Term Rewriting System R:

   [x, y, v, w, z]
   sort(nil) -> nil
   sort(cons(x, y)) -> insert(x, sort(y))
   insert(x, nil) -> cons(x, nil)
   insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
   choose(x, cons(v, w), y, 0) -> cons(x, cons(v, w))
   choose(x, cons(v, w), 0, s(z)) -> cons(v, insert(x, w))
   choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, 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: 


   CHOOSE(x, cons(v, w), s(y), s(z)) -> CHOOSE(x, cons(v, w), y, z)
   CHOOSE(x, cons(v, w), 0, s(z)) -> INSERT(x, w)
   INSERT(x, cons(v, w)) -> CHOOSE(x, cons(v, w), x, v)
   SORT(cons(x, y)) -> INSERT(x, sort(y))
   SORT(cons(x, y)) -> SORT(y)

Furthermore, R contains two SCCs.


SCC1:

INSERT(x, cons(v, w)) -> CHOOSE(x, cons(v, w), x, v)
CHOOSE(x, cons(v, w), 0, s(z)) -> INSERT(x, w)
CHOOSE(x, cons(v, w), s(y), s(z)) -> CHOOSE(x, cons(v, w), 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(CHOOSE(x_1, x_2, x_3, x_4)) = x_2
POL(s(x_1)) = 0
POL(INSERT(x_1, x_2)) = 1 + x_2
POL(0) = 0
POL(cons(x_1, x_2)) = 1 + x_2

 resulting in one subcycle.


SCC1.Polo1:

CHOOSE(x, cons(v, w), s(y), s(z)) -> CHOOSE(x, cons(v, w), y, 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(CHOOSE(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4
POL(s(x_1)) = 1 + x_1
POL(cons(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   CHOOSE(x, cons(v, w), s(y), s(z)) -> CHOOSE(x, cons(v, w), y, z)



 This transformation is resulting in no new subcycles.


SCC2:

SORT(cons(x, y)) -> SORT(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(SORT(x_1)) = 1 + x_1
POL(cons(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   SORT(cons(x, y)) -> SORT(y)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.682 seconds.