Term Rewriting System R:
[x, y]
app(f, 0) -> true
app(f, 1) -> false
app(f, app(s, x)) -> app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) -> app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) -> app(s, y)
app(app(g, x), app(c, y)) -> app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) -> app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, 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(app(g, x), app(c, y)) -> APP(c, app(app(g, x), y))
APP(app(g, x), app(c, y)) -> APP(app(g, x), y)
APP(app(g, x), app(c, y)) -> APP(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y)))
APP(app(g, x), app(c, y)) -> APP(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, y))
APP(app(g, x), app(c, y)) -> APP(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y)))
APP(app(g, x), app(c, y)) -> APP(if, app(f, x))
APP(app(g, x), app(c, y)) -> APP(f, x)
APP(app(g, x), app(c, y)) -> APP(c, app(app(g, app(s, x)), y))
APP(app(g, x), app(c, y)) -> APP(app(g, app(s, x)), y)
APP(app(g, x), app(c, y)) -> APP(g, app(s, x))
APP(app(g, x), app(c, y)) -> APP(s, x)

Furthermore, R contains two SCCs.


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


Dependency Pair:

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


Rules:


app(f, 0) -> true
app(f, 1) -> false
app(f, app(s, x)) -> app(f, x)
app(app(app(if, true), app(s, x)), app(s, y)) -> app(s, x)
app(app(app(if, false), app(s, x)), app(s, y)) -> app(s, y)
app(app(g, x), app(c, y)) -> app(c, app(app(g, x), y))
app(app(g, x), app(c, y)) -> app(app(g, x), app(app(app(if, app(f, x)), app(c, app(app(g, app(s, x)), y))), app(c, 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)




The following remains to be proven:


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




The following remains to be proven:

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