Term Rewriting System R:

   [x, y, z]
   msort(nil) -> nil
   msort(.(x, y)) -> .(min(x, y), msort(del(min(x, y), .(x, y))))
   min(x, nil) -> x
   min(x, .(y, z)) -> if(<=(x, y), min(x, z), min(y, z))
   del(x, nil) -> nil
   del(x, .(y, z)) -> if(=(x, y), z, .(y, del(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: 


   DEL(x, .(y, z)) -> DEL(x, z)
   MIN(x, .(y, z)) -> MIN(x, z)
   MIN(x, .(y, z)) -> MIN(y, z)
   MSORT(.(x, y)) -> MIN(x, y)
   MSORT(.(x, y)) -> MSORT(del(min(x, y), .(x, y)))
   MSORT(.(x, y)) -> DEL(min(x, y), .(x, y))

Furthermore, R contains three SCCs.


SCC1:

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

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



 This transformation is resulting in no new subcycles.


SCC2:

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

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



 This transformation is resulting in no new subcycles.


SCC3:

MSORT(.(x, y)) -> MSORT(del(min(x, y), .(x, y)))

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

   MSORT(.(x, y)) -> MSORT(del(min(x, y), .(x, y)))
one new Dependency Pair
is created:


   MSORT(.(x, y)) -> MSORT(if(=(min(x, y), x), y, .(x, del(min(x, y), y))))

The transformation is resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.628 seconds.