S(add(X, Y))
   ADD(s(X), Y) -> ADD(X, Y)
   SQR(s(X)) -> S(add(sqr(X), dbl(X)))
   SQR(s(X)) -> ADD(sqr(X), dbl(X))
   SQR(s(X)) -> SQR(X)
   SQR(s(X)) -> DBL(X)
   DBL(s(X)) -> S(s(dbl(X)))
   DBL(s(X)) -> S(dbl(X))
   DBL(s(X)) -> DBL(X)
   FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

Furthermore, R contains four SCCs.


SCC1:

ADD(s(X), Y) -> ADD(X, Y)

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(ADD(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   ADD(s(X), Y) -> ADD(X, Y)



 This transformation is resulting in no new subcycles.


SCC2:

DBL(s(X)) -> DBL(X)

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(DBL(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   DBL(s(X)) -> DBL(X)



 This transformation is resulting in no new subcycles.


SCC3:

SQR(s(X)) -> SQR(X)

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(SQR(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   SQR(s(X)) -> SQR(X)



 This transformation is resulting in no new subcycles.


SCC4:

ACTIVATE(n__first(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n__first(X1, X2)) -> ACTIVATE(X1)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n__first(X1, X2)) -> FIRST(activate(X1), activate(X2))
ACTIVATE(n__s(X)) -> ACTIVATE(X)
ACTIVATE(n__terms(X)) -> ACTIVATE(X)

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rules can be oriented: 

   first(0, X) -> nil
   first(X1, X2) -> n__first(X1, X2)
   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
   activate(n__terms(X)) -> terms(activate(X))
   activate(X) -> X
   activate(n__s(X)) -> s(activate(X))
   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
   terms(X) -> n__terms(X)
   terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N)))
   s(X) -> n__s(X)


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(n__first(x_1, x_2)) = 1 + x_1 + x_2
POL(nil) = 0
POL(add(x_1, x_2)) = 0
POL(dbl(x_1)) = 0
POL(activate(x_1)) = x_1
POL(s(x_1)) = x_1
POL(n__terms(x_1)) = x_1
POL(FIRST(x_1, x_2)) = x_2
POL(0) = 0
POL(cons(x_1, x_2)) = x_2
POL(first(x_1, x_2)) = 1 + x_1 + x_2
POL(terms(x_1)) = x_1
POL(ACTIVATE(x_1)) = x_1
POL(sqr(x_1)) = 0
POL(recip(x_1)) = 0
POL(n__s(x_1)) = x_1

 resulting in one subcycle.


SCC4.Polo1:

ACTIVATE(n__s(X)) -> ACTIVATE(X)
ACTIVATE(n__terms(X)) -> ACTIVATE(X)

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(n__terms(x_1)) = 1 + x_1
POL(ACTIVATE(x_1)) = 1 + x_1
POL(n__s(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   ACTIVATE(n__s(X)) -> ACTIVATE(X)
   ACTIVATE(n__terms(X)) -> ACTIVATE(X)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 1.270 seconds.