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. 



   R
     ↳Removing Redundant Rules



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

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

where the Polynomial interpretation:
  POL(0)=  0  
  POL(L(x1))=  x1  
  POL(N(x1, x2))=  1 + x1 + x2  
  POL(max(x1))=  x1  
  POL(s(x1))=  x1  
was used.

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


   R
     ↳RRRPolo
       →TRS2
         ↳Removing Redundant Rules



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(x1))=  x1  
  POL(N(x1, x2))=  x1 + x2  
  POL(max(x1))=  x1  
  POL(s(x1))=  1 + x1  
was used.

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


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳OC
             ...
               →TRS4
                 ↳Dependency Pair Analysis



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. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳OC
             ...
               →DP Problem 1
                 ↳Usable Rules (Innermost)


Dependency Pair:

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


Rules:


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


Strategy:

innermost




As we are in the innermost case, we can delete all 2 non-usable-rules.


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳OC
             ...
               →DP Problem 2
                 ↳Size-Change Principle


Dependency Pair:

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


Rule:

none


Strategy:

innermost




We number the DPs as follows:
  1. MAX(N(L(x), N(y, z))) -> MAX(N(y, z))
and get the following Size-Change Graph(s):
{1} , {1}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
L(x1) -> L(x1)
N(x1, x2) -> N(x1, x2)

We obtain no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes