Term Rewriting System R:
[x, y]
\(x, x) -> e
\(e, x) -> x
\(x, .(x, y)) -> y
\(/(x, y), x) -> y
/(x, x) -> e
/(x, e) -> x
/(.(y, x), x) -> y
/(x, \(y, x)) -> y
.(e, x) -> x
.(x, e) -> x
.(x, \(x, y)) -> y
.(/(y, x), x) -> y
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
\(x, x) -> e
\(e, x) -> x
\(x, .(x, y)) -> y
\(/(x, y), x) -> y
/(x, \(y, x)) -> y
.(x, \(x, y)) -> y
where the Polynomial interpretation:
| POL(e) | = 0 |
| POL(.(x1, x2)) | = x1 + x2 |
| POL(/(x1, x2)) | = x1 + x2 |
| 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:
/(x, x) -> e
/(.(y, x), x) -> y
/(x, e) -> x
.(/(y, x), x) -> y
where the Polynomial interpretation:
| POL(e) | = 0 |
| POL(.(x1, x2)) | = x1 + x2 |
| 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:
.(e, x) -> x
.(x, e) -> x
where the Polynomial interpretation:
| POL(e) | = 0 |
| POL(.(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