Term Rewriting System R:
[x]
g(h(g(x))) -> g(x)
g(g(x)) -> g(h(g(x)))
h(h(x)) -> h(f(h(x), x))
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
G(g(x)) -> G(h(g(x)))
G(g(x)) -> H(g(x))
H(h(x)) -> H(f(h(x), x))
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Polynomial Ordering
Dependency Pair:
G(g(x)) -> G(h(g(x)))
Rules:
g(h(g(x))) -> g(x)
g(g(x)) -> g(h(g(x)))
h(h(x)) -> h(f(h(x), x))
Strategy:
innermost
The following dependency pair can be strictly oriented:
G(g(x)) -> G(h(g(x)))
Additionally, the following usable rules for innermost can be oriented:
h(h(x)) -> h(f(h(x), x))
g(h(g(x))) -> g(x)
g(g(x)) -> g(h(g(x)))
Used ordering: Polynomial ordering with Polynomial interpretation:
| POL(g(x1)) | = 1 |
| POL(G(x1)) | = x1 |
| POL(h(x1)) | = 0 |
| POL(f(x1, x2)) | = 0 |
resulting in one new DP problem.
R
↳DPs
→DP Problem 1
↳Polo
→DP Problem 2
↳Dependency Graph
Dependency Pair:
Rules:
g(h(g(x))) -> g(x)
g(g(x)) -> g(h(g(x)))
h(h(x)) -> h(f(h(x), x))
Strategy:
innermost
Using the Dependency Graph resulted in no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes