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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


The following usable rules for innermost can be oriented:

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


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{a, c, b}

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> F(x1, x2)
f(x1, x2) -> f(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

F(a, f(b, x)) -> F(b, f(a, x))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 3
                 ↳Narrowing Transformation


Dependency Pairs:

F(b, f(c, x)) -> F(c, f(b, x))
F(a, f(b, f(b, x''))) -> F(b, f(b, f(a, x'')))
F(c, f(a, x)) -> F(a, f(c, x))


Rules:


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


Strategy:

innermost




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

F(b, f(c, x)) -> F(c, f(b, x))
one new Dependency Pair is created:

F(b, f(c, f(c, x''))) -> F(c, f(c, f(b, x'')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Narrowing Transformation


Dependency Pairs:

F(c, f(a, x)) -> F(a, f(c, x))
F(b, f(c, f(c, x''))) -> F(c, f(c, f(b, x'')))
F(a, f(b, f(b, x''))) -> F(b, f(b, f(a, x'')))


Rules:


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


Strategy:

innermost




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

F(c, f(a, x)) -> F(a, f(c, x))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 5
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

F(a, f(b, f(b, x''))) -> F(b, f(b, f(a, x'')))
F(c, f(a, f(a, x''))) -> F(a, f(a, f(c, x'')))
F(b, f(c, f(c, x''))) -> F(c, f(c, f(b, x'')))


Rules:


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


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:01 minutes