Term Rewriting System R:

   [x, y, z]
   purge(nil) -> nil
   purge(.(x, y)) -> .(x, purge(remove(x, y)))
   remove(x, nil) -> nil
   remove(x, .(y, z)) -> if(=(x, y), remove(x, z), .(y, remove(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: 


   REMOVE(x, .(y, z)) -> REMOVE(x, z)
   PURGE(.(x, y)) -> PURGE(remove(x, y))
   PURGE(.(x, y)) -> REMOVE(x, y)

Furthermore, R contains one SCC.


SCC1:

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

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



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.500 seconds.