Term Rewriting System R:
[x]
f(a, h(x)) -> f(g(x), h(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, h(x)) -> F(g(x), h(x))
F(a, h(x)) -> G(x)
H(g(x)) -> H(a)
G(h(x)) -> G(x)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
Dependency Pair:
F(a, h(x)) -> F(g(x), h(x))
Rules:
f(a, h(x)) -> f(g(x), h(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, h(x)) -> F(g(x), h(x))
The following usable rule for innermost w.r.t. to the AFS can be oriented:
g(h(x)) -> g(x)
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
a > g
resulting in one new DP problem.
Used Argument Filtering System: F(x1, x2) -> x1
g(x1) -> g
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Dependency Graph
Dependency Pair:
Rules:
f(a, h(x)) -> f(g(x), h(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