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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Instantiation Transformation


Dependency Pair:

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


Rule:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Instantiation Transformation


Dependency Pair:

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


Rule:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 3
                 ↳Instantiation Transformation


Dependency Pair:

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


Rule:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 4
                 ↳Instantiation Transformation


Dependency Pair:

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


Rule:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 5
                 ↳Instantiation Transformation


Dependency Pair:

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


Rule:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 6
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

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


Rule:


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


Strategy:

innermost



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