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