Termination Proof using AProVE

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

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

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

Rule:

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

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

Rule:

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Narrowing Transformation

Dependency Pairs:

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

Rule:

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Narrowing Transformation

Dependency Pairs:

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

Rule:

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 5

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Rule:

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

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

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

no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 6

                 ↳Narrowing Transformation

Dependency Pair:

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

Rule:

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 7

                 ↳Narrowing Transformation

Dependency Pair:

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

Rule:

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 8

                 ↳Narrowing Transformation

Dependency Pair:

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

Rule:

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

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

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

one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 9

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pair:

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

Rule:

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

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