Term Rewriting System R:

   [x, y, z]
   :(x, x) -> e
   :(x, e) -> x
   :(:(x, y), z) -> :(x, :(z, i(y)))
   :(e, x) -> i(x)
   :(x, :(y, i(x))) -> i(y)
   :(x, :(y, :(i(x), z))) -> :(i(z), y)
   :(i(x), :(y, x)) -> i(y)
   :(i(x), :(y, :(x, z))) -> :(i(z), y)
   i(:(x, y)) -> :(y, x)
   i(i(x)) -> x
   i(e) -> e

Termination of R to be shown.


Removing the following rules from R which fullfill a polynomial ordering: 

   :(x, x) -> e
   :(x, e) -> x
   :(e, x) -> i(x)
   :(x, :(y, i(x))) -> i(y)
   :(x, :(y, :(i(x), z))) -> :(i(z), y)
   :(i(x), :(y, x)) -> i(y)
   :(i(x), :(y, :(x, z))) -> :(i(z), y)

where the Polynomial interpretation:
POL(e) = 0
POL(i(x_1)) = x_1
POL(:(x_1, x_2)) = 1 + x_1 + x_2
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


R contains the following Dependency Pairs: 


   I(:(x, y)) -> :'(y, x)
   :'(:(x, y), z) -> :'(x, :(z, i(y)))
   :'(:(x, y), z) -> :'(z, i(y))
   :'(:(x, y), z) -> I(y)

Furthermore, R contains one SCC.


SCC1:

:'(:(x, y), z) -> I(y)
:'(:(x, y), z) -> :'(z, i(y))
:'(:(x, y), z) -> :'(x, :(z, i(y)))
I(:(x, y)) -> :'(y, x)

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(e) = 0
POL(I(x_1)) = x_1
POL(i(x_1)) = x_1
POL(:(x_1, x_2)) = 1 + x_1 + x_2
POL(:'(x_1, x_2)) = x_1 + x_2

The following Dependency Pairs can be deleted:

   :'(:(x, y), z) -> I(y)
   :'(:(x, y), z) -> :'(z, i(y))
   I(:(x, y)) -> :'(y, x)

No rules of R can be deleted.


 This transformation is resulting in one new subcycle:


SCC1.MRR1:

:'(:(x, y), z) -> :'(x, :(z, i(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(e) = 0
POL(i(x_1)) = 0
POL(:(x_1, x_2)) = 1 + x_1
POL(:'(x_1, x_2)) = x_1

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.616 seconds.