Term Rewriting System R:
[X, Y]
f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
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
if(true, X, Y) -> X
R
↳RRRI
→TRS2
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(X) -> nf(X)
where the Polynomial interpretation:
| POL(n__f(x1)) | = 1 + x1 |
| POL(n__true) | = 0 |
| POL(activate(x1)) | = 2·x1 |
| POL(if(x1, x2, x3)) | = x1 + x2 + 2·x3 |
| POL(c) | = 0 |
| POL(false) | = 0 |
| POL(true) | = 0 |
| 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.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
if(false, X, Y) -> activate(Y)
where the Polynomial interpretation:
| POL(activate(x1)) | = x1 |
| POL(n__true) | = 0 |
| POL(n__f(x1)) | = x1 |
| POL(if(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(c) | = 0 |
| POL(false) | = 1 |
| POL(true) | = 0 |
| 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
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
true -> ntrue
activate(X) -> X
where the Polynomial interpretation:
| POL(n__f(x1)) | = x1 |
| POL(n__true) | = 0 |
| POL(activate(x1)) | = 1 + x1 |
| POL(if(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(c) | = 0 |
| POL(true) | = 1 |
| 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
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS5
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(X) -> if(X, c, nf(ntrue))
where the Polynomial interpretation:
| POL(activate(x1)) | = 2·x1 |
| POL(n__true) | = 0 |
| POL(n__f(x1)) | = 1 + x1 |
| POL(if(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(c) | = 0 |
| POL(true) | = 0 |
| 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.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS6
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
activate(ntrue) -> true
where the Polynomial interpretation:
| POL(activate(x1)) | = x1 |
| POL(n__f(x1)) | = x1 |
| POL(n__true) | = 1 |
| POL(true) | = 0 |
| 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
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS7
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
activate(nf(X)) -> f(activate(X))
where the Polynomial interpretation:
| POL(activate(x1)) | = x1 |
| POL(n__f(x1)) | = 1 + x1 |
| POL(f(x1)) | = x1 |
was used.
All Rules of R can be deleted.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS8
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes