Term Rewriting System R:
[X, Y, X1, X2, X3, Z]
aand(true, X) -> mark(X)
aand(false, Y) -> false
aand(X1, X2) -> and(X1, X2)
aif(true, X, Y) -> mark(X)
aif(false, X, Y) -> mark(Y)
aif(X1, X2, X3) -> if(X1, X2, X3)
aadd(0, X) -> mark(X)
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
afirst(0, X) -> nil
afirst(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
afirst(X1, X2) -> first(X1, X2)
afrom(X) -> cons(X, from(s(X)))
afrom(X) -> from(X)
mark(and(X1, X2)) -> aand(mark(X1), X2)
mark(if(X1, X2, X3)) -> aif(mark(X1), X2, X3)
mark(add(X1, X2)) -> aadd(mark(X1), X2)
mark(first(X1, X2)) -> afirst(mark(X1), mark(X2))
mark(from(X)) -> afrom(X)
mark(true) -> true
mark(false) -> false
mark(0) -> 0
mark(s(X)) -> s(X)
mark(nil) -> nil
mark(cons(X1, X2)) -> cons(X1, X2)
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
aand(true, X) -> mark(X)
aand(false, Y) -> false
mark(and(X1, X2)) -> aand(mark(X1), X2)
where the Polynomial interpretation:
| POL(from(x1)) | = x1 |
| POL(and(x1, x2)) | = 1 + x1 + x2 |
| POL(a__and(x1, x2)) | = 1 + x1 + 2·x2 |
| POL(false) | = 0 |
| POL(true) | = 0 |
| POL(mark(x1)) | = 2·x1 |
| POL(a__from(x1)) | = 2·x1 |
| POL(a__add(x1, x2)) | = x1 + 2·x2 |
| POL(a__first(x1, x2)) | = x1 + x2 |
| POL(add(x1, x2)) | = x1 + x2 |
| POL(if(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(first(x1, x2)) | = x1 + x2 |
| POL(0) | = 0 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(a__if(x1, x2, x3)) | = x1 + 2·x2 + 2·x3 |
| POL(nil) | = 0 |
| POL(s(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:
afirst(0, X) -> nil
mark(0) -> 0
aadd(0, X) -> mark(X)
where the Polynomial interpretation:
| POL(from(x1)) | = x1 |
| POL(and(x1, x2)) | = x1 + x2 |
| POL(a__and(x1, x2)) | = x1 + x2 |
| POL(false) | = 0 |
| POL(true) | = 0 |
| POL(mark(x1)) | = 2·x1 |
| POL(a__add(x1, x2)) | = x1 + 2·x2 |
| POL(a__from(x1)) | = 2·x1 |
| POL(a__first(x1, x2)) | = x1 + x2 |
| POL(add(x1, x2)) | = x1 + x2 |
| POL(if(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(0) | = 1 |
| POL(first(x1, x2)) | = x1 + x2 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(a__if(x1, x2, x3)) | = x1 + 2·x2 + 2·x3 |
| POL(nil) | = 0 |
| POL(s(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:
aif(false, X, Y) -> mark(Y)
mark(false) -> false
where the Polynomial interpretation:
| POL(from(x1)) | = x1 |
| POL(and(x1, x2)) | = x1 + x2 |
| POL(a__and(x1, x2)) | = x1 + x2 |
| POL(false) | = 1 |
| POL(true) | = 0 |
| POL(mark(x1)) | = 2·x1 |
| POL(a__from(x1)) | = 2·x1 |
| POL(a__add(x1, x2)) | = x1 + x2 |
| POL(a__first(x1, x2)) | = x1 + x2 |
| POL(add(x1, x2)) | = x1 + x2 |
| POL(if(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(first(x1, x2)) | = x1 + x2 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(a__if(x1, x2, x3)) | = x1 + 2·x2 + 2·x3 |
| POL(nil) | = 0 |
| POL(s(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
...
→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(from(x1)) | = x1 |
| POL(and(x1, x2)) | = x1 + x2 |
| POL(a__and(x1, x2)) | = x1 + x2 |
| POL(true) | = 0 |
| POL(mark(x1)) | = 2·x1 |
| POL(a__from(x1)) | = 2·x1 |
| POL(a__add(x1, x2)) | = x1 + x2 |
| POL(a__first(x1, x2)) | = x1 + x2 |
| POL(add(x1, x2)) | = x1 + x2 |
| POL(if(x1, x2, x3)) | = 1 + x1 + x2 + x3 |
| POL(first(x1, x2)) | = x1 + x2 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(a__if(x1, x2, x3)) | = 2 + x1 + 2·x2 + x3 |
| POL(nil) | = 0 |
| POL(s(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:
afrom(X) -> cons(X, from(s(X)))
afrom(X) -> from(X)
where the Polynomial interpretation:
| POL(from(x1)) | = 1 + x1 |
| POL(and(x1, x2)) | = x1 + x2 |
| POL(a__and(x1, x2)) | = x1 + x2 |
| POL(true) | = 0 |
| POL(mark(x1)) | = 2·x1 |
| POL(a__from(x1)) | = 2 + 2·x1 |
| POL(a__add(x1, x2)) | = x1 + x2 |
| POL(a__first(x1, x2)) | = x1 + x2 |
| POL(add(x1, x2)) | = x1 + x2 |
| POL(if(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(first(x1, x2)) | = x1 + x2 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(a__if(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(nil) | = 0 |
| POL(s(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(first(X1, X2)) -> afirst(mark(X1), mark(X2))
where the Polynomial interpretation:
| POL(from(x1)) | = x1 |
| POL(and(x1, x2)) | = x1 + x2 |
| POL(a__and(x1, x2)) | = x1 + x2 |
| POL(true) | = 0 |
| POL(mark(x1)) | = 2·x1 |
| POL(a__from(x1)) | = x1 |
| POL(a__add(x1, x2)) | = x1 + x2 |
| POL(a__first(x1, x2)) | = 1 + x1 + x2 |
| POL(add(x1, x2)) | = x1 + x2 |
| POL(if(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(first(x1, x2)) | = 1 + x1 + x2 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(a__if(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(nil) | = 0 |
| POL(s(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:
afirst(X1, X2) -> first(X1, X2)
afirst(s(X), cons(Y, Z)) -> cons(Y, first(X, Z))
where the Polynomial interpretation:
| POL(from(x1)) | = x1 |
| POL(and(x1, x2)) | = x1 + x2 |
| POL(a__and(x1, x2)) | = x1 + x2 |
| POL(true) | = 0 |
| POL(mark(x1)) | = x1 |
| POL(a__add(x1, x2)) | = x1 + x2 |
| POL(a__from(x1)) | = x1 |
| POL(a__first(x1, x2)) | = 1 + x1 + x2 |
| POL(add(x1, x2)) | = x1 + x2 |
| POL(if(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(first(x1, x2)) | = x1 + x2 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(a__if(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(nil) | = 0 |
| POL(s(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:
mark(if(X1, X2, X3)) -> aif(mark(X1), X2, X3)
where the Polynomial interpretation:
| POL(from(x1)) | = x1 |
| POL(and(x1, x2)) | = x1 + x2 |
| POL(a__and(x1, x2)) | = x1 + x2 |
| POL(true) | = 0 |
| POL(mark(x1)) | = x1 |
| POL(a__add(x1, x2)) | = x1 + x2 |
| POL(a__from(x1)) | = x1 |
| POL(add(x1, x2)) | = x1 + x2 |
| POL(if(x1, x2, x3)) | = 1 + x1 + x2 + x3 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(a__if(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(nil) | = 0 |
| POL(s(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:
mark(true) -> true
where the Polynomial interpretation:
| POL(from(x1)) | = x1 |
| POL(and(x1, x2)) | = x1 + x2 |
| POL(a__and(x1, x2)) | = x1 + x2 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(nil) | = 0 |
| POL(true) | = 1 |
| POL(s(x1)) | = x1 |
| POL(mark(x1)) | = 2·x1 |
| POL(a__add(x1, x2)) | = x1 + x2 |
| POL(a__from(x1)) | = x1 |
| POL(add(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
...
→TRS10
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
mark(add(X1, X2)) -> aadd(mark(X1), X2)
where the Polynomial interpretation:
| POL(from(x1)) | = x1 |
| POL(and(x1, x2)) | = x1 + x2 |
| POL(a__and(x1, x2)) | = x1 + x2 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(nil) | = 0 |
| POL(s(x1)) | = x1 |
| POL(mark(x1)) | = 2·x1 |
| POL(a__from(x1)) | = x1 |
| POL(a__add(x1, x2)) | = 1 + x1 + x2 |
| POL(add(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
...
→TRS11
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
mark(nil) -> nil
where the Polynomial interpretation:
| POL(from(x1)) | = x1 |
| POL(and(x1, x2)) | = x1 + x2 |
| POL(a__and(x1, x2)) | = x1 + x2 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(nil) | = 1 |
| POL(s(x1)) | = x1 |
| POL(mark(x1)) | = 2·x1 |
| POL(a__from(x1)) | = x1 |
| POL(a__add(x1, x2)) | = x1 + x2 |
| POL(add(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
...
→TRS12
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
mark(s(X)) -> s(X)
mark(cons(X1, X2)) -> cons(X1, X2)
mark(from(X)) -> afrom(X)
where the Polynomial interpretation:
| POL(from(x1)) | = x1 |
| POL(and(x1, x2)) | = x1 + x2 |
| POL(a__and(x1, x2)) | = x1 + x2 |
| POL(cons(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(mark(x1)) | = 1 + x1 |
| POL(a__from(x1)) | = x1 |
| POL(a__add(x1, x2)) | = x1 + x2 |
| POL(add(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
...
→TRS13
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
aadd(s(X), Y) -> s(add(X, Y))
aadd(X1, X2) -> add(X1, X2)
where the Polynomial interpretation:
| POL(and(x1, x2)) | = x1 + x2 |
| POL(a__and(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(a__add(x1, x2)) | = 1 + x1 + x2 |
| POL(add(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
...
→TRS14
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
aand(X1, X2) -> and(X1, X2)
where the Polynomial interpretation:
| POL(and(x1, x2)) | = x1 + x2 |
| POL(a__and(x1, x2)) | = 1 + x1 + x2 |
was used.
All Rules of R can be deleted.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS15
↳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
...
→TRS16
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:01 minutes