Term Rewriting System R:
[x, y]
f(x, y) -> g1(x, x, y)
f(x, y) -> g1(y, x, x)
f(x, y) -> g2(x, y, y)
f(x, y) -> g2(y, y, x)
g1(x, x, y) -> h(x, y)
g1(y, x, x) -> h(x, y)
g2(x, y, y) -> h(x, y)
g2(y, y, x) -> h(x, y)
h(x, x) -> 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) -> g1(x, x, y)
f(x, y) -> g1(y, x, x)
f(x, y) -> g2(x, y, y)
f(x, y) -> g2(y, y, x)
where the Polynomial interpretation:
| POL(g2(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(h(x1, x2)) | = x1 + x2 |
| POL(f(x1, x2)) | = 1 + 2·x1 + 2·x2 |
| POL(g1(x1, x2, x3)) | = x1 + x2 + x3 |
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:
h(x, x) -> x
where the Polynomial interpretation:
| POL(g2(x1, x2, x3)) | = 1 + x1 + x2 + x3 |
| POL(h(x1, x2)) | = 1 + x1 + x2 |
| POL(g1(x1, x2, x3)) | = 1 + x1 + x2 + x3 |
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:
g1(y, x, x) -> h(x, y)
g1(x, x, y) -> h(x, y)
where the Polynomial interpretation:
| POL(g2(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(h(x1, x2)) | = x1 + x2 |
| POL(g1(x1, x2, x3)) | = 1 + x1 + x2 + x3 |
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:
g2(x, y, y) -> h(x, y)
g2(y, y, x) -> h(x, y)
where the Polynomial interpretation:
| POL(g2(x1, x2, x3)) | = 1 + x1 + x2 + x3 |
| POL(h(x1, x2)) | = 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