Term Rewriting System R:

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

Termination of R to be shown.


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


R contains the following Dependency Pairs: 


   QUOT(x, 0, s(z)) -> QUOT(x, plus(z, s(0)), s(z))
   QUOT(x, 0, s(z)) -> PLUS(z, s(0))
   QUOT(s(x), s(y), z) -> QUOT(x, y, z)
   PLUS(s(x), y) -> PLUS(x, y)

Furthermore, R contains two 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:

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

 resulting in one subcycle.


SCC2.Polo1:

QUOT(x, 0, s(z)) -> QUOT(x, plus(z, s(0)), s(z))

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

   QUOT(x, 0, s(z)) -> QUOT(x, plus(z, s(0)), s(z))
two new Dependency Pairs
are created:


   QUOT(x, 0, s(s(x''))) -> QUOT(x, s(plus(x'', s(0))), s(s(x'')))
   QUOT(x, 0, s(0)) -> QUOT(x, s(0), s(0))

The transformation is resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.627 seconds.