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)
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
↳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