Term Rewriting System R:
[X]
f(0) -> cons(0, nf(ns(n0)))
f(s(0)) -> f(p(s(0)))
f(X) -> nf(X)
p(s(0)) -> 0
s(X) -> ns(X)
0 -> n0
activate(nf(X)) -> f(activate(X))
activate(ns(X)) -> s(activate(X))
activate(n0) -> 0
activate(X) -> X
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(0) -> cons(0, nf(ns(n0)))
f(X) -> nf(X)
where the Polynomial interpretation:
| POL(n__f(x1)) | = 1 + x1 |
| POL(activate(x1)) | = 2·x1 |
| POL(0) | = 0 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(n__s(x1)) | = x1 |
| POL(s(x1)) | = x1 |
| POL(n__0) | = 0 |
| POL(f(x1)) | = 2 + x1 |
| POL(p(x1)) | = x1 |
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:
0 -> n0
where the Polynomial interpretation:
| POL(activate(x1)) | = 2·x1 |
| POL(n__f(x1)) | = x1 |
| POL(0) | = 2 |
| POL(n__s(x1)) | = x1 |
| POL(s(x1)) | = x1 |
| POL(n__0) | = 1 |
| POL(f(x1)) | = x1 |
| POL(p(x1)) | = x1 |
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:
s(X) -> ns(X)
p(s(0)) -> 0
where the Polynomial interpretation:
| POL(activate(x1)) | = 2·x1 |
| POL(n__f(x1)) | = x1 |
| POL(0) | = 0 |
| POL(n__s(x1)) | = 1 + x1 |
| POL(s(x1)) | = 2 + x1 |
| POL(n__0) | = 0 |
| POL(f(x1)) | = x1 |
| POL(p(x1)) | = x1 |
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:
activate(ns(X)) -> s(activate(X))
where the Polynomial interpretation:
| POL(activate(x1)) | = x1 |
| POL(n__f(x1)) | = x1 |
| POL(0) | = 0 |
| POL(n__s(x1)) | = 1 + x1 |
| POL(s(x1)) | = x1 |
| POL(n__0) | = 0 |
| POL(f(x1)) | = x1 |
| POL(p(x1)) | = x1 |
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:
activate(n0) -> 0
activate(X) -> X
where the Polynomial interpretation:
| POL(activate(x1)) | = 1 + x1 |
| POL(n__f(x1)) | = x1 |
| POL(0) | = 0 |
| POL(s(x1)) | = x1 |
| POL(n__0) | = 0 |
| POL(f(x1)) | = x1 |
| POL(p(x1)) | = x1 |
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:
activate(nf(X)) -> f(activate(X))
where the Polynomial interpretation:
| POL(activate(x1)) | = x1 |
| POL(n__f(x1)) | = 1 + x1 |
| POL(0) | = 0 |
| POL(s(x1)) | = x1 |
| POL(f(x1)) | = x1 |
| POL(p(x1)) | = x1 |
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
...
→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 the following Dependency Pairs:
F(s(0)) -> F(p(s(0)))
R contains no SCCs.
Termination of R successfully shown.
Duration:
0:00 minutes