Term Rewriting System R:

   [x, y, z]
   bsort(nil) -> nil
   bsort(.(x, y)) -> last(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
   bubble(nil) -> nil
   bubble(.(x, nil)) -> .(x, nil)
   bubble(.(x, .(y, z))) -> if(<=(x, y), .(y, bubble(.(x, z))), .(x, bubble(.(y, z))))
   last(nil) -> 0
   last(.(x, nil)) -> x
   last(.(x, .(y, z))) -> last(.(y, z))
   butlast(nil) -> nil
   butlast(.(x, nil)) -> nil
   butlast(.(x, .(y, z))) -> .(x, butlast(.(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: 


   BUBBLE(.(x, .(y, z))) -> BUBBLE(.(x, z))
   BUBBLE(.(x, .(y, z))) -> BUBBLE(.(y, z))
   BSORT(.(x, y)) -> LAST(.(bubble(.(x, y)), bsort(butlast(bubble(.(x, y))))))
   BSORT(.(x, y)) -> BUBBLE(.(x, y))
   BSORT(.(x, y)) -> BSORT(butlast(bubble(.(x, y))))
   BSORT(.(x, y)) -> BUTLAST(bubble(.(x, y)))
   BUTLAST(.(x, .(y, z))) -> BUTLAST(.(y, z))
   LAST(.(x, .(y, z))) -> LAST(.(y, z))

Furthermore, R contains four SCCs.


SCC1:

BUBBLE(.(x, .(y, z))) -> BUBBLE(.(y, z))
BUBBLE(.(x, .(y, z))) -> BUBBLE(.(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(BUBBLE(x_1)) = 1 + x_1
POL(.(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC2:

LAST(.(x, .(y, z))) -> LAST(.(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(LAST(x_1)) = 1 + x_1
POL(.(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC3:

BUTLAST(.(x, .(y, z))) -> BUTLAST(.(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(BUTLAST(x_1)) = 1 + x_1
POL(.(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC4:

BSORT(.(x, y)) -> BSORT(butlast(bubble(.(x, y))))

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

   BSORT(.(x, y)) -> BSORT(butlast(bubble(.(x, y))))
two new Dependency Pairs
are created:


   BSORT(.(x'', .(y'', z'))) -> BSORT(butlast(if(<=(x'', y''), .(y'', bubble(.(x'', z'))), .(x'', bubble(.(y'', z'))))))
   BSORT(.(x'', nil)) -> BSORT(butlast(.(x'', nil)))

The transformation is resulting in one subcycle:


SCC4.Nar1:

BSORT(.(x'', nil)) -> BSORT(butlast(.(x'', nil)))
BSORT(.(x'', .(y'', z'))) -> BSORT(butlast(if(<=(x'', y''), .(y'', bubble(.(x'', z'))), .(x'', bubble(.(y'', z'))))))

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

   BSORT(.(x'', nil)) -> BSORT(butlast(.(x'', nil)))
one new Dependency Pair
is created:


   BSORT(.(x'', nil)) -> BSORT(nil)

The transformation is resulting in one subcycle:


SCC4.Nar1.Rewr1:

BSORT(.(x'', .(y'', z'))) -> BSORT(butlast(if(<=(x'', y''), .(y'', bubble(.(x'', z'))), .(x'', bubble(.(y'', z'))))))

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rules can be oriented: 

   butlast(.(x, .(y, z))) -> .(x, butlast(.(y, z)))
   butlast(nil) -> nil
   butlast(.(x, nil)) -> nil


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(bubble(x_1)) = 0
POL(nil) = 0
POL(<=(x_1, x_2)) = 0
POL(BSORT(x_1)) = 1 + x_1
POL(butlast(x_1)) = x_1
POL(if(x_1, x_2, x_3)) = 0
POL(.(x_1, x_2)) = 1

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 11.201 seconds.