Term Rewriting System R:
[y, x, f]
app(app(plus, 0), y) -> y
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(app(curry, f), x), y) -> app(app(f, x), y)
add -> app(curry, plus)
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
app(app(plus, 0), y) -> y
where the Polynomial interpretation:
| POL(plus) | = 0 |
| POL(0) | = 1 |
| POL(curry) | = 0 |
| POL(s) | = 0 |
| POL(app(x1, x2)) | = x1 + x2 |
| POL(add) | = 0 |
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:
add -> app(curry, plus)
where the Polynomial interpretation:
| POL(plus) | = 0 |
| POL(curry) | = 0 |
| POL(s) | = 0 |
| POL(app(x1, x2)) | = x1 + x2 |
| POL(add) | = 1 |
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:
app(app(plus, app(s, x)), y) -> app(s, app(app(plus, x), y))
app(app(app(curry, f), x), y) -> app(app(f, x), y)
where the Polynomial interpretation:
| POL(plus) | = 0 |
| POL(curry) | = 4 |
| POL(s) | = 0 |
| POL(app(x1, x2)) | = 1 + 2·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