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
             ↳Forward 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, a Forward 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(X''''), s(s(Y'''')))) -> F(c(s(s(X'''')), s(Y'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

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