Term Rewriting System R:
[X, Y]
f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(c(X, s(Y))) -> F(c(s(X), Y))
G(c(s(X), Y)) -> F(c(X, s(Y)))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Instantiation Transformation


Dependency Pair:

F(c(X, s(Y))) -> F(c(s(X), Y))


Rules:


f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))





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

F(c(X, s(Y))) -> F(c(s(X), Y))
one new Dependency Pair is created:

F(c(s(X''), s(Y''))) -> F(c(s(s(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(c(s(X''), s(Y''))) -> F(c(s(s(X'')), Y''))


Rules:


f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))





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

F(c(s(X''), s(Y''))) -> F(c(s(s(X'')), Y''))
one new Dependency Pair is created:

F(c(s(s(X'''')), s(Y''''))) -> F(c(s(s(s(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(c(s(s(X'''')), s(Y''''))) -> F(c(s(s(s(X''''))), Y''''))


Rules:


f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))





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

F(c(s(s(X'''')), s(Y''''))) -> F(c(s(s(s(X''''))), Y''''))
one new Dependency Pair is created:

F(c(s(s(s(X''''''))), s(Y''''''))) -> F(c(s(s(s(s(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(c(s(s(s(X''''''))), s(Y''''''))) -> F(c(s(s(s(s(X'''''')))), Y''''''))


Rules:


f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))





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

F(c(s(s(s(X''''''))), s(Y''''''))) -> F(c(s(s(s(s(X'''''')))), Y''''''))
one new Dependency Pair is created:

F(c(s(s(s(s(X'''''''')))), s(Y''''''''))) -> F(c(s(s(s(s(s(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(c(s(s(s(s(X'''''''')))), s(Y''''''''))) -> F(c(s(s(s(s(s(X''''''''))))), Y''''''''))


Rules:


f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))





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

F(c(s(s(s(s(X'''''''')))), s(Y''''''''))) -> F(c(s(s(s(s(s(X''''''''))))), Y''''''''))
one new Dependency Pair is created:

F(c(s(s(s(s(s(X''''''''''))))), s(Y''''''''''))) -> F(c(s(s(s(s(s(s(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(c(s(s(s(s(s(X''''''''''))))), s(Y''''''''''))) -> F(c(s(s(s(s(s(s(X'''''''''')))))), Y''''''''''))


Rules:


f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))




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