Term Rewriting System R:

   [X, Z, N, Y]
   from(X) -> cons(X, n__from(s(X)))
   from(X) -> n__from(X)
   2ndspos(0, Z) -> rnil
   2ndspos(s(N), cons(X, Z)) -> 2ndspos(s(N), cons2(X, activate(Z)))
   2ndspos(s(N), cons2(X, cons(Y, Z))) -> rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
   2ndsneg(0, Z) -> rnil
   2ndsneg(s(N), cons(X, Z)) -> 2ndsneg(s(N), cons2(X, activate(Z)))
   2ndsneg(s(N), cons2(X, cons(Y, Z))) -> rcons(negrecip(Y), 2ndspos(N, activate(Z)))
   pi(X) -> 2ndspos(X, from(0))
   plus(0, Y) -> Y
   plus(s(X), Y) -> s(plus(X, Y))
   times(0, Y) -> 0
   times(s(X), Y) -> plus(Y, times(X, Y))
   square(X) -> times(X, X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

Termination of R to be shown.


R contains the following Dependency Pairs: 


   TIMES(s(X), Y) -> PLUS(Y, times(X, Y))
   TIMES(s(X), Y) -> TIMES(X, Y)
   PLUS(s(X), Y) -> PLUS(X, Y)
   ACTIVATE(n__from(X)) -> FROM(X)
   2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, activate(Z))
   2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> ACTIVATE(Z)
   2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, activate(Z)))
   2NDSPOS(s(N), cons(X, Z)) -> ACTIVATE(Z)
   2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
   2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> ACTIVATE(Z)
   2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
   2NDSNEG(s(N), cons(X, Z)) -> ACTIVATE(Z)
   SQUARE(X) -> TIMES(X, X)
   PI(X) -> 2NDSPOS(X, from(0))
   PI(X) -> FROM(0)

Furthermore, R contains three SCCs.


SCC1:

PLUS(s(X), Y) -> PLUS(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(PLUS(x_1, x_2)) = 1 + x_1 + x_2
POL(s(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC2:

2NDSNEG(s(N), cons(X, Z)) -> 2NDSNEG(s(N), cons2(X, activate(Z)))
2NDSPOS(s(N), cons(X, Z)) -> 2NDSPOS(s(N), cons2(X, activate(Z)))
2NDSNEG(s(N), cons2(X, cons(Y, Z))) -> 2NDSPOS(N, activate(Z))
2NDSPOS(s(N), cons2(X, cons(Y, Z))) -> 2NDSNEG(N, activate(Z))

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

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(n__from(x_1)) = 0
POL(2NDSNEG(x_1, x_2)) = x_1
POL(cons2(x_1, x_2)) = 0
POL(activate(x_1)) = 0
POL(s(x_1)) = 1 + x_1
POL(from(x_1)) = 0
POL(2NDSPOS(x_1, x_2)) = x_1
POL(cons(x_1, x_2)) = 0

 resulting in no subcycles.


SCC3:

TIMES(s(X), Y) -> TIMES(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(TIMES(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 1.102 seconds.