Term Rewriting System R:
[x]
half(0) -> 0
half(s(s(x))) -> s(half(x))
log(s(0)) -> 0
log(s(s(x))) -> s(log(s(half(x))))

Innermost Termination of R to be shown.



   TRS
Removing Redundant Rules



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

log(s(0)) -> 0

where the Polynomial interpretation:
  POL(0)=  0  
  POL(log(x1))=  1 + x1  
  POL(s(x1))=  x1  
  POL(half(x1))=  x1  
was used.

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


   TRS
RRRPolo
       →TRS2
Removing Redundant Rules



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

half(s(s(x))) -> s(half(x))

where the Polynomial interpretation:
  POL(0)=  0  
  POL(log(x1))=  x1  
  POL(s(x1))=  1 + x1  
  POL(half(x1))=  x1  
was used.

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


   TRS
RRRPolo
       →TRS2
RRRPolo
           →TRS3
Removing Redundant Rules



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

log(s(s(x))) -> s(log(s(half(x))))

where the Polynomial interpretation:
  POL(0)=  0  
  POL(log(x1))=  2·x1  
  POL(s(x1))=  1 + x1  
  POL(half(x1))=  x1  
was used.

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


   TRS
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS4
Removing Redundant Rules



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

half(0) -> 0

where the Polynomial interpretation:
  POL(0)=  0  
  POL(half(x1))=  1 + x1  
was used.

All Rules of R can be deleted.


   TRS
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS5
Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

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