Term Rewriting System R:

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

Termination of R to be shown.


R contains the following Dependency Pairs: 


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

Furthermore, R contains five 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:

HALF(s(s(X))) -> HALF(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(HALF(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   HALF(s(s(X))) -> HALF(X)



 This transformation is resulting in no new subcycles.


SCC3:

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.


SCC4:

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.


SCC5:

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

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__first(x_1, x_2)) = x_1 + x_2
POL(s(x_1)) = x_1
POL(ACTIVATE(x_1)) = x_1
POL(FIRST(x_1, x_2)) = x_1 + x_2
POL(cons(x_1, x_2)) = x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.908 seconds.