Term Rewriting System R:

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

Termination of R to be shown.


R contains the following Dependency Pairs: 


   PLUS(s(s(x)), y) -> PLUS(x, s(y))
   PLUS(x, s(s(y))) -> PLUS(s(x), y)
   ACK(s(x), 0) -> ACK(x, s(0))
   ACK(s(x), s(y)) -> ACK(x, plus(y, ack(s(x), y)))
   ACK(s(x), s(y)) -> PLUS(y, ack(s(x), y))
   ACK(s(x), s(y)) -> ACK(s(x), y)

Furthermore, R contains two SCCs.


SCC1:

PLUS(x, s(s(y))) -> PLUS(s(x), y)
PLUS(s(s(x)), y) -> PLUS(x, s(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)) = x_1 + x_2
POL(s(x_1)) = x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   plus(s(0), y) -> s(y)
   plus(s(s(x)), y) -> s(plus(x, s(y)))
   plus(x, s(s(y))) -> s(plus(s(x), y))
   plus(0, y) -> y
   ack(s(x), 0) -> ack(x, s(0))
   ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))
   ack(0, y) -> s(y)


 This transformation is resulting in one new subcycle:


SCC1.MRR1:

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



 This transformation is resulting in no new subcycles.


SCC2:

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

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(plus(x_1, x_2)) = 0
POL(s(x_1)) = 1 + x_1
POL(ack(x_1, x_2)) = 0
POL(ACK(x_1, x_2)) = x_1
POL(0) = 0

 resulting in one subcycle.


SCC2.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: 0.819 seconds.