Term Rewriting System R:

   [y, x]
   ack(0, y) -> s(y)
   ack(s(x), 0) -> ack(x, s(0))
   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
   ack(s(x), y) -> f(x, x)
   f(s(x), y) -> f(x, s(x))
   f(x, s(y)) -> f(y, x)
   f(x, y) -> ack(x, y)

Termination of R to be shown.


R contains the following Dependency Pairs: 


   ACK(s(x), s(y)) -> ACK(x, ack(s(x), y))
   ACK(s(x), s(y)) -> ACK(s(x), y)
   ACK(s(x), y) -> F(x, x)
   ACK(s(x), 0) -> ACK(x, s(0))
   F(x, s(y)) -> F(y, x)
   F(x, y) -> ACK(x, y)
   F(s(x), y) -> F(x, s(x))

Furthermore, R contains one SCC.


SCC1:

F(s(x), y) -> F(x, s(x))
ACK(s(x), 0) -> ACK(x, s(0))
F(x, y) -> ACK(x, y)
F(x, s(y)) -> F(y, x)
ACK(s(x), y) -> F(x, x)
ACK(s(x), s(y)) -> ACK(s(x), y)
ACK(s(x), s(y)) -> ACK(x, ack(s(x), y))

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(s(x_1)) = 1 + 2*x_1
POL(ack(x_1, x_2)) = 0
POL(ACK(x_1, x_2)) = 2*x_1
POL(F(x_1, x_2)) = 2*x_1 + x_2
POL(f(x_1, x_2)) = 0
POL(0) = 0

 resulting in one subcycle.


SCC1.Polo1:

ACK(s(x), s(y)) -> ACK(s(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(ACK(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   ACK(s(x), s(y)) -> ACK(s(x), y)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 11.532 seconds.