Term Rewriting System R:

   [n, m]
   ack_in(0, n) -> ack_out(s(n))
   ack_in(s(m), 0) -> u11(ack_in(m, s(0)))
   ack_in(s(m), s(n)) -> u21(ack_in(s(m), n), m)
   u11(ack_out(n)) -> ack_out(n)
   u21(ack_out(n), m) -> u22(ack_in(m, n))
   u22(ack_out(n)) -> ack_out(n)

Termination of R to be shown.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   U21(ack_out(n), m) -> U22(ack_in(m, n))
   U21(ack_out(n), m) -> ACK_IN(m, n)
   ACK_IN(s(m), s(n)) -> U21(ack_in(s(m), n), m)
   ACK_IN(s(m), s(n)) -> ACK_IN(s(m), n)
   ACK_IN(s(m), 0) -> U11(ack_in(m, s(0)))
   ACK_IN(s(m), 0) -> ACK_IN(m, s(0))

Furthermore, R contains one SCC.


SCC1:

ACK_IN(s(m), 0) -> ACK_IN(m, s(0))
ACK_IN(s(m), s(n)) -> ACK_IN(s(m), n)
ACK_IN(s(m), s(n)) -> U21(ack_in(s(m), n), m)
U21(ack_out(n), m) -> ACK_IN(m, n)

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 + x_1
POL(U21(x_1, x_2)) = x_2
POL(u11(x_1)) = 0
POL(ack_in(x_1, x_2)) = 0
POL(u22(x_1)) = 0
POL(u21(x_1, x_2)) = 0
POL(ACK_IN(x_1, x_2)) = x_1
POL(0) = 0
POL(ack_out(x_1)) = 0

 resulting in one subcycle.


SCC1.Polo1:

ACK_IN(s(m), s(n)) -> ACK_IN(s(m), n)

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_IN(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   ACK_IN(s(m), s(n)) -> ACK_IN(s(m), n)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.765 seconds.