Term Rewriting System R:
[X, Y, X1, X2]
anats -> aadx(azeros)
anats -> nats
azeros -> cons(0, zeros)
azeros -> zeros
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
aincr(X) -> incr(X)
aadx(cons(X, Y)) -> aincr(cons(X, adx(Y)))
aadx(X) -> adx(X)
ahd(cons(X, Y)) -> mark(X)
ahd(X) -> hd(X)
atl(cons(X, Y)) -> mark(Y)
atl(X) -> tl(X)
mark(nats) -> anats
mark(adx(X)) -> aadx(mark(X))
mark(zeros) -> azeros
mark(incr(X)) -> aincr(mark(X))
mark(hd(X)) -> ahd(mark(X))
mark(tl(X)) -> atl(mark(X))
mark(cons(X1, X2)) -> cons(X1, X2)
mark(0) -> 0
mark(s(X)) -> s(X)
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
anats -> aadx(azeros)
where the Polynomial interpretation:
| POL(a__nats) | = 1 |
| POL(adx(x1)) | = x1 |
| POL(a__zeros) | = 0 |
| POL(incr(x1)) | = x1 |
| POL(a__hd(x1)) | = x1 |
| POL(tl(x1)) | = x1 |
| POL(a__tl(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(a__adx(x1)) | = x1 |
| POL(0) | = 0 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(hd(x1)) | = x1 |
| POL(nats) | = 1 |
| POL(s(x1)) | = x1 |
| POL(zeros) | = 0 |
| POL(a__incr(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
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
mark(nats) -> anats
where the Polynomial interpretation:
| POL(a__nats) | = 1 |
| POL(adx(x1)) | = x1 |
| POL(a__zeros) | = 0 |
| POL(incr(x1)) | = x1 |
| POL(a__hd(x1)) | = 2·x1 |
| POL(mark(x1)) | = 2·x1 |
| POL(tl(x1)) | = 2·x1 |
| POL(a__tl(x1)) | = 2·x1 |
| POL(a__adx(x1)) | = x1 |
| POL(0) | = 0 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(hd(x1)) | = 2·x1 |
| POL(nats) | = 1 |
| POL(s(x1)) | = x1 |
| POL(zeros) | = 0 |
| POL(a__incr(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
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
aincr(X) -> incr(X)
aincr(cons(X, Y)) -> cons(s(X), incr(Y))
ahd(cons(X, Y)) -> mark(X)
mark(cons(X1, X2)) -> cons(X1, X2)
aadx(X) -> adx(X)
azeros -> zeros
atl(cons(X, Y)) -> mark(Y)
where the Polynomial interpretation:
| POL(a__nats) | = 0 |
| POL(adx(x1)) | = 1 + 2·x1 |
| POL(a__zeros) | = 2 |
| POL(incr(x1)) | = 1 + x1 |
| POL(a__hd(x1)) | = 2·x1 |
| POL(mark(x1)) | = 2·x1 |
| POL(tl(x1)) | = 2·x1 |
| POL(a__tl(x1)) | = 2·x1 |
| POL(a__adx(x1)) | = 2 + 2·x1 |
| POL(0) | = 0 |
| POL(cons(x1, x2)) | = 1 + x1 + x2 |
| POL(hd(x1)) | = 2·x1 |
| POL(nats) | = 0 |
| POL(s(x1)) | = x1 |
| POL(zeros) | = 1 |
| POL(a__incr(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
↳RRRPolo
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
azeros -> cons(0, zeros)
mark(s(X)) -> s(X)
mark(0) -> 0
where the Polynomial interpretation:
| POL(a__nats) | = 0 |
| POL(adx(x1)) | = x1 |
| POL(a__zeros) | = 1 |
| POL(incr(x1)) | = x1 |
| POL(a__hd(x1)) | = x1 |
| POL(mark(x1)) | = 1 + x1 |
| POL(tl(x1)) | = x1 |
| POL(a__tl(x1)) | = x1 |
| POL(a__adx(x1)) | = x1 |
| POL(0) | = 0 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(hd(x1)) | = x1 |
| POL(nats) | = 0 |
| POL(s(x1)) | = x1 |
| POL(zeros) | = 0 |
| POL(a__incr(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:
anats -> nats
where the Polynomial interpretation:
| POL(a__nats) | = 1 |
| POL(adx(x1)) | = x1 |
| POL(a__zeros) | = 0 |
| POL(incr(x1)) | = x1 |
| POL(a__hd(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(tl(x1)) | = x1 |
| POL(a__tl(x1)) | = x1 |
| POL(a__adx(x1)) | = x1 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(hd(x1)) | = x1 |
| POL(nats) | = 0 |
| POL(zeros) | = 0 |
| POL(a__incr(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:
ahd(X) -> hd(X)
where the Polynomial interpretation:
| POL(adx(x1)) | = x1 |
| POL(a__zeros) | = 0 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(hd(x1)) | = 1 + x1 |
| POL(incr(x1)) | = x1 |
| POL(a__hd(x1)) | = 2 + x1 |
| POL(mark(x1)) | = 2·x1 |
| POL(tl(x1)) | = x1 |
| POL(a__tl(x1)) | = x1 |
| POL(zeros) | = 0 |
| POL(a__adx(x1)) | = x1 |
| POL(a__incr(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(incr(X)) -> aincr(mark(X))
where the Polynomial interpretation:
| POL(adx(x1)) | = x1 |
| POL(a__zeros) | = 0 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(hd(x1)) | = x1 |
| POL(incr(x1)) | = 1 + x1 |
| POL(a__hd(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(tl(x1)) | = x1 |
| POL(a__tl(x1)) | = x1 |
| POL(zeros) | = 0 |
| POL(a__adx(x1)) | = x1 |
| POL(a__incr(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
...
→TRS8
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
aadx(cons(X, Y)) -> aincr(cons(X, adx(Y)))
where the Polynomial interpretation:
| POL(adx(x1)) | = 1 + x1 |
| POL(a__zeros) | = 0 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(hd(x1)) | = x1 |
| POL(a__hd(x1)) | = x1 |
| POL(a__tl(x1)) | = x1 |
| POL(tl(x1)) | = x1 |
| POL(mark(x1)) | = 2·x1 |
| POL(zeros) | = 0 |
| POL(a__adx(x1)) | = 2 + x1 |
| POL(a__incr(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
...
→TRS9
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
atl(X) -> tl(X)
where the Polynomial interpretation:
| POL(adx(x1)) | = x1 |
| POL(a__zeros) | = 0 |
| POL(hd(x1)) | = x1 |
| POL(a__hd(x1)) | = x1 |
| POL(mark(x1)) | = 2·x1 |
| POL(tl(x1)) | = 1 + x1 |
| POL(a__tl(x1)) | = 2 + x1 |
| POL(zeros) | = 0 |
| POL(a__adx(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
...
→TRS10
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
mark(hd(X)) -> ahd(mark(X))
where the Polynomial interpretation:
| POL(adx(x1)) | = x1 |
| POL(a__zeros) | = 0 |
| POL(hd(x1)) | = 1 + x1 |
| POL(a__hd(x1)) | = x1 |
| POL(mark(x1)) | = x1 |
| POL(tl(x1)) | = x1 |
| POL(a__tl(x1)) | = x1 |
| POL(zeros) | = 0 |
| POL(a__adx(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
...
→TRS11
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
mark(tl(X)) -> atl(mark(X))
where the Polynomial interpretation:
| POL(adx(x1)) | = x1 |
| POL(a__zeros) | = 0 |
| POL(mark(x1)) | = x1 |
| POL(zeros) | = 0 |
| POL(tl(x1)) | = 1 + x1 |
| POL(a__tl(x1)) | = x1 |
| POL(a__adx(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
...
→TRS12
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
mark(zeros) -> azeros
where the Polynomial interpretation:
| POL(adx(x1)) | = x1 |
| POL(a__zeros) | = 0 |
| POL(mark(x1)) | = x1 |
| POL(zeros) | = 1 |
| POL(a__adx(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
...
→TRS13
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
mark(adx(X)) -> aadx(mark(X))
where the Polynomial interpretation:
| POL(adx(x1)) | = 1 + x1 |
| POL(mark(x1)) | = x1 |
| POL(a__adx(x1)) | = x1 |
was used.
All Rules of R can be deleted.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS14
↳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
...
→TRS15
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:01 minutes