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