Term Rewriting System R:
[x, y, z, x1, x2, x3, x4]
f1 -> g1
f1 -> g2
f2 -> g1
f2 -> g2
g1 -> h1
g1 -> h2
g2 -> h1
g2 -> h2
h1 -> i
h2 -> i
e1(h1, h2, x, y, z) -> e2(x, x, y, z, z)
e1(x1, x1, x, y, z) -> e5(x1, x, y, z)
e2(f1, x, y, z, f2) -> e3(x, y, x, y, y, z, y, z, x, y, z)
e2(x, x, y, z, z) -> e6(x, y, z)
e2(i, x, y, z, i) -> e6(x, y, z)
e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) -> e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
e3(x, y, x, y, y, z, y, z, x, y, z) -> e6(x, y, z)
e4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) -> e1(x1, x1, x, y, z)
e4(i, x1, i, x1, i, x1, i, x1, x, y, z) -> e5(x1, x, y, z)
e4(x, x, x, x, x, x, x, x, x, x, x) -> e6(x, x, x)
e5(i, x, y, z) -> e6(x, y, z)
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
e1(h1, h2, x, y, z) -> e2(x, x, y, z, z)
e2(f1, x, y, z, f2) -> e3(x, y, x, y, y, z, y, z, x, y, z)
e4(g1, x1, g2, x1, g1, x1, g2, x1, x, y, z) -> e1(x1, x1, x, y, z)
R
↳RRRI
→TRS2
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f1 -> g1
f1 -> g2
where the Polynomial interpretation:
| POL(f2) | = 0 |
| POL(i) | = 0 |
| POL(h1) | = 0 |
| POL(e1(x1, x2, x3, x4, x5)) | = x1 + x2 + x3 + x4 + x5 |
| POL(e6(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(f1) | = 1 |
| POL(h2) | = 0 |
| POL(g1) | = 0 |
| POL(e5(x1, x2, x3, x4)) | = x1 + x2 + x3 + x4 |
| POL(e3(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)) | = x1 + x10 + x11 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 |
| POL(e4(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)) | = x1 + x10 + x11 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 |
| POL(g2) | = 0 |
| POL(e2(x1, x2, x3, x4, x5)) | = x1 + x2 + x3 + x4 + x5 |
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:
f2 -> g1
f2 -> g2
where the Polynomial interpretation:
| POL(f2) | = 1 |
| POL(i) | = 0 |
| POL(h1) | = 0 |
| POL(e6(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(e1(x1, x2, x3, x4, x5)) | = x1 + x2 + x3 + x4 + x5 |
| POL(e3(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)) | = x1 + x10 + x11 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 |
| POL(e4(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)) | = x1 + x10 + x11 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 |
| POL(g2) | = 0 |
| POL(h2) | = 0 |
| POL(g1) | = 0 |
| POL(e5(x1, x2, x3, x4)) | = x1 + x2 + x3 + x4 |
| POL(e2(x1, x2, x3, x4, x5)) | = x1 + x2 + x3 + x4 + x5 |
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:
h1 -> i
g1 -> h2
g2 -> h2
where the Polynomial interpretation:
| POL(i) | = 0 |
| POL(e6(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(e1(x1, x2, x3, x4, x5)) | = x1 + x2 + x3 + x4 + x5 |
| POL(h1) | = 1 |
| POL(e3(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)) | = x1 + x10 + x11 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 |
| POL(e4(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)) | = x1 + x10 + x11 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 |
| POL(g2) | = 1 |
| POL(h2) | = 0 |
| POL(g1) | = 1 |
| POL(e2(x1, x2, x3, x4, x5)) | = x1 + x2 + x3 + x4 + x5 |
| POL(e5(x1, x2, x3, x4)) | = x1 + x2 + x3 + x4 |
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:
e3(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z) -> e4(x1, x1, x2, x2, x3, x3, x4, x4, x, y, z)
e3(x, y, x, y, y, z, y, z, x, y, z) -> e6(x, y, z)
where the Polynomial interpretation:
| POL(i) | = 0 |
| POL(e6(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(e1(x1, x2, x3, x4, x5)) | = x1 + x2 + x3 + x4 + x5 |
| POL(h1) | = 0 |
| POL(e3(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)) | = 1 + x1 + x10 + x11 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 |
| POL(e4(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)) | = x1 + x10 + x11 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 |
| POL(g2) | = 0 |
| POL(h2) | = 0 |
| POL(g1) | = 0 |
| POL(e5(x1, x2, x3, x4)) | = x1 + x2 + x3 + x4 |
| POL(e2(x1, x2, x3, x4, x5)) | = x1 + x2 + x3 + x4 + x5 |
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:
e1(x1, x1, x, y, z) -> e5(x1, x, y, z)
where the Polynomial interpretation:
| POL(i) | = 0 |
| POL(e6(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(h1) | = 0 |
| POL(e1(x1, x2, x3, x4, x5)) | = 1 + x1 + x2 + x3 + x4 + x5 |
| POL(e4(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)) | = x1 + x10 + x11 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 |
| POL(g2) | = 0 |
| POL(h2) | = 0 |
| POL(g1) | = 0 |
| POL(e5(x1, x2, x3, x4)) | = x1 + x2 + x3 + x4 |
| POL(e2(x1, x2, x3, x4, x5)) | = x1 + x2 + x3 + x4 + x5 |
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:
e2(i, x, y, z, i) -> e6(x, y, z)
e2(x, x, y, z, z) -> e6(x, y, z)
where the Polynomial interpretation:
| POL(i) | = 0 |
| POL(e6(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(h1) | = 0 |
| POL(e4(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)) | = x1 + x10 + x11 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 |
| POL(g2) | = 0 |
| POL(h2) | = 0 |
| POL(g1) | = 0 |
| POL(e2(x1, x2, x3, x4, x5)) | = 1 + x1 + x2 + x3 + x4 + x5 |
| POL(e5(x1, x2, x3, x4)) | = x1 + x2 + x3 + x4 |
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
...
→TRS8
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
h2 -> i
where the Polynomial interpretation:
| POL(i) | = 0 |
| POL(e6(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(h1) | = 0 |
| POL(e4(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)) | = x1 + x10 + x11 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 |
| POL(g2) | = 0 |
| POL(h2) | = 1 |
| POL(g1) | = 0 |
| POL(e5(x1, x2, x3, x4)) | = x1 + x2 + x3 + x4 |
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
...
→TRS9
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
e4(x, x, x, x, x, x, x, x, x, x, x) -> e6(x, x, x)
e4(i, x1, i, x1, i, x1, i, x1, x, y, z) -> e5(x1, x, y, z)
where the Polynomial interpretation:
| POL(i) | = 0 |
| POL(e6(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(h1) | = 0 |
| POL(e4(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11)) | = 1 + x1 + x10 + x11 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 |
| POL(g2) | = 0 |
| POL(g1) | = 0 |
| POL(e5(x1, x2, x3, x4)) | = x1 + x2 + x3 + x4 |
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
...
→TRS10
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
g1 -> h1
where the Polynomial interpretation:
| POL(i) | = 0 |
| POL(e6(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(h1) | = 0 |
| POL(g2) | = 0 |
| POL(g1) | = 1 |
| POL(e5(x1, x2, x3, x4)) | = x1 + x2 + x3 + x4 |
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
...
→TRS11
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
e5(i, x, y, z) -> e6(x, y, z)
where the Polynomial interpretation:
| POL(i) | = 0 |
| POL(h1) | = 0 |
| POL(e6(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(g2) | = 0 |
| POL(e5(x1, x2, x3, x4)) | = 1 + x1 + x2 + x3 + x4 |
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
...
→TRS12
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
g2 -> h1
where the Polynomial interpretation:
was used.
All Rules of R can be deleted.
R
↳RRRI
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS13
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes