Term Rewriting System R:
[x]
f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(s(s(x))) -> F(f(x))
F(s(s(x))) -> F(x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

F(s(s(x))) -> F(x)
F(s(s(x))) -> F(f(x))


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))


Strategy:

innermost




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

F(s(s(x))) -> F(f(x))
two new Dependency Pairs are created:

F(s(s(x''))) -> F(s(x''))
F(s(s(s(s(x''))))) -> F(s(f(f(x''))))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(s(s(s(s(x''))))) -> F(s(f(f(x''))))
F(s(s(x''))) -> F(s(x''))
F(s(s(x))) -> F(x)


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))


Strategy:

innermost




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

F(s(s(x))) -> F(x)
three new Dependency Pairs are created:

F(s(s(s(s(x''))))) -> F(s(s(x'')))
F(s(s(s(s(x''''))))) -> F(s(s(x'''')))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(x'''')))))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(x'''')))))
F(s(s(s(s(x''''))))) -> F(s(s(x'''')))
F(s(s(s(s(x''))))) -> F(s(s(x'')))
F(s(s(x''))) -> F(s(x''))
F(s(s(s(s(x''))))) -> F(s(f(f(x''))))


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))


Strategy:

innermost




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

F(s(s(x''))) -> F(s(x''))
four new Dependency Pairs are created:

F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(s(s(s(x'''')))))) -> F(s(s(s(s(x'''')))))
F(s(s(s(s(s(x'''''')))))) -> F(s(s(s(s(x'''''')))))
F(s(s(s(s(s(s(s(x'''''')))))))) -> F(s(s(s(s(s(s(x'''''')))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 4
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

F(s(s(s(s(s(s(s(x'''''')))))))) -> F(s(s(s(s(s(s(x'''''')))))))
F(s(s(s(s(s(x'''''')))))) -> F(s(s(s(s(x'''''')))))
F(s(s(s(s(s(x'''')))))) -> F(s(s(s(s(x'''')))))
F(s(s(s(x'''')))) -> F(s(s(x'''')))
F(s(s(s(s(x''''))))) -> F(s(s(x'''')))
F(s(s(s(s(x''))))) -> F(s(s(x'')))
F(s(s(s(s(x''))))) -> F(s(f(f(x''))))
F(s(s(s(s(s(s(x''''))))))) -> F(s(s(s(s(x'''')))))


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))


Strategy:

innermost



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