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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




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

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

F(s(y''), s(y0)) -> F(y0, s(y''))
F(x0, s(s(x''))) -> F(s(x''), x0)

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(x0, s(s(x''))) -> F(s(x''), x0)
F(s(y''), s(y0)) -> F(y0, s(y''))
F(s(x), y) -> F(x, s(x))


Rules:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:
Dependency Pairs:

F(s(s(x'''')), y) -> F(s(x''''), s(s(x'''')))
F(s(s(y'''')), y) -> F(s(y''''), s(s(y'''')))
F(s(s(x'')), y) -> F(s(x''), s(s(x'')))
F(s(y''), s(y0)) -> F(y0, s(y''))
F(x0, s(s(x''))) -> F(s(x''), x0)


Rules:


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


Strategy:

innermost



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