Term Rewriting System R:
[X, Y, X1, X2]
active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
active(f(X1, X2)) -> f(active(X1), X2)
active(g(X)) -> g(active(X))
f(mark(X1), X2) -> mark(f(X1, X2))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
g(mark(X)) -> mark(g(X))
g(ok(X)) -> ok(g(X))
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
proper(g(X)) -> g(proper(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
active(f(g(X), Y)) -> mark(f(X, f(g(X), Y)))
f(ok(X1), ok(X2)) -> ok(f(X1, X2))
where the Polynomial interpretation:
| POL(top(x1)) | = 1 + x1 |
| POL(active(x1)) | = 1 + 2·x1 |
| POL(proper(x1)) | = x1 |
| POL(g(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(ok(x1)) | = 1 + 2·x1 |
| POL(f(x1, x2)) | = 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:
f(mark(X1), X2) -> mark(f(X1, X2))
top(mark(X)) -> top(proper(X))
where the Polynomial interpretation:
| POL(top(x1)) | = x1 |
| POL(proper(x1)) | = x1 |
| POL(active(x1)) | = x1 |
| POL(g(x1)) | = x1 |
| POL(mark(x1)) | = 1 + x1 |
| POL(ok(x1)) | = x1 |
| POL(f(x1, x2)) | = 2·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:
proper(f(X1, X2)) -> f(proper(X1), proper(X2))
where the Polynomial interpretation:
| POL(top(x1)) | = x1 |
| POL(proper(x1)) | = 2·x1 |
| POL(active(x1)) | = x1 |
| POL(g(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(ok(x1)) | = x1 |
| POL(f(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
↳RRRPolo
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
active(g(X)) -> g(active(X))
top(ok(X)) -> top(active(X))
where the Polynomial interpretation:
| POL(top(x1)) | = x1 |
| POL(active(x1)) | = 2·x1 |
| POL(proper(x1)) | = x1 |
| POL(g(x1)) | = 1 + 2·x1 |
| POL(mark(x1)) | = x1 |
| POL(ok(x1)) | = 1 + 2·x1 |
| POL(f(x1, x2)) | = 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
↳RRRPolo
...
→TRS5
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
g(mark(X)) -> mark(g(X))
where the Polynomial interpretation:
| POL(proper(x1)) | = x1 |
| POL(active(x1)) | = x1 |
| POL(g(x1)) | = 2·x1 |
| POL(mark(x1)) | = 1 + x1 |
| POL(ok(x1)) | = x1 |
| POL(f(x1, x2)) | = 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
↳RRRPolo
...
→TRS6
↳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(active(x1)) | = x1 |
| POL(proper(x1)) | = 2·x1 |
| POL(g(x1)) | = 1 + x1 |
| POL(ok(x1)) | = x1 |
| POL(f(x1, x2)) | = 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
↳RRRPolo
...
→TRS7
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
active(f(X1, X2)) -> f(active(X1), X2)
where the Polynomial interpretation:
| POL(active(x1)) | = 2·x1 |
| POL(g(x1)) | = x1 |
| POL(ok(x1)) | = x1 |
| POL(f(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
↳RRRPolo
...
→TRS8
↳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(ok(x1)) | = 1 + x1 |
was used.
All Rules of R can be deleted.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS9
↳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
...
→TRS10
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:00 minutes