Term Rewriting System R:
[X]
f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))
Innermost Termination of R to be shown.
R
↳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
↳DPs
→DP Problem 1
↳Polynomial Ordering
Dependency Pairs:
F(g(X)) -> F(X)
F(g(X)) -> F(f(X))
Rules:
f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))
Strategy:
innermost
The following dependency pairs can be strictly oriented:
F(g(X)) -> F(X)
F(g(X)) -> F(f(X))
Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:
f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(g(x1)) | = 1 + x1 |
| POL(h(x1)) | = 0 |
| POL(f(x1)) | = x1 |
| POL(F(x1)) | = 1 + x1 |
resulting in one new DP problem.
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Dependency Graph
Dependency Pair:
Rules:
f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))
Strategy:
innermost
Using the Dependency Graph resulted in no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes