Term Rewriting System R:
[z, x, y]
app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))
APP(app(h, z), app(e, x)) -> APP(h, app(c, z))
APP(app(h, z), app(e, x)) -> APP(c, z)
APP(app(h, z), app(e, x)) -> APP(app(d, z), x)
APP(app(h, z), app(e, x)) -> APP(d, z)
APP(app(d, z), app(app(g, 0), 0)) -> APP(e, 0)
APP(app(d, z), app(app(g, x), y)) -> APP(app(g, app(e, x)), app(app(d, z), y))
APP(app(d, z), app(app(g, x), y)) -> APP(g, app(e, x))
APP(app(d, z), app(app(g, x), y)) -> APP(e, x)
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(g, app(app(d, app(c, z)), app(app(g, x), y)))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(d, z)
APP(app(g, app(e, x)), app(e, y)) -> APP(e, app(app(g, x), y))
APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
APP(app(g, app(e, x)), app(e, y)) -> APP(g, x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
APP(app(d, z), app(app(g, x), y)) -> APP(app(g, app(e, x)), app(app(d, z), y))
APP(app(h, z), app(e, x)) -> APP(app(d, z), x)
APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))


Rules:


app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))





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

APP(app(d, z), app(app(g, x), y)) -> APP(app(g, app(e, x)), app(app(d, z), y))
three new Dependency Pairs are created:

APP(app(d, z''), app(app(g, x), app(app(g, 0), 0))) -> APP(app(g, app(e, x)), app(e, 0))
APP(app(d, z''), app(app(g, x), app(app(g, x''), y''))) -> APP(app(g, app(e, x)), app(app(g, app(e, x'')), app(app(d, z''), y'')))
APP(app(d, app(c, z'')), app(app(g, x), app(app(g, app(app(g, x''), y'')), 0))) -> APP(app(g, app(e, x)), app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y''))), app(app(d, z''), app(app(g, x''), y''))))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

APP(app(d, app(c, z'')), app(app(g, x), app(app(g, app(app(g, x''), y'')), 0))) -> APP(app(g, app(e, x)), app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y''))), app(app(d, z''), app(app(g, x''), y''))))
APP(app(d, z''), app(app(g, x), app(app(g, x''), y''))) -> APP(app(g, app(e, x)), app(app(g, app(e, x'')), app(app(d, z''), y'')))
APP(app(d, z''), app(app(g, x), app(app(g, 0), 0))) -> APP(app(g, app(e, x)), app(e, 0))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
APP(app(h, z), app(e, x)) -> APP(app(d, z), x)
APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))
APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)


Rules:


app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))





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

APP(app(d, z''), app(app(g, x), app(app(g, x''), y''))) -> APP(app(g, app(e, x)), app(app(g, app(e, x'')), app(app(d, z''), y'')))
three new Dependency Pairs are created:

APP(app(d, z'''), app(app(g, x), app(app(g, x''), app(app(g, 0), 0)))) -> APP(app(g, app(e, x)), app(app(g, app(e, x'')), app(e, 0)))
APP(app(d, z'''), app(app(g, x), app(app(g, x''), app(app(g, x'''), y')))) -> APP(app(g, app(e, x)), app(app(g, app(e, x'')), app(app(g, app(e, x''')), app(app(d, z'''), y'))))
APP(app(d, app(c, z')), app(app(g, x), app(app(g, x''), app(app(g, app(app(g, x'''), y')), 0)))) -> APP(app(g, app(e, x)), app(app(g, app(e, x'')), app(app(g, app(app(d, app(c, z')), app(app(g, x'''), y'))), app(app(d, z'), app(app(g, x'''), y')))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 3
                 ↳Polynomial Ordering


Dependency Pairs:

APP(app(d, app(c, z')), app(app(g, x), app(app(g, x''), app(app(g, app(app(g, x'''), y')), 0)))) -> APP(app(g, app(e, x)), app(app(g, app(e, x'')), app(app(g, app(app(d, app(c, z')), app(app(g, x'''), y'))), app(app(d, z'), app(app(g, x'''), y')))))
APP(app(d, z'''), app(app(g, x), app(app(g, x''), app(app(g, x'''), y')))) -> APP(app(g, app(e, x)), app(app(g, app(e, x'')), app(app(g, app(e, x''')), app(app(d, z'''), y'))))
APP(app(d, z'''), app(app(g, x), app(app(g, x''), app(app(g, 0), 0)))) -> APP(app(g, app(e, x)), app(app(g, app(e, x'')), app(e, 0)))
APP(app(d, z''), app(app(g, x), app(app(g, 0), 0))) -> APP(app(g, app(e, x)), app(e, 0))
APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
APP(app(h, z), app(e, x)) -> APP(app(d, z), x)
APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))
APP(app(d, app(c, z'')), app(app(g, x), app(app(g, app(app(g, x''), y'')), 0))) -> APP(app(g, app(e, x)), app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y''))), app(app(d, z''), app(app(g, x''), y''))))


Rules:


app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))





The following dependency pair can be strictly oriented:

APP(app(h, z), app(e, x)) -> APP(app(d, z), x)


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

app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(c)=  1  
  POL(0)=  0  
  POL(g)=  0  
  POL(e)=  0  
  POL(d)=  0  
  POL(h)=  1  
  POL(app(x1, x2))=  x1  
  POL(APP(x1, x2))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Polynomial Ordering


Dependency Pairs:

APP(app(d, app(c, z')), app(app(g, x), app(app(g, x''), app(app(g, app(app(g, x'''), y')), 0)))) -> APP(app(g, app(e, x)), app(app(g, app(e, x'')), app(app(g, app(app(d, app(c, z')), app(app(g, x'''), y'))), app(app(d, z'), app(app(g, x'''), y')))))
APP(app(d, z'''), app(app(g, x), app(app(g, x''), app(app(g, x'''), y')))) -> APP(app(g, app(e, x)), app(app(g, app(e, x'')), app(app(g, app(e, x''')), app(app(d, z'''), y'))))
APP(app(d, z'''), app(app(g, x), app(app(g, x''), app(app(g, 0), 0)))) -> APP(app(g, app(e, x)), app(app(g, app(e, x'')), app(e, 0)))
APP(app(d, z''), app(app(g, x), app(app(g, 0), 0))) -> APP(app(g, app(e, x)), app(e, 0))
APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))
APP(app(d, app(c, z'')), app(app(g, x), app(app(g, app(app(g, x''), y'')), 0))) -> APP(app(g, app(e, x)), app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y''))), app(app(d, z''), app(app(g, x''), y''))))


Rules:


app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))





The following dependency pairs can be strictly oriented:

APP(app(d, app(c, z')), app(app(g, x), app(app(g, x''), app(app(g, app(app(g, x'''), y')), 0)))) -> APP(app(g, app(e, x)), app(app(g, app(e, x'')), app(app(g, app(app(d, app(c, z')), app(app(g, x'''), y'))), app(app(d, z'), app(app(g, x'''), y')))))
APP(app(d, z'''), app(app(g, x), app(app(g, x''), app(app(g, x'''), y')))) -> APP(app(g, app(e, x)), app(app(g, app(e, x'')), app(app(g, app(e, x''')), app(app(d, z'''), y'))))
APP(app(d, z'''), app(app(g, x), app(app(g, x''), app(app(g, 0), 0)))) -> APP(app(g, app(e, x)), app(app(g, app(e, x'')), app(e, 0)))
APP(app(d, z''), app(app(g, x), app(app(g, 0), 0))) -> APP(app(g, app(e, x)), app(e, 0))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
APP(app(d, app(c, z'')), app(app(g, x), app(app(g, app(app(g, x''), y'')), 0))) -> APP(app(g, app(e, x)), app(app(g, app(app(d, app(c, z'')), app(app(g, x''), y''))), app(app(d, z''), app(app(g, x''), y''))))


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

app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(c)=  1  
  POL(0)=  0  
  POL(g)=  0  
  POL(e)=  0  
  POL(d)=  1  
  POL(h)=  1  
  POL(app(x1, x2))=  x1  
  POL(APP(x1, x2))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 5
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

APP(app(g, app(e, x)), app(e, y)) -> APP(app(g, x), y)
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, z), app(app(g, x), y))
APP(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> APP(app(d, app(c, z)), app(app(g, x), y))
APP(app(d, z), app(app(g, x), y)) -> APP(app(d, z), y)
APP(app(h, z), app(e, x)) -> APP(app(h, app(c, z)), app(app(d, z), x))


Rules:


app(app(h, z), app(e, x)) -> app(app(h, app(c, z)), app(app(d, z), x))
app(app(d, z), app(app(g, 0), 0)) -> app(e, 0)
app(app(d, z), app(app(g, x), y)) -> app(app(g, app(e, x)), app(app(d, z), y))
app(app(d, app(c, z)), app(app(g, app(app(g, x), y)), 0)) -> app(app(g, app(app(d, app(c, z)), app(app(g, x), y))), app(app(d, z), app(app(g, x), y)))
app(app(g, app(e, x)), app(e, y)) -> app(e, app(app(g, x), y))




Termination of R could not be shown.
Duration:
0:01 minutes