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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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))) -> G(c(x, y))
two new Dependency Pairs are created:

F(c(s(x''), s(s(y'')))) -> G(c(x'', s(y'')))
F(c(s(s(x'')), s(s(y'')))) -> G(c(s(x''), s(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 Pairs:

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


Rules:


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


Strategy:

innermost




On this DP problem, a Forward 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(x'', s(s(y'')))) -> G(c(s(x''), s(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 Pairs:

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


Rules:


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


Strategy:

innermost




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

G(c(s(x), s(y))) -> F(c(x, y))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 5
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rules:


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


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:01 minutes