Term Rewriting System R:
[x, y]
app(f, app(s, x)) -> app(f, x)
app(g, app(app(cons, 0), y)) -> app(g, y)
app(g, app(app(cons, app(s, x)), y)) -> app(s, x)
app(h, app(app(cons, x), y)) -> app(h, app(g, app(app(cons, x), y)))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(f, app(s, x)) -> APP(f, x)
APP(g, app(app(cons, 0), y)) -> APP(g, y)
APP(h, app(app(cons, x), y)) -> APP(h, app(g, app(app(cons, x), y)))
APP(h, app(app(cons, x), y)) -> APP(g, app(app(cons, x), y))

Furthermore, R contains three SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation
       →DP Problem 2
         ↳Remaining
       →DP Problem 3
         ↳Remaining


Dependency Pair:

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


Rules:


app(f, app(s, x)) -> app(f, x)
app(g, app(app(cons, 0), y)) -> app(g, y)
app(g, app(app(cons, app(s, x)), y)) -> app(s, x)
app(h, app(app(cons, x), y)) -> app(h, app(g, app(app(cons, x), y)))


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:


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




The following remains to be proven:


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




The following remains to be proven:

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