Term Rewriting System R:
[x, y, z]
norm(nil) -> 0
norm(g(x, y)) -> s(norm(x))
f(x, nil) -> g(nil, x)
f(x, g(y, z)) -> g(f(x, y), z)
rem(nil, y) -> nil
rem(g(x, y), 0) -> g(x, y)
rem(g(x, y), s(z)) -> rem(x, z)
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
norm(nil) -> 0
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(g(x1, x2)) | = x1 + x2 |
| POL(nil) | = 0 |
| POL(s(x1)) | = x1 |
| POL(rem(x1, x2)) | = x1 + x2 |
| POL(f(x1, x2)) | = x1 + x2 |
| POL(norm(x1)) | = 1 + 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:
norm(g(x, y)) -> s(norm(x))
rem(g(x, y), s(z)) -> rem(x, z)
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(g(x1, x2)) | = 1 + x1 + x2 |
| POL(nil) | = 0 |
| POL(rem(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(f(x1, x2)) | = 1 + x1 + x2 |
| POL(norm(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:
rem(nil, y) -> nil
rem(g(x, y), 0) -> g(x, y)
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(g(x1, x2)) | = x1 + x2 |
| POL(nil) | = 0 |
| POL(rem(x1, x2)) | = 1 + x1 + x2 |
| POL(f(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
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(x, nil) -> g(nil, x)
where the Polynomial interpretation:
| POL(g(x1, x2)) | = x1 + x2 |
| POL(nil) | = 1 |
| POL(f(x1, x2)) | = x1 + 2·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
...
→TRS5
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(x, g(y, z)) -> g(f(x, y), z)
where the Polynomial interpretation:
| POL(g(x1, x2)) | = 1 + x1 + x2 |
| POL(f(x1, x2)) | = x1 + 2·x2 |
was used.
All Rules of R can be deleted.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS6
↳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
...
→TRS7
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:00 minutes