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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pairs:

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


Rules:


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





The following dependency pair can be strictly oriented:

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


The following usable rules w.r.t. to the AFS can be oriented:

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


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{f, g}

resulting in one new DP problem.
Used Argument Filtering System:
APP(x1, x2) -> APP(x1, x2)
app(x1, x2) -> app(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Dependency Graph


Dependency Pair:

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


Rules:


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





Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes