Term Rewriting System R:
[X, XS, X1, X2]
azeros -> cons(0, zeros)
azeros -> zeros
atail(cons(X, XS)) -> mark(XS)
atail(X) -> tail(X)
mark(zeros) -> azeros
mark(tail(X)) -> atail(mark(X))
mark(cons(X1, X2)) -> cons(mark(X1), X2)
mark(0) -> 0

Innermost Termination of R to be shown.



   R
Removing Redundant Rules



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

azeros -> cons(0, zeros)
azeros -> zeros
mark(0) -> 0

where the Polynomial interpretation:
  POL(0)=  0  
  POL(a__zeros)=  1  
  POL(cons(x1, x2))=  x1 + x2  
  POL(tail(x1))=  1 + x1  
  POL(a__tail(x1))=  1 + x1  
  POL(zeros)=  0  
  POL(mark(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
Removing Redundant Rules



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

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

where the Polynomial interpretation:
  POL(cons(x1, x2))=  x1 + 2·x2  
  POL(a__zeros)=  0  
  POL(tail(x1))=  1 + x1  
  POL(a__tail(x1))=  2 + x1  
  POL(mark(x1))=  2·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
RRRPolo
           →TRS3
Removing Redundant Rules



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

mark(tail(X)) -> atail(mark(X))

where the Polynomial interpretation:
  POL(a__zeros)=  0  
  POL(cons(x1, x2))=  x1 + x2  
  POL(tail(x1))=  1 + x1  
  POL(a__tail(x1))=  x1  
  POL(mark(x1))=  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
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS4
Removing Redundant Rules



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

mark(zeros) -> azeros

where the Polynomial interpretation:
  POL(cons(x1, x2))=  x1 + x2  
  POL(a__zeros)=  0  
  POL(mark(x1))=  x1  
  POL(zeros)=  1  
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
RRRPolo
             ...
               →TRS5
Removing Redundant Rules



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

mark(cons(X1, X2)) -> cons(mark(X1), X2)

where the Polynomial interpretation:
  POL(cons(x1, x2))=  1 + x1 + x2  
  POL(mark(x1))=  2·x1  
was used.

All Rules of R can be deleted.


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS6
Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

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