Term Rewriting System R:
[x]
f(g(x)) -> f(a(g(g(f(x))), g(f(x))))
Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(g(x)) -> F(a(g(g(f(x))), g(f(x))))
F(g(x)) -> F(x)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
Dependency Pair:
F(g(x)) -> F(x)
Rule:
f(g(x)) -> f(a(g(g(f(x))), g(f(x))))
The following dependency pair can be strictly oriented:
F(g(x)) -> F(x)
The following rule can be oriented:
f(g(x)) -> f(a(g(g(f(x))), g(f(x))))
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(g(x1)) | = 1 + x1 |
| POL(F(x1)) | = 1 + x1 |
| POL(f(x1)) | = x1 |
resulting in one new DP problem.
Used Argument Filtering System: F(x1) -> F(x1)
g(x1) -> g(x1)
f(x1) -> f(x1)
a(x1, x2) -> x2
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Dependency Graph
Dependency Pair:
Rule:
f(g(x)) -> f(a(g(g(f(x))), g(f(x))))
Using the Dependency Graph resulted in no new DP problems.
Termination of R successfully shown.
Duration:
0:00 minutes