Term Rewriting System R:
[x, y]
app(id, x) -> x
app(plus, 0) -> id
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))

Termination of R to be shown.



   R
Removing Redundant Rules



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

app(id, x) -> x

where the Polynomial interpretation:
  POL(plus)=  0  
  POL(0)=  1  
  POL(s)=  0  
  POL(app(x1, x2))=  x1 + x2  
  POL(id)=  1  
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:

app(plus, 0) -> id

where the Polynomial interpretation:
  POL(plus)=  0  
  POL(0)=  1  
  POL(s)=  0  
  POL(app(x1, x2))=  x1 + x2  
  POL(id)=  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:

app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))

where the Polynomial interpretation:
  POL(plus)=  0  
  POL(s)=  0  
  POL(app(x1, x2))=  1 + 2·x1 + x2  
was used.

All Rules of R can be deleted.


   R
RRRPolo
       →TRS2
RRRPolo
           →TRS3
RRRPolo
             ...
               →TRS4
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
             ...
               →TRS5
Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

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