Term Rewriting System R:
[X, XS]
zeros -> cons(0, zeros)
tail(cons(X, XS)) -> XS

Termination of R to be shown.



   R
Removing Redundant Rules



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

tail(cons(X, XS)) -> XS

where the Polynomial interpretation:
  POL(0)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(tail(x1))=  1 + x1  
  POL(zeros)=  0  
was used.

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


   R
RRRPolo
       →TRS2
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
OC
           →TRS3
Dependency Pair Analysis



R contains the following Dependency Pairs:

ZEROS -> ZEROS

Furthermore, R contains one SCC.


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


Dependency Pair:

ZEROS -> ZEROS


Rule:


zeros -> cons(0, zeros)


Strategy:

innermost




As we are in the innermost case, we can delete all 1 non-usable-rules.


   R
RRRPolo
       →TRS2
OC
           →TRS3
DPs
             ...
               →DP Problem 2
Non Termination


Dependency Pair:

ZEROS -> ZEROS


Rule:

none


Strategy:

innermost




Found an infinite P-chain over R:
P =

ZEROS -> ZEROS

R = none

s = ZEROS
evaluates to t =ZEROS

Thus, s starts an infinite chain.

Non-Termination of R could be shown.
Duration:
0:05 minutes