Term Rewriting System R:
[x, y]
f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))
Termination of R to be shown.
R
↳Removing Redundant Rules
Removing the following rules from R which fullfill a polynomial ordering:
f'(s(x), y, y) -> f'(y, x, s(x))
where the Polynomial interpretation:
| POL(g(x1)) | = x1 |
| POL(h(x1)) | = x1 |
| POL(f'(x1, x2, x3)) | = 1 + 2·x1 + x2 + x3 |
| POL(s(x1)) | = 1 + x1 |
| 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
↳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
↳OC
→TRS3
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(g(x)) -> F(f(x))
F(g(x)) -> F(x)
Furthermore, R contains one SCC.
R
↳RRRPolo
→TRS2
↳OC
→TRS3
↳DPs
...
→DP Problem 1
↳Negative Polynomial Order
Dependency Pairs:
F(g(x)) -> F(x)
F(g(x)) -> F(f(x))
Rules:
f(h(x)) -> h(g(x))
f(g(x)) -> g(f(f(x)))
Strategy:
innermost
The following Dependency Pairs can be strictly oriented using the given order.
F(g(x)) -> F(x)
F(g(x)) -> F(f(x))
Moreover, the following usable rules (regarding the implicit AFS) are oriented.
f(h(x)) -> h(g(x))
f(g(x)) -> g(f(f(x)))
Used ordering:
Polynomial Order with Interpretation:
POL( F(x1) ) = x1
POL( g(x1) ) = x1 + 1
POL( f(x1) ) = x1
POL( h(x1) ) = 0
This results in one new DP problem.
R
↳RRRPolo
→TRS2
↳OC
→TRS3
↳DPs
...
→DP Problem 2
↳Dependency Graph
Dependency Pair:
Rules:
f(h(x)) -> h(g(x))
f(g(x)) -> g(f(f(x)))
Strategy:
innermost
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes