Term Rewriting System R:

   [x, y, z]
   max(L(x)) -> x
   max(N(L(0), L(y))) -> y
   max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))
   max(N(L(x), N(y, z))) -> max(N(L(x), L(max(N(y, z)))))

Termination of R to be shown.


Removing the following rules from R which fullfill a polynomial ordering: 

   max(N(L(0), L(y))) -> y

where the Polynomial interpretation:
POL(L(x_1)) = x_1
POL(s(x_1)) = x_1
POL(max(x_1)) = x_1
POL(0) = 0
POL(N(x_1, x_2)) = 1 + x_1 + x_2
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


Removing the following rules from R which fullfill a polynomial ordering: 

   max(N(L(s(x)), L(s(y)))) -> s(max(N(L(x), L(y))))

where the Polynomial interpretation:
POL(L(x_1)) = x_1
POL(s(x_1)) = 1 + x_1
POL(max(x_1)) = x_1
POL(N(x_1, x_2)) = x_1 + x_2
was used. 



Not all Rules of R can be deleted, so we still have to regard a part of R.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   MAX(N(L(x), N(y, z))) -> MAX(N(L(x), L(max(N(y, z)))))
   MAX(N(L(x), N(y, z))) -> MAX(N(y, z))

Furthermore, R contains one SCC.


SCC1:

MAX(N(L(x), N(y, z))) -> MAX(N(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(L(x_1)) = 1 + x_1
POL(MAX(x_1)) = 1 + x_1
POL(N(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   MAX(N(L(x), N(y, z))) -> MAX(N(y, z))



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.588 seconds.