Term Rewriting System R:
[x, y]
f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(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)
F'(s(x), y, y) -> F'(y, x, s(x))
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Argument Filtering and 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))
f'(s(x), y, y) -> f'(y, x, s(x))
Strategy:
innermost
The following dependency pairs can be strictly oriented:
F(g(x)) -> F(x)
F(g(x)) -> F(f(x))
The following usable rules for innermost can be oriented:
f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System: F(x1) -> F(x1)
g(x1) -> g(x1)
f(x1) -> x1
h(x1) -> h
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Dependency Graph
Dependency Pair:
Rules:
f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))
Strategy:
innermost
Using the Dependency Graph resulted in no new DP problems.
Innermost Termination of R successfully shown.
Duration:
0:00 minutes