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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(x, g(y)) -> F(g(x), y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Instantiation Transformation


Dependency Pair:

F(x, g(y)) -> F(g(x), y)


Rules:


f(x, a) -> x
f(x, g(y)) -> f(g(x), y)





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

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

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

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


Rules:


f(x, a) -> x
f(x, g(y)) -> f(g(x), y)





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

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

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

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


Rules:


f(x, a) -> x
f(x, g(y)) -> f(g(x), y)





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

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

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

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


Rules:


f(x, a) -> x
f(x, g(y)) -> f(g(x), y)





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

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

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

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


Rules:


f(x, a) -> x
f(x, g(y)) -> f(g(x), y)





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

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

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

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


Rules:


f(x, a) -> x
f(x, g(y)) -> f(g(x), y)




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