Term Rewriting System R:
[x]
f(a, f(x, a)) -> f(a, f(f(a, x), f(a, a)))

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

F(a, f(x, a)) -> F(a, f(f(a, x), f(a, a)))
F(a, f(x, a)) -> F(f(a, x), f(a, a))
F(a, f(x, a)) -> F(a, x)
F(a, f(x, a)) -> F(a, a)

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Polynomial Ordering

Dependency Pairs:

F(a, f(x, a)) -> F(a, x)
F(a, f(x, a)) -> F(f(a, x), f(a, a))
F(a, f(x, a)) -> F(a, f(f(a, x), f(a, a)))

Rule:

f(a, f(x, a)) -> f(a, f(f(a, x), f(a, a)))

Strategy:

innermost

The following dependency pair can be strictly oriented:

F(a, f(x, a)) -> F(f(a, x), f(a, a))

Additionally, the following usable rule for innermost can be oriented:

f(a, f(x, a)) -> f(a, f(f(a, x), f(a, a)))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(a) =  1 POL(f(x1, x2)) =  0 POL(F(x1, x2)) =  x1

resulting in one new DP problem.

R
DPs
→DP Problem 1
Polo
→DP Problem 2
Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

F(a, f(x, a)) -> F(a, x)
F(a, f(x, a)) -> F(a, f(f(a, x), f(a, a)))

Rule:

f(a, f(x, a)) -> f(a, f(f(a, x), f(a, a)))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:00 minutes