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
         ↳Narrowing Transformation


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)





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

APP(f, app(f, x)) -> APP(g, app(f, x))
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Polynomial Ordering


Dependency Pairs:

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


Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g)=  0  
  POL(app(x1, x2))=  1 + x2  
  POL(f)=  0  
  POL(APP(x1, x2))=  x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pair:

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