Term Rewriting System R:
[f, g, x, l, xs]
app(app(app(compose, f), g), x) -> app(g, app(f, x))
app(reverse, l) -> app(app(reverse2, l), nil)
app(app(reverse2, nil), l) -> l
app(app(reverse2, app(app(cons, x), xs)), l) -> app(app(reverse2, xs), app(app(cons, x), l))
app(hd, app(app(cons, x), xs)) -> x
app(tl, app(app(cons, x), xs)) -> xs
last -> app(app(compose, hd), reverse)
init -> app(app(compose, reverse), app(app(compose, tl), reverse))

Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



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

app(app(app(compose, f), g), x) -> app(g, app(f, x))

where the Polynomial interpretation:
  POL(last)=  1  
  POL(reverse)=  0  
  POL(cons)=  0  
  POL(hd)=  0  
  POL(nil)=  0  
  POL(tl)=  0  
  POL(compose)=  1  
  POL(init)=  2  
  POL(app(x1, x2))=  x1 + x2  
  POL(reverse2)=  0  
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:

last -> app(app(compose, hd), reverse)

where the Polynomial interpretation:
  POL(last)=  1  
  POL(reverse)=  0  
  POL(cons)=  0  
  POL(hd)=  0  
  POL(nil)=  0  
  POL(tl)=  0  
  POL(init)=  0  
  POL(compose)=  0  
  POL(app(x1, x2))=  x1 + x2  
  POL(reverse2)=  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(reverse, l) -> app(app(reverse2, l), nil)
app(hd, app(app(cons, x), xs)) -> x

where the Polynomial interpretation:
  POL(reverse)=  1  
  POL(cons)=  0  
  POL(hd)=  1  
  POL(nil)=  0  
  POL(tl)=  0  
  POL(init)=  2  
  POL(compose)=  0  
  POL(app(x1, x2))=  x1 + x2  
  POL(reverse2)=  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
             ↳RRRPolo
             ...
               →TRS4
                 ↳Removing Redundant Rules



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

init -> app(app(compose, reverse), app(app(compose, tl), reverse))
app(app(reverse2, nil), l) -> l

where the Polynomial interpretation:
  POL(reverse)=  0  
  POL(cons)=  0  
  POL(nil)=  0  
  POL(tl)=  0  
  POL(init)=  1  
  POL(compose)=  0  
  POL(app(x1, x2))=  x1 + x2  
  POL(reverse2)=  1  
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:

app(app(reverse2, app(app(cons, x), xs)), l) -> app(app(reverse2, xs), app(app(cons, x), l))
app(tl, app(app(cons, x), xs)) -> xs

where the Polynomial interpretation:
  POL(cons)=  1  
  POL(tl)=  2  
  POL(app(x1, x2))=  2·x1 + x2  
  POL(reverse2)=  2  
was used.

All Rules of R can be deleted. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳RRRPolo
             ...
               →TRS6
                 ↳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
             ...
               →TRS7
                 ↳Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

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