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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(g(x)) -> F(x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pair:

F(g(x)) -> F(x)


Rules:


f(g(x)) -> g(g(f(x)))
f(g(x)) -> g(g(g(x)))


Strategy:

innermost




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

F(g(x)) -> F(x)
one new Dependency Pair is created:

F(g(g(x''))) -> F(g(x''))

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

F(g(g(x''))) -> F(g(x''))


Rules:


f(g(x)) -> g(g(f(x)))
f(g(x)) -> g(g(g(x)))


Strategy:

innermost




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

F(g(g(x''))) -> F(g(x''))
one new Dependency Pair is created:

F(g(g(g(x'''')))) -> F(g(g(x'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Argument Filtering and Ordering


Dependency Pair:

F(g(g(g(x'''')))) -> F(g(g(x'''')))


Rules:


f(g(x)) -> g(g(f(x)))
f(g(x)) -> g(g(g(x)))


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(g(g(g(x'''')))) -> F(g(g(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(x1))=  1 + x1  
  POL(F(x1))=  1 + x1  

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


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


Dependency Pair:


Rules:


f(g(x)) -> g(g(f(x)))
f(g(x)) -> g(g(g(x)))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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