Term Rewriting System R:
[x, y, z]
rev(nil) -> nil
rev(rev(x)) -> x
rev(++(x, y)) -> ++(rev(y), rev(x))
++(nil, y) -> y
++(x, nil) -> x
++(.(x, y), z) -> .(x, ++(y, z))
++(x, ++(y, z)) -> ++(++(x, y), z)
make(x) -> .(x, nil)

Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



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

rev(nil) -> nil
++(nil, y) -> y
++(x, nil) -> x

where the Polynomial interpretation:
  POL(make(x1))=  1 + x1  
  POL(rev(x1))=  2·x1  
  POL(++(x1, x2))=  x1 + x2  
  POL(nil)=  1  
  POL(.(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
         ↳Removing Redundant Rules



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

rev(++(x, y)) -> ++(rev(y), rev(x))

where the Polynomial interpretation:
  POL(make(x1))=  x1  
  POL(rev(x1))=  2·x1  
  POL(++(x1, x2))=  1 + x1 + x2  
  POL(nil)=  0  
  POL(.(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
             ↳Removing Redundant Rules



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

rev(rev(x)) -> x

where the Polynomial interpretation:
  POL(make(x1))=  x1  
  POL(rev(x1))=  1 + x1  
  POL(++(x1, x2))=  x1 + x2  
  POL(nil)=  0  
  POL(.(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:

++(x, ++(y, z)) -> ++(++(x, y), z)

where the Polynomial interpretation:
  POL(make(x1))=  x1  
  POL(++(x1, x2))=  1 + x1 + 2·x2  
  POL(nil)=  0  
  POL(.(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
             ...
               →TRS5
                 ↳Removing Redundant Rules



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

++(.(x, y), z) -> .(x, ++(y, z))

where the Polynomial interpretation:
  POL(make(x1))=  1 + x1  
  POL(++(x1, x2))=  2·x1 + x2  
  POL(nil)=  0  
  POL(.(x1, x2))=  1 + 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
             ...
               →TRS6
                 ↳Removing Redundant Rules



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

make(x) -> .(x, nil)

where the Polynomial interpretation:
  POL(make(x1))=  1 + x1  
  POL(nil)=  0  
  POL(.(x1, x2))=  x1 + x2  
was used.

All Rules of R can be deleted. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳RRRPolo
             ...
               →TRS7
                 ↳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
             ↳RRRPolo
             ...
               →TRS8
                 ↳Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

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