Term Rewriting System R:
[x, y, z]
+(x, 0) -> x
+(x, s(y)) -> s(+(x, y))
+(0, y) -> y
+(s(x), y) -> s(+(x, y))
+(x, +(y, z)) -> +(+(x, y), z)
f(g(f(x))) -> f(h(s(0), x))
f(g(h(x, y))) -> f(h(s(x), y))
f(h(x, h(y, z))) -> f(h(+(x, y), z))
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
+(x, 0) -> x
+(0, y) -> y
where the Polynomial interpretation:
| POL(0) | = 1 |
| POL(g(x1)) | = x1 |
| POL(s(x1)) | = x1 |
| POL(h(x1, x2)) | = x1 + x2 |
| POL(+(x1, x2)) | = x1 + x2 |
| POL(f(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:
+(x, s(y)) -> s(+(x, y))
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(g(x1)) | = 1 + 2·x1 |
| POL(s(x1)) | = 1 + x1 |
| POL(h(x1, x2)) | = x1 + 2·x2 |
| POL(+(x1, x2)) | = x1 + 2·x2 |
| POL(f(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:
+(x, +(y, z)) -> +(+(x, y), z)
f(g(h(x, y))) -> f(h(s(x), y))
f(g(f(x))) -> f(h(s(0), x))
f(h(x, h(y, z))) -> f(h(+(x, y), z))
where the Polynomial interpretation:
| POL(0) | = 0 |
| POL(g(x1)) | = 2·x1 |
| POL(s(x1)) | = x1 |
| POL(h(x1, x2)) | = 1 + x1 + 2·x2 |
| POL(+(x1, x2)) | = 1 + x1 + 2·x2 |
| POL(f(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
↳RRRPolo
...
→TRS4
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
+(s(x), y) -> s(+(x, y))
where the Polynomial interpretation:
| POL(s(x1)) | = 1 + x1 |
| POL(+(x1, x2)) | = 2·x1 + x2 |
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