Term Rewriting System R:
[x]
g(s(x)) -> f(x)
g(0) -> 0
f(0) -> s(0)
f(s(x)) -> s(s(g(x)))
Innermost Termination of R to be shown.
TRS
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
g(0) -> 0
f(0) -> s(0)
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(g(x1)) | = 1 + x1 |
| POL(s(x1)) | = x1 |
| POL(f(x1)) | = 1 + x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
TRS
↳RRRPolo
→TRS2
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
g(s(x)) -> f(x)
where the Polynomial interpretation:
| POL(g(x1)) | = 2·x1 |
| POL(s(x1)) | = 1 + x1 |
| POL(f(x1)) | = 2·x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
TRS
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(s(x)) -> s(s(g(x)))
where the Polynomial interpretation:
| POL(g(x1)) | = x1 |
| POL(s(x1)) | = x1 |
| POL(f(x1)) | = 1 + x1 |
was used.
All Rules of R can be deleted.
TRS
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS4
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes