Term Rewriting System R:

   [x, y, z]
   rev(nil) -> nil
   rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))
   rev1(x, nil) -> x
   rev1(x, ++(y, z)) -> rev1(y, z)
   rev2(x, nil) -> nil
   rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))

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: 


   REV2(x, ++(y, z)) -> REV(++(x, rev(rev2(y, z))))
   REV2(x, ++(y, z)) -> REV(rev2(y, z))
   REV2(x, ++(y, z)) -> REV2(y, z)
   REV(++(x, y)) -> REV1(x, y)
   REV(++(x, y)) -> REV2(x, y)
   REV1(x, ++(y, z)) -> REV1(y, z)

Furthermore, R contains two SCCs.


SCC1:

REV1(x, ++(y, z)) -> REV1(y, z)

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(++(x_1, x_2)) = 1 + x_1 + x_2
POL(REV1(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   REV1(x, ++(y, z)) -> REV1(y, z)



 This transformation is resulting in no new subcycles.


SCC2:

REV2(x, ++(y, z)) -> REV2(y, z)
REV2(x, ++(y, z)) -> REV(rev2(y, z))
REV(++(x, y)) -> REV2(x, y)
REV2(x, ++(y, z)) -> REV(++(x, rev(rev2(y, z))))

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

Additionally, the following rules can be oriented: 

   rev2(x, ++(y, z)) -> rev(++(x, rev(rev2(y, z))))
   rev2(x, nil) -> nil
   rev(nil) -> nil
   rev(++(x, y)) -> ++(rev1(x, y), rev2(x, y))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(nil) = 0
POL(++(x_1, x_2)) = 1 + x_2
POL(rev2(x_1, x_2)) = x_2
POL(REV(x_1)) = x_1
POL(rev1(x_1, x_2)) = 0
POL(rev(x_1)) = x_1
POL(REV2(x_1, x_2)) = x_2

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.729 seconds.