Term Rewriting System R:
[x, y]
f(0) -> s(0)
f(s(0)) -> s(s(0))
f(s(0)) -> *(s(s(0)), f(0))
f(+(x, s(0))) -> +(s(s(0)), f(x))
f(+(x, y)) -> *(f(x), f(y))
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(0) -> s(0)
f(s(0)) -> s(s(0))
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(*(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(f(x1)) | = 1 + x1 |
| POL(+(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
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(+(x, y)) -> *(f(x), f(y))
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(*(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(f(x1)) | = x1 |
| POL(+(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:
f(+(x, s(0))) -> +(s(s(0)), f(x))
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(*(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(f(x1)) | = 2·x1 |
| POL(+(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
↳RRRPolo
...
→TRS4
↳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
...
→TRS5
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(s(0)) -> F(0)
R contains no SCCs.
Termination of R successfully shown.
Duration:
0:00 minutes