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
Argument Filtering and Ordering


Dependency Pair:

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


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

A(b(x)) -> A(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(b(x1))=  1 + x1  
  POL(A(x1))=  x1  

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


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 2
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