Term Rewriting System R:
[X, Y, X1, X2]
f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
f(X1, X2) -> nf(X1, X2)
g(X) -> ng(X)
activate(nf(X1, X2)) -> f(activate(X1), X2)
activate(ng(X)) -> g(activate(X))
activate(X) -> X
Innermost Termination of R to be shown.
R
↳Removing Redundant Rules for Innermost Termination
Removing the following rules from R which left hand sides contain non normal subterms
f(g(X), Y) -> f(X, nf(ng(X), activate(Y)))
R
↳RRRI
→TRS2
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(X1, X2) -> nf(X1, X2)
where the Polynomial interpretation:
| POL(activate(x1)) | = 2·x1 |
| POL(n__f(x1, x2)) | = 1 + x1 + x2 |
| POL(g(x1)) | = x1 |
| POL(n__g(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
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
activate(nf(X1, X2)) -> f(activate(X1), X2)
where the Polynomial interpretation:
| POL(activate(x1)) | = x1 |
| POL(n__f(x1, x2)) | = 1 + x1 + x2 |
| POL(g(x1)) | = x1 |
| POL(n__g(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
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
activate(ng(X)) -> g(activate(X))
where the Polynomial interpretation:
| POL(activate(x1)) | = 2·x1 |
| POL(g(x1)) | = 1 + x1 |
| POL(n__g(x1)) | = 1 + x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS5
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
g(X) -> ng(X)
where the Polynomial interpretation:
| POL(activate(x1)) | = x1 |
| POL(g(x1)) | = 1 + x1 |
| POL(n__g(x1)) | = x1 |
was used.
Not all Rules of R can be deleted, so we still have to regard a part of R.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS6
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
activate(X) -> X
where the Polynomial interpretation:
| POL(activate(x1)) | = 1 + x1 |
was used.
All Rules of R can be deleted.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS7
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes