Term Rewriting System R:

   [y, x, z]
   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   quot(0, s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   DIV(x, y) -> QUOT(x, y, y)
   QUOT(x, 0, s(z)) -> DIV(x, s(z))
   QUOT(s(x), s(y), z) -> QUOT(x, y, z)

Furthermore, R contains one SCC.


SCC1:

QUOT(s(x), s(y), z) -> QUOT(x, y, z)
QUOT(x, 0, s(z)) -> DIV(x, s(z))
DIV(x, y) -> QUOT(x, y, 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(s(x_1)) = 1 + x_1
POL(DIV(x_1, x_2)) = x_1
POL(QUOT(x_1, x_2, x_3)) = x_1
POL(0) = 0

 resulting in one subcycle.


SCC1.Polo1:

DIV(x, y) -> QUOT(x, y, y)
QUOT(x, 0, s(z)) -> DIV(x, s(z))

On this Scc, a Instantiation SCC transformation can be performed.
 
As a result of transforming the rule 

   DIV(x, y) -> QUOT(x, y, y)
one new Dependency Pair
is created:


   DIV(x'', s(z'')) -> QUOT(x'', s(z''), s(z''))

The transformation is resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.546 seconds.