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