Term Rewriting System R:
[X]
active(f(X)) -> mark(g(h(f(X))))
active(f(X)) -> f(active(X))
active(h(X)) -> h(active(X))
f(mark(X)) -> mark(f(X))
f(ok(X)) -> ok(f(X))
h(mark(X)) -> mark(h(X))
h(ok(X)) -> ok(h(X))
proper(f(X)) -> f(proper(X))
proper(g(X)) -> g(proper(X))
proper(h(X)) -> h(proper(X))
g(ok(X)) -> ok(g(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Termination of R to be shown.
TRS
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
top(ok(X)) -> top(active(X))
where the Polynomial interpretation:
| POL(top(x1)) | = 1 + x1 |
| POL(active(x1)) | = x1 |
| POL(proper(x1)) | = x1 |
| POL(g(x1)) | = x1 |
| POL(h(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(ok(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.
TRS
↳RRRPolo
→TRS2
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
active(f(X)) -> f(active(X))
active(f(X)) -> mark(g(h(f(X))))
where the Polynomial interpretation:
| POL(top(x1)) | = 1 + x1 |
| POL(active(x1)) | = 2·x1 |
| POL(proper(x1)) | = x1 |
| POL(g(x1)) | = x1 |
| POL(h(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(ok(x1)) | = x1 |
| POL(f(x1)) | = 1 + x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
TRS
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
proper(g(X)) -> g(proper(X))
where the Polynomial interpretation:
| POL(top(x1)) | = x1 |
| POL(proper(x1)) | = 2·x1 |
| POL(active(x1)) | = x1 |
| POL(g(x1)) | = 1 + x1 |
| POL(h(x1)) | = x1 |
| POL(mark(x1)) | = 2·x1 |
| POL(ok(x1)) | = 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.
TRS
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
active(h(X)) -> h(active(X))
where the Polynomial interpretation:
| POL(top(x1)) | = x1 |
| POL(proper(x1)) | = x1 |
| POL(active(x1)) | = 2·x1 |
| POL(g(x1)) | = x1 |
| POL(h(x1)) | = 1 + x1 |
| POL(mark(x1)) | = x1 |
| POL(ok(x1)) | = 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.
TRS
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS5
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
top(mark(X)) -> top(proper(X))
where the Polynomial interpretation:
| POL(top(x1)) | = x1 |
| POL(proper(x1)) | = x1 |
| POL(g(x1)) | = x1 |
| POL(h(x1)) | = x1 |
| POL(mark(x1)) | = 1 + x1 |
| POL(ok(x1)) | = 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.
TRS
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS6
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
h(mark(X)) -> mark(h(X))
where the Polynomial interpretation:
| POL(proper(x1)) | = x1 |
| POL(g(x1)) | = x1 |
| POL(h(x1)) | = 2·x1 |
| POL(mark(x1)) | = 1 + x1 |
| POL(ok(x1)) | = 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.
TRS
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS7
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
proper(h(X)) -> h(proper(X))
where the Polynomial interpretation:
| POL(proper(x1)) | = 2·x1 |
| POL(g(x1)) | = x1 |
| POL(h(x1)) | = 1 + x1 |
| POL(mark(x1)) | = x1 |
| POL(ok(x1)) | = 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.
TRS
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS8
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(mark(X)) -> mark(f(X))
where the Polynomial interpretation:
| POL(proper(x1)) | = x1 |
| POL(g(x1)) | = x1 |
| POL(h(x1)) | = x1 |
| POL(mark(x1)) | = 1 + x1 |
| POL(ok(x1)) | = x1 |
| POL(f(x1)) | = 2·x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
TRS
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS9
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
proper(f(X)) -> f(proper(X))
where the Polynomial interpretation:
| POL(proper(x1)) | = 2·x1 |
| POL(g(x1)) | = x1 |
| POL(h(x1)) | = x1 |
| POL(ok(x1)) | = x1 |
| POL(f(x1)) | = 1 + x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
TRS
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS10
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(ok(X)) -> ok(f(X))
where the Polynomial interpretation:
| POL(g(x1)) | = x1 |
| POL(h(x1)) | = x1 |
| POL(ok(x1)) | = 1 + x1 |
| POL(f(x1)) | = 2·x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
TRS
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS11
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
g(ok(X)) -> ok(g(X))
where the Polynomial interpretation:
| POL(g(x1)) | = 2·x1 |
| POL(h(x1)) | = x1 |
| POL(ok(x1)) | = 1 + x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
TRS
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS12
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
h(ok(X)) -> ok(h(X))
where the Polynomial interpretation:
| POL(h(x1)) | = 2·x1 |
| POL(ok(x1)) | = 1 + x1 |
was used.
All Rules of R can be deleted.
TRS
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS13
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:02 minutes