Term Rewriting System R:
[x]
f(a, f(x, a)) -> f(f(a, f(f(a, a), x)), a)
Innermost Termination of R to be shown.
R
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
F(a, f(x, a)) -> F(f(a, f(f(a, a), x)), a)
F(a, f(x, a)) -> F(a, f(f(a, a), x))
F(a, f(x, a)) -> F(f(a, a), x)
F(a, f(x, a)) -> F(a, a)
Furthermore, R contains one SCC.
R
↳DPs
→DP Problem 1
↳Argument Filtering and Ordering
Dependency Pairs:
F(a, f(x, a)) -> F(f(a, a), x)
F(a, f(x, a)) -> F(a, f(f(a, a), x))
Rule:
f(a, f(x, a)) -> f(f(a, f(f(a, a), x)), a)
Strategy:
innermost
The following dependency pair can be strictly oriented:
F(a, f(x, a)) -> F(f(a, a), x)
The following usable rule for innermost w.r.t. to the AFS can be oriented:
f(a, f(x, a)) -> f(f(a, f(f(a, a), x)), a)
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
a > f
resulting in one new DP problem.
Used Argument Filtering System: F(x1, x2) -> x1
f(x1, x2) -> f
R
↳DPs
→DP Problem 1
↳AFS
→DP Problem 2
↳Remaining Obligation(s)
The following remains to be proven:
Dependency Pair:
F(a, f(x, a)) -> F(a, f(f(a, a), x))
Rule:
f(a, f(x, a)) -> f(f(a, f(f(a, a), x)), a)
Strategy:
innermost
Innermost Termination of R could not be shown.
Duration:
0:00 minutes