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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

G(c(x, s(y))) -> G(c(s(x), y))
F(c(s(x), y)) -> F(c(x, s(y)))
F(f(x)) -> F(d(f(x)))

Furthermore, R contains two SCCs. 


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


Dependency Pair:

G(c(x, s(y))) -> G(c(s(x), y))


Rules:


g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))
f(f(x)) -> f(d(f(x)))
f(x) -> x





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

G(c(x, s(y))) -> G(c(s(x), y))
one new Dependency Pair is created:

G(c(s(x''), s(y''))) -> G(c(s(s(x'')), y''))

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

G(c(s(x''), s(y''))) -> G(c(s(s(x'')), y''))


Rules:


g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))
f(f(x)) -> f(d(f(x)))
f(x) -> x





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

G(c(s(x''), s(y''))) -> G(c(s(s(x'')), y''))
one new Dependency Pair is created:

G(c(s(s(x'''')), s(y''''))) -> G(c(s(s(s(x''''))), y''''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 3
             ↳Inst
             ...
               →DP Problem 4
                 ↳Instantiation Transformation
       →DP Problem 2
         ↳Remaining


Dependency Pair:

G(c(s(s(x'''')), s(y''''))) -> G(c(s(s(s(x''''))), y''''))


Rules:


g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))
f(f(x)) -> f(d(f(x)))
f(x) -> x





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

G(c(s(s(x'''')), s(y''''))) -> G(c(s(s(s(x''''))), y''''))
one new Dependency Pair is created:

G(c(s(s(s(x''''''))), s(y''''''))) -> G(c(s(s(s(s(x'''''')))), y''''''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 3
             ↳Inst
             ...
               →DP Problem 5
                 ↳Instantiation Transformation
       →DP Problem 2
         ↳Remaining


Dependency Pair:

G(c(s(s(s(x''''''))), s(y''''''))) -> G(c(s(s(s(s(x'''''')))), y''''''))


Rules:


g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))
f(f(x)) -> f(d(f(x)))
f(x) -> x





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

G(c(s(s(s(x''''''))), s(y''''''))) -> G(c(s(s(s(s(x'''''')))), y''''''))
one new Dependency Pair is created:

G(c(s(s(s(s(x'''''''')))), s(y''''''''))) -> G(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 3
             ↳Inst
             ...
               →DP Problem 6
                 ↳Instantiation Transformation
       →DP Problem 2
         ↳Remaining


Dependency Pair:

G(c(s(s(s(s(x'''''''')))), s(y''''''''))) -> G(c(s(s(s(s(s(x''''''''))))), y''''''''))


Rules:


g(c(x, s(y))) -> g(c(s(x), y))
f(c(s(x), y)) -> f(c(x, s(y)))
f(f(x)) -> f(d(f(x)))
f(x) -> x





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

G(c(s(s(s(s(x'''''''')))), s(y''''''''))) -> G(c(s(s(s(s(s(x''''''''))))), y''''''''))
one new Dependency Pair is created:

G(c(s(s(s(s(s(x''''''''''))))), s(y''''''''''))) -> G(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
         ↳Remaining Obligation(s)




The following remains to be proven:


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




The following remains to be proven:

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