Term Rewriting System R:

   [x, y, z]
   +(x, 0) -> x
   +(x, i(x)) -> 0
   +(+(x, y), z) -> +(x, +(y, z))
   *(x, +(y, z)) -> +(*(x, y), *(x, z))
   *(+(x, y), z) -> +(*(x, z), *(y, z))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   +'(+(x, y), z) -> +'(x, +(y, z))
   +'(+(x, y), z) -> +'(y, z)
   *'(+(x, y), z) -> +'(*(x, z), *(y, z))
   *'(+(x, y), z) -> *'(x, z)
   *'(+(x, y), z) -> *'(y, z)
   *'(x, +(y, z)) -> +'(*(x, y), *(x, z))
   *'(x, +(y, z)) -> *'(x, y)
   *'(x, +(y, z)) -> *'(x, z)

Furthermore, R contains two SCCs.


SCC1:

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

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   *(+(x, y), z) -> +(*(x, z), *(y, z))
   *(x, +(y, z)) -> +(*(x, y), *(x, z))
   +(x, i(x)) -> 0
   +(x, 0) -> x


 This transformation is resulting in one new subcycle:


SCC1.MRR1:

+'(+(x, y), z) -> +'(x, +(y, z))
+'(+(x, y), z) -> +'(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(+(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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

No rules of R can be deleted.


 This transformation is resulting in one new subcycle:


SCC1.MRR1.MRR1:

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

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
POL(+'(x_1, x_2)) = x_1

 resulting in no subcycles.


SCC2:

*'(x, +(y, z)) -> *'(x, z)
*'(x, +(y, z)) -> *'(x, y)
*'(+(x, y), z) -> *'(y, z)
*'(+(x, y), z) -> *'(x, 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(+(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   *'(x, +(y, z)) -> *'(x, z)
   *'(x, +(y, z)) -> *'(x, y)
   *'(+(x, y), z) -> *'(y, z)
   *'(+(x, y), z) -> *'(x, z)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.610 seconds.