Term Rewriting System R:

   [X, Y]
   ackin(0, X) -> ackout(s(X))
   ackin(s(X), 0) -> u11(ackin(X, s(0)))
   ackin(s(X), s(Y)) -> u21(ackin(s(X), Y), X)
   u11(ackout(X)) -> ackout(X)
   u21(ackout(X), Y) -> u22(ackin(Y, X))
   u22(ackout(X)) -> ackout(X)

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: 


   ACKIN(s(X), 0) -> U11(ackin(X, s(0)))
   ACKIN(s(X), 0) -> ACKIN(X, s(0))
   ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)
   ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
   U21(ackout(X), Y) -> U22(ackin(Y, X))
   U21(ackout(X), Y) -> ACKIN(Y, X)

Furthermore, R contains one SCC.


SCC1:

ACKIN(s(X), s(Y)) -> ACKIN(s(X), Y)
U21(ackout(X), Y) -> ACKIN(Y, X)
ACKIN(s(X), s(Y)) -> U21(ackin(s(X), Y), X)
ACKIN(s(X), 0) -> ACKIN(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(ACKIN(x_1, x_2)) = x_1
POL(s(x_1)) = 1 + x_1
POL(U21(x_1, x_2)) = 1 + x_2
POL(u11(x_1)) = 0
POL(u22(x_1)) = 0
POL(ackout(x_1)) = 0
POL(u21(x_1, x_2)) = 0
POL(0) = 0
POL(ackin(x_1, x_2)) = 0

 resulting in one subcycle.


SCC1.Polo1:

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

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.771 seconds.