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

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

A(b(x)) -> A(x)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Forward Instantiation Transformation


Dependency Pair:

A(b(x)) -> A(x)


Rules:


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


Strategy:

innermost




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

A(b(x)) -> A(x)
one new Dependency Pair is created:

A(b(b(x''))) -> A(b(x''))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
Forward Instantiation Transformation


Dependency Pair:

A(b(b(x''))) -> A(b(x''))


Rules:


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


Strategy:

innermost




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

A(b(b(x''))) -> A(b(x''))
one new Dependency Pair is created:

A(b(b(b(x'''')))) -> A(b(b(x'''')))

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 3
Polynomial Ordering


Dependency Pair:

A(b(b(b(x'''')))) -> A(b(b(x'''')))


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

A(b(b(b(x'''')))) -> A(b(b(x'''')))


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(b(x1))=  1 + x1  
  POL(A(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
FwdInst
             ...
               →DP Problem 4
Dependency Graph


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes