Term Rewriting System R:
[x, y]
f(x, y) -> x
g(a) -> h(a, b, a)
i(x) -> f(x, x)
h(x, x, y) -> g(x)
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(x, y) -> x
where the Polynomial interpretation:
| POL(i(x1)) | = 1 + 2·x1 |
| POL(g(x1)) | = x1 |
| POL(b) | = 0 |
| POL(h(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(a) | = 0 |
| 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:
i(x) -> f(x, x)
where the Polynomial interpretation:
| POL(i(x1)) | = 1 + 2·x1 |
| POL(g(x1)) | = x1 |
| POL(b) | = 0 |
| POL(h(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(a) | = 0 |
| 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
↳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
↳OC
...
→TRS4
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
H(x, x, y) -> G(x)
G(a) -> H(a, b, a)
R contains no SCCs.
Termination of R successfully shown.
Duration:
0:00 minutes