Term Rewriting System R:
[X, Y, X1, X2, X3]
af(X) -> aif(mark(X), c, f(true))
af(X) -> f(X)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
mark(f(X)) -> af(mark(X))
mark(if(X1, X2, X3)) -> aif(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
af(X) -> f(X)
where the Polynomial interpretation:
| POL(c) | = 0 |
| POL(if(x1, x2, x3)) | = x1 + 2·x2 + x3 |
| POL(false) | = 0 |
| POL(a__if(x1, x2, x3)) | = x1 + 2·x2 + 2·x3 |
| POL(true) | = 0 |
| POL(mark(x1)) | = 2·x1 |
| POL(f(x1)) | = 1 + 2·x1 |
| POL(a__f(x1)) | = 2 + 2·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:
mark(false) -> false
aif(false, X, Y) -> mark(Y)
where the Polynomial interpretation:
| POL(if(x1, x2, x3)) | = x1 + 2·x2 + x3 |
| POL(c) | = 0 |
| POL(false) | = 1 |
| POL(a__if(x1, x2, x3)) | = x1 + 2·x2 + 2·x3 |
| POL(true) | = 0 |
| POL(mark(x1)) | = 2·x1 |
| POL(f(x1)) | = 2·x1 |
| POL(a__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
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
af(X) -> aif(mark(X), c, f(true))
where the Polynomial interpretation:
| POL(if(x1, x2, x3)) | = x1 + 2·x2 + x3 |
| POL(c) | = 0 |
| POL(a__if(x1, x2, x3)) | = x1 + 2·x2 + x3 |
| POL(true) | = 0 |
| POL(mark(x1)) | = 2·x1 |
| POL(f(x1)) | = 1 + 2·x1 |
| POL(a__f(x1)) | = 2 + 2·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
↳RRRPolo
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
aif(X1, X2, X3) -> if(X1, X2, X3)
aif(true, X, Y) -> mark(X)
where the Polynomial interpretation:
| POL(if(x1, x2, x3)) | = 1 + x1 + 2·x2 + x3 |
| POL(c) | = 0 |
| POL(a__if(x1, x2, x3)) | = 2 + x1 + 2·x2 + x3 |
| POL(true) | = 0 |
| POL(mark(x1)) | = 2·x1 |
| POL(f(x1)) | = x1 |
| POL(a__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
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS5
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
mark(if(X1, X2, X3)) -> aif(mark(X1), mark(X2), X3)
where the Polynomial interpretation:
| POL(c) | = 0 |
| POL(if(x1, x2, x3)) | = 1 + x1 + x2 + x3 |
| POL(a__if(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(true) | = 0 |
| POL(mark(x1)) | = x1 |
| POL(f(x1)) | = x1 |
| POL(a__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
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS6
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
mark(f(X)) -> af(mark(X))
where the Polynomial interpretation:
| POL(c) | = 0 |
| POL(true) | = 0 |
| POL(mark(x1)) | = x1 |
| POL(f(x1)) | = 1 + x1 |
| POL(a__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
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS7
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
mark(true) -> true
mark(c) -> c
where the Polynomial interpretation:
| POL(c) | = 0 |
| POL(true) | = 0 |
| POL(mark(x1)) | = 1 + x1 |
was used.
All Rules of R can be deleted.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS8
↳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
...
→TRS9
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:00 minutes