Term Rewriting System R:

   [x, y, z]
   *(x, 1) -> x
   *(1, y) -> y
   *(i(x), x) -> 1
   *(x, i(x)) -> 1
   *(x, *(y, z)) -> *(*(x, y), z)
   *(*(x, y), i(y)) -> x
   *(*(x, i(y)), y) -> x
   *(k(x, y), k(y, x)) -> 1
   *(*(i(x), k(y, z)), x) -> k(*(*(i(x), y), x), *(*(i(x), z), x))
   i(1) -> 1
   i(i(x)) -> x
   i(*(x, y)) -> *(i(y), i(x))
   k(x, 1) -> 1
   k(x, x) -> 1
   k(*(x, i(y)), *(y, i(x))) -> 1

Termination of R to be shown.


R contains the following Dependency Pairs: 


   *'(x, *(y, z)) -> *'(*(x, y), z)
   *'(x, *(y, z)) -> *'(x, y)
   *'(*(i(x), k(y, z)), x) -> K(*(*(i(x), y), x), *(*(i(x), z), x))
   *'(*(i(x), k(y, z)), x) -> *'(*(i(x), y), x)
   *'(*(i(x), k(y, z)), x) -> *'(i(x), y)
   *'(*(i(x), k(y, z)), x) -> *'(*(i(x), z), x)
   *'(*(i(x), k(y, z)), x) -> *'(i(x), z)
   I(*(x, y)) -> *'(i(y), i(x))
   I(*(x, y)) -> I(y)
   I(*(x, y)) -> I(x)

Furthermore, R contains two SCCs.


SCC1:

*'(*(i(x), k(y, z)), x) -> *'(i(x), z)
*'(*(i(x), k(y, z)), x) -> *'(*(i(x), z), x)
*'(*(i(x), k(y, z)), x) -> *'(i(x), y)
*'(*(i(x), k(y, z)), x) -> *'(*(i(x), y), x)
*'(x, *(y, z)) -> *'(x, y)
*'(x, *(y, z)) -> *'(*(x, 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: 

   *(*(x, i(y)), y) -> x
   *(*(x, y), i(y)) -> x
   *(x, *(y, z)) -> *(*(x, y), z)
   *(x, 1) -> x
   *(x, i(x)) -> 1
   *(i(x), x) -> 1
   *(*(i(x), k(y, z)), x) -> k(*(*(i(x), y), x), *(*(i(x), z), x))
   *(k(x, y), k(y, x)) -> 1
   *(1, y) -> y
   i(i(x)) -> x
   i(1) -> 1
   i(*(x, y)) -> *(i(y), i(x))
   k(*(x, i(y)), *(y, i(x))) -> 1
   k(x, x) -> 1
   k(x, 1) -> 1


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(*(x_1, x_2)) = x_1 + x_1*x_2 + x_2
POL(1) = 0
POL(i(x_1)) = x_1
POL(k(x_1, x_2)) = 1 + x_1 + x_2
POL(*'(x_1, x_2)) = x_1 + x_1*x_2 + x_2

 resulting in one subcycle.


SCC1.Polo1:

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

 resulting in no subcycles.


SCC2:

I(*(x, y)) -> I(x)
I(*(x, y)) -> I(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(*(x_1, x_2)) = 1 + x_1 + x_2
POL(I(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   I(*(x, y)) -> I(x)
   I(*(x, y)) -> I(y)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 11.688 seconds.