Term Rewriting System R:

   [x, y, z]
   qsort(nil) -> nil
   qsort(.(x, y)) -> ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
   lowers(x, nil) -> nil
   lowers(x, .(y, z)) -> if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
   greaters(x, nil) -> nil
   greaters(x, .(y, z)) -> if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

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: 


   GREATERS(x, .(y, z)) -> GREATERS(x, z)
   QSORT(.(x, y)) -> QSORT(lowers(x, y))
   QSORT(.(x, y)) -> LOWERS(x, y)
   QSORT(.(x, y)) -> QSORT(greaters(x, y))
   QSORT(.(x, y)) -> GREATERS(x, y)
   LOWERS(x, .(y, z)) -> LOWERS(x, z)

Furthermore, R contains two SCCs.


SCC1:

GREATERS(x, .(y, z)) -> GREATERS(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(GREATERS(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:

   GREATERS(x, .(y, z)) -> GREATERS(x, z)



 This transformation is resulting in no new subcycles.


SCC2:

LOWERS(x, .(y, z)) -> LOWERS(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(LOWERS(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:

   LOWERS(x, .(y, z)) -> LOWERS(x, z)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.614 seconds.