Term Rewriting System R:

   [N, X, Y]
   terms(N) -> cons(recip(sqr(N)))
   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)) -> cons(Y)
   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(X))) -> s(half(X))
   half(dbl(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)

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:

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.




Termination of R successfully shown.


Duration: 2.385 seconds.