Term Rewriting System R:
[x, y, u, v]
s(a) -> a
s(s(x)) -> x
s(f(x, y)) -> f(s(y), s(x))
s(g(x, y)) -> g(s(x), s(y))
f(x, a) -> x
f(a, y) -> y
f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
g(a, a) -> a
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
s(a) -> a
f(x, a) -> x
f(a, y) -> y
g(a, a) -> a
where the Polynomial interpretation:
| POL(g(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = 2·x1 |
| POL(a) | = 1 |
| 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
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
s(g(x, y)) -> g(s(x), s(y))
f(g(x, y), g(u, v)) -> g(f(x, u), f(y, v))
where the Polynomial interpretation:
| POL(g(x1, x2)) | = 1 + x1 + x2 |
| POL(s(x1)) | = 2·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:
s(s(x)) -> x
where the Polynomial interpretation:
| POL(s(x1)) | = 1 + 2·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
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
s(f(x, y)) -> f(s(y), s(x))
where the Polynomial interpretation:
| POL(s(x1)) | = 2·x1 |
| POL(f(x1, x2)) | = 1 + x1 + x2 |
was used.
All Rules of R can be deleted.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS5
↳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
...
→TRS6
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:00 minutes