Term Rewriting System R:

   [x, y, z]
   minus(0) -> 0
   minus(minus(x)) -> x
   +(x, 0) -> x
   +(0, y) -> y
   +(minus(1), 1) -> 0
   +(x, minus(y)) -> minus(+(minus(x), y))
   +(x, +(y, z)) -> +(+(x, y), z)
   +(minus(+(x, 1)), 1) -> minus(x)

Termination of R to be shown.


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

   minus(0) -> 0
   minus(minus(x)) -> x
   +(minus(1), 1) -> 0

where the Polynomial interpretation:
POL(1) = 0
POL(minus(x_1)) = 1 + x_1
POL(+(x_1, x_2)) = x_1 + 2*x_2
POL(0) = 0
was used. 



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


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

   +(x, 0) -> x
   +(0, y) -> y

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



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


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

   +(x, +(y, z)) -> +(+(x, y), z)
   +(minus(+(x, 1)), 1) -> minus(x)

where the Polynomial interpretation:
POL(1) = 0
POL(minus(x_1)) = x_1
POL(+(x_1, x_2)) = 1 + x_1 + 2*x_2
was used. 



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


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


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

Furthermore, R contains one SCC.


SCC1:

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

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.512 seconds.