Term Rewriting System R:
[x, y, z, u]
f(j(x, y), y) -> g(f(x, k(y)))
f(x, h1(y, z)) -> h2(0, x, h1(y, z))
g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))
h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
i(f(x, h(y))) -> y
i(h2(s(x), y, h1(x, z))) -> z
k(h(x)) -> h1(0, x)
k(h1(x, y)) -> h1(s(x), y)
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(j(x, y), y) -> g(f(x, k(y)))
h2(x, j(y, h1(z, u)), h1(z, u)) -> h2(s(x), y, h1(s(z), u))
i(f(x, h(y))) -> y
i(h2(s(x), y, h1(x, z))) -> z
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(i(x1)) | = 1 + x1 |
| POL(g(x1)) | = x1 |
| POL(h1(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(h(x1)) | = x1 |
| POL(j(x1, x2)) | = 1 + x1 + x2 |
| POL(f(x1, x2)) | = x1 + x2 |
| POL(h2(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(k(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:
k(h1(x, y)) -> h1(s(x), y)
k(h(x)) -> h1(0, x)
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(g(x1)) | = x1 |
| POL(h1(x1, x2)) | = x1 + x2 |
| POL(h(x1)) | = x1 |
| POL(s(x1)) | = x1 |
| POL(f(x1, x2)) | = x1 + x2 |
| POL(h2(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(k(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
↳RRRPolo
→TRS3
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f(x, h1(y, z)) -> h2(0, x, h1(y, z))
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(g(x1)) | = x1 |
| POL(h1(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(h2(x1, x2, x3)) | = x1 + x2 + x3 |
| POL(f(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
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
g(h2(x, y, h1(z, u))) -> h2(s(x), y, h1(z, u))
where the Polynomial interpretation:
| POL(g(x1)) | = 1 + x1 |
| POL(h1(x1, x2)) | = x1 + x2 |
| POL(s(x1)) | = x1 |
| POL(h2(x1, x2, x3)) | = x1 + x2 + x3 |
was used.
All Rules of R can be deleted.
R
↳RRRPolo
→TRS2
↳RRRPolo
→TRS3
↳RRRPolo
...
→TRS5
↳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
...
→TRS6
↳Dependency Pair Analysis
R contains no Dependency Pairs and therefore no SCCs.
Termination of R successfully shown.
Duration:
0:00 minutes