Term Rewriting System R:
[x, y]
g(0, f(x, x)) -> x
g(x, s(y)) -> g(f(x, y), 0)
g(s(x), y) -> g(f(x, y), 0)
g(f(x, y), 0) -> f(g(x, 0), g(y, 0))
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
g(0, f(x, x)) -> x
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(g(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = 1 + x1 |
| POL(f(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:
g(s(x), y) -> g(f(x, y), 0)
g(x, s(y)) -> g(f(x, y), 0)
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(g(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = 1 + x1 |
| POL(f(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
↳RRRPolo
→TRS3
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
g(f(x, y), 0) -> f(g(x, 0), g(y, 0))
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(g(x1, x2)) | = 2·x1 + x2 |
| POL(f(x1, x2)) | = 1 + x1 + x2 |
was used.
All Rules of R can be deleted.
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 no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:00 minutes