Term Rewriting System R:

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


   TIMES(s(x), y) -> PLUS(y, times(x, y))
   TIMES(s(x), y) -> TIMES(x, y)
   PLUS(s(x), y) -> PLUS(x, y)
   DIV(x, y) -> QUOT(x, y, y)
   DIV(div(x, y), z) -> DIV(x, times(y, z))
   DIV(div(x, y), z) -> TIMES(y, z)
   QUOT(x, 0, s(z)) -> DIV(x, s(z))
   QUOT(s(x), s(y), z) -> QUOT(x, y, z)

Furthermore, R contains three SCCs.


SCC1:

PLUS(s(x), y) -> PLUS(x, 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(PLUS(x_1, x_2)) = 1 + x_1 + x_2
POL(s(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   PLUS(s(x), y) -> PLUS(x, y)



 This transformation is resulting in no new subcycles.


SCC2:

TIMES(s(x), y) -> TIMES(x, 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(s(x_1)) = 1 + x_1
POL(TIMES(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   TIMES(s(x), y) -> TIMES(x, y)



 This transformation is resulting in no new subcycles.


SCC3:

QUOT(s(x), s(y), z) -> QUOT(x, y, z)
DIV(div(x, y), z) -> DIV(x, times(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(plus(x_1, x_2)) = 0
POL(DIV(x_1, x_2)) = x_1
POL(QUOT(x_1, x_2, x_3)) = x_1
POL(div(x_1, x_2)) = x_1
POL(times(x_1, x_2)) = 0
POL(0) = 0

 resulting in one subcycle.


SCC3.Polo1:

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

 resulting in one subcycle.


SCC3.Polo1.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: 0.776 seconds.