Term Rewriting System R:
[x, y, z]
norm(nil) -> 0
norm(g(x, y)) -> s(norm(x))
f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
rem(nil, y) -> nil
rem(g(x, y), 0) -> g(x, y)
rem(g(x, y), s(z)) -> rem(x, z)

Innermost Termination of R to be shown.



   R
Removing Redundant Rules



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

norm(nil) -> 0

where the Polynomial interpretation:
  POL(0)=  0  
  POL(g(x1, x2))=  x1 + x2  
  POL(nil)=  0  
  POL(s(x1))=  x1  
  POL(rem(x1, x2))=  x1 + x2  
  POL(f(x1, x2))=  x1 + x2  
  POL(norm(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:

norm(g(x, y)) -> s(norm(x))
rem(g(x, y), s(z)) -> rem(x, z)

where the Polynomial interpretation:
  POL(0)=  0  
  POL(g(x1, x2))=  1 + x1 + x2  
  POL(nil)=  0  
  POL(rem(x1, x2))=  x1 + x2  
  POL(s(x1))=  x1  
  POL(f(x1, x2))=  1 + x1 + x2  
  POL(norm(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
RRRPolo
           →TRS3
Removing Redundant Rules



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

rem(nil, y) -> nil
rem(g(x, y), 0) -> g(x, y)

where the Polynomial interpretation:
  POL(0)=  0  
  POL(g(x1, x2))=  x1 + x2  
  POL(nil)=  0  
  POL(rem(x1, x2))=  1 + x1 + x2  
  POL(f(x1, x2))=  x1 + x2  
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:

f(x, nil) -> g(nil, x)

where the Polynomial interpretation:
  POL(g(x1, x2))=  x1 + x2  
  POL(nil)=  1  
  POL(f(x1, x2))=  x1 + 2·x2  
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:

f(x, g(y, z)) -> g(f(x, y), z)

where the Polynomial interpretation:
  POL(g(x1, x2))=  1 + x1 + x2  
  POL(f(x1, x2))=  x1 + 2·x2  
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