Term Rewriting System R:
[x]
h(f(f(x))) -> h(f(g(f(x))))
f(g(f(x))) -> f(f(x))
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
H(f(f(x))) -> H(f(g(f(x))))
H(f(f(x))) -> F(g(f(x)))
F(g(f(x))) -> F(f(x))
Furthermore, R contains two SCCs.
R
↳DPs
→DP Problem 1
↳Polynomial Ordering
→DP Problem 2
↳Remaining
Dependency Pair:
F(g(f(x))) -> F(f(x))
Rules:
h(f(f(x))) -> h(f(g(f(x))))
f(g(f(x))) -> f(f(x))
The following dependency pair can be strictly oriented:
F(g(f(x))) -> F(f(x))
Additionally, the following usable rule using the Ce-refinement can be oriented:
f(g(f(x))) -> f(f(x))
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(g(x1)) | = 1 |
| POL(f(x1)) | = 0 |
| POL(F(x1)) | = x1 |
resulting in one new DP problem.
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 3
↳Dependency Graph
→DP Problem 2
↳Remaining
Dependency Pair:
Rules:
h(f(f(x))) -> h(f(g(f(x))))
f(g(f(x))) -> f(f(x))
Using the Dependency Graph resulted in no new DP problems.
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Remaining Obligation(s)
The following remains to be proven:
Dependency Pair:
H(f(f(x))) -> H(f(g(f(x))))
Rules:
h(f(f(x))) -> h(f(g(f(x))))
f(g(f(x))) -> f(f(x))
Termination of R could not be shown.
Duration:
0:00 minutes