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

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Narrowing Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

A(x, g) -> A(f, a(g, a(f, x)))
two new Dependency Pairs are created:

A(a(f, x''), g) -> A(f, a(g, a(x'', g)))
A(g, g) -> A(f, a(g, a(f, a(g, a(f, f)))))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Narrowing Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

A(x, g) -> A(g, a(f, x))
two new Dependency Pairs are created:

A(a(f, x''), g) -> A(g, a(x'', g))
A(g, g) -> A(g, a(f, a(g, a(f, f))))

The transformation is resulting in two new DP problems:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 3
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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


Dependency Pair:

A(g, g) -> A(g, a(f, a(g, a(f, f))))


Rules:


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


Strategy:

innermost




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

A(g, g) -> A(g, a(f, a(g, a(f, f))))
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.


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