Term Rewriting System R:
[N, X, Y]
terms(N) -> cons(recip(sqr(N)))
sqr(0) -> 0
sqr(s) -> s
dbl(0) -> 0
dbl(s) -> s
add(0, X) -> X
add(s, Y) -> s
first(0, X) -> nil
first(s, cons(Y)) -> cons(Y)

Innermost Termination of R to be shown.



   R
Removing Redundant Rules



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

terms(N) -> cons(recip(sqr(N)))

where the Polynomial interpretation:
  POL(0)=  0  
  POL(first(x1, x2))=  x1 + x2  
  POL(cons(x1))=  x1  
  POL(sqr(x1))=  x1  
  POL(dbl(x1))=  x1  
  POL(nil)=  0  
  POL(s)=  0  
  POL(terms(x1))=  1 + x1  
  POL(recip(x1))=  x1  
  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
Removing Redundant Rules



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

add(0, X) -> X
add(s, Y) -> s

where the Polynomial interpretation:
  POL(0)=  0  
  POL(first(x1, x2))=  x1 + x2  
  POL(cons(x1))=  x1  
  POL(sqr(x1))=  x1  
  POL(dbl(x1))=  x1  
  POL(nil)=  0  
  POL(s)=  0  
  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:

sqr(s) -> s
sqr(0) -> 0

where the Polynomial interpretation:
  POL(0)=  0  
  POL(first(x1, x2))=  x1 + x2  
  POL(cons(x1))=  x1  
  POL(sqr(x1))=  1 + x1  
  POL(dbl(x1))=  x1  
  POL(nil)=  0  
  POL(s)=  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:

dbl(s) -> s
dbl(0) -> 0

where the Polynomial interpretation:
  POL(0)=  0  
  POL(first(x1, x2))=  x1 + x2  
  POL(cons(x1))=  x1  
  POL(dbl(x1))=  1 + x1  
  POL(nil)=  0  
  POL(s)=  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
             ...
               →TRS5
Removing Redundant Rules



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

first(0, X) -> nil

where the Polynomial interpretation:
  POL(first(x1, x2))=  x1 + x2  
  POL(0)=  1  
  POL(cons(x1))=  x1  
  POL(nil)=  0  
  POL(s)=  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
             ...
               →TRS6
Removing Redundant Rules



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

first(s, cons(Y)) -> cons(Y)

where the Polynomial interpretation:
  POL(first(x1, x2))=  1 + x1 + x2  
  POL(cons(x1))=  x1  
  POL(s)=  0  
was used.

All Rules of R can be deleted.


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



R contains no Dependency Pairs and therefore no SCCs.

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