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

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Narrowing Transformation

Dependency Pairs:

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

Rule:

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

Strategy:

innermost

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(f(y, z), f(x, f(a, x))) -> F(f(a, z), f(x, a))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

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

The following remains to be proven:
Dependency Pair:

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

Rule:

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

Strategy:

innermost

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