Term Rewriting System R:
[x]
app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(g, app(g, x)) -> APP(g, app(h, app(g, x)))
APP(g, app(g, x)) -> APP(h, app(g, x))
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
APP(h, app(h, x)) -> APP(f, app(h, x))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(g, app(g, x)) -> APP(h, app(g, x))
APP(g, app(g, x)) -> APP(g, app(h, app(g, x)))


Rules:


app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))





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

APP(g, app(g, x)) -> APP(g, app(h, app(g, x)))
two new Dependency Pairs are created:

APP(g, app(g, app(h, app(g, x'')))) -> APP(g, app(h, app(g, x'')))
APP(g, app(g, app(g, x''))) -> APP(g, app(h, app(g, app(h, app(g, x'')))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pairs:

APP(g, app(g, app(g, x''))) -> APP(g, app(h, app(g, app(h, app(g, x'')))))
APP(g, app(g, app(h, app(g, x'')))) -> APP(g, app(h, app(g, x'')))
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(g, app(g, x)) -> APP(h, app(g, x))
APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)


Rules:


app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))





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

APP(g, app(g, x)) -> APP(h, app(g, x))
two new Dependency Pairs are created:

APP(g, app(g, app(h, app(g, x'')))) -> APP(h, app(g, x''))
APP(g, app(g, app(g, x''))) -> APP(h, app(g, app(h, app(g, x''))))

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:
Dependency Pairs:

APP(g, app(g, app(g, x''))) -> APP(h, app(g, app(h, app(g, x''))))
APP(h, app(h, x)) -> APP(app(f, app(h, x)), x)
APP(h, app(h, x)) -> APP(h, app(app(f, app(h, x)), x))
APP(g, app(g, app(h, app(g, x'')))) -> APP(h, app(g, x''))
APP(g, app(g, app(h, app(g, x'')))) -> APP(g, app(h, app(g, x'')))
APP(g, app(g, app(g, x''))) -> APP(g, app(h, app(g, app(h, app(g, x'')))))


Rules:


app(g, app(h, app(g, x))) -> app(g, x)
app(g, app(g, x)) -> app(g, app(h, app(g, x)))
app(h, app(h, x)) -> app(h, app(app(f, app(h, x)), x))




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