Term Rewriting System R:
[y, n, x]
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(nil) -> nil
shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))

Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



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

reverse(nil) -> nil

where the Polynomial interpretation:
  POL(reverse(x1))=  1 + x1  
  POL(shuffle(x1))=  2·x1  
  POL(nil)=  0  
  POL(app(x1, x2))=  x1 + x2  
  POL(add(x1, x2))=  2 + 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:

shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))

where the Polynomial interpretation:
  POL(reverse(x1))=  x1  
  POL(shuffle(x1))=  2·x1  
  POL(nil)=  0  
  POL(app(x1, x2))=  x1 + x2  
  POL(add(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
             ↳Removing Redundant Rules



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

app(nil, y) -> y

where the Polynomial interpretation:
  POL(reverse(x1))=  2·x1  
  POL(shuffle(x1))=  x1  
  POL(nil)=  1  
  POL(app(x1, x2))=  x1 + x2  
  POL(add(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
             ...
               →TRS4
                 ↳Removing Redundant Rules



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

shuffle(nil) -> nil

where the Polynomial interpretation:
  POL(reverse(x1))=  x1  
  POL(shuffle(x1))=  1 + x1  
  POL(nil)=  0  
  POL(app(x1, x2))=  x1 + x2  
  POL(add(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:

reverse(add(n, x)) -> app(reverse(x), add(n, nil))

where the Polynomial interpretation:
  POL(reverse(x1))=  2·x1  
  POL(nil)=  0  
  POL(app(x1, x2))=  x1 + x2  
  POL(add(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:

app(add(n, x), y) -> add(n, app(x, y))

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