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
                 ↳Argument Filtering and Ordering


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




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g)=  1  
  POL(a(x1, x2))=  1 + x1 + x2  
  POL(A(x1, x2))=  x1 + x2  
  POL(f)=  0  

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


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 5
                 ↳Dependency Graph


Dependency Pair:

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




Using the Dependency Graph resulted in no new DP problems.


   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 successfully shown.
Duration:
0:00 minutes