Term Rewriting System R:
[x]
s1(s1(s0(s0(x)))) -> s0(s0(s0(s1(s1(s1(x))))))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

S1(s1(s0(s0(x)))) -> S1(s1(s1(x)))
S1(s1(s0(s0(x)))) -> S1(s1(x))
S1(s1(s0(s0(x)))) -> S1(x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

S1(s1(s0(s0(x)))) -> S1(x)
S1(s1(s0(s0(x)))) -> S1(s1(x))
S1(s1(s0(s0(x)))) -> S1(s1(s1(x)))


Rule:


s1(s1(s0(s0(x)))) -> s0(s0(s0(s1(s1(s1(x))))))


Strategy:

innermost




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

S1(s1(s0(s0(x)))) -> S1(s1(s1(x)))
two new Dependency Pairs are created:

S1(s1(s0(s0(s0(s0(x'')))))) -> S1(s0(s0(s0(s1(s1(s1(x'')))))))
S1(s1(s0(s0(s1(s0(s0(x''))))))) -> S1(s1(s0(s0(s0(s1(s1(s1(x''))))))))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

S1(s1(s0(s0(s1(s0(s0(x''))))))) -> S1(s1(s0(s0(s0(s1(s1(s1(x''))))))))
S1(s1(s0(s0(x)))) -> S1(s1(x))
S1(s1(s0(s0(x)))) -> S1(x)


Rule:


s1(s1(s0(s0(x)))) -> s0(s0(s0(s1(s1(s1(x))))))


Strategy:

innermost




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

S1(s1(s0(s0(x)))) -> S1(x)
two new Dependency Pairs are created:

S1(s1(s0(s0(s1(s0(s0(x''))))))) -> S1(s1(s0(s0(x''))))
S1(s1(s0(s0(s1(s0(s0(s1(s0(s0(x'''')))))))))) -> S1(s1(s0(s0(s1(s0(s0(x'''')))))))

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:
Dependency Pairs:

S1(s1(s0(s0(s1(s0(s0(s1(s0(s0(x'''')))))))))) -> S1(s1(s0(s0(s1(s0(s0(x'''')))))))
S1(s1(s0(s0(s1(s0(s0(x''))))))) -> S1(s1(s0(s0(x''))))
S1(s1(s0(s0(x)))) -> S1(s1(x))
S1(s1(s0(s0(s1(s0(s0(x''))))))) -> S1(s1(s0(s0(s0(s1(s1(s1(x''))))))))


Rule:


s1(s1(s0(s0(x)))) -> s0(s0(s0(s1(s1(s1(x))))))


Strategy:

innermost



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