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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pair:

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


Rules:


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





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

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

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

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:
Dependency Pair:

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


Rules:


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




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