Term Rewriting System R:
[p, x, xs]
app(app(and, true), true) -> true
app(app(and, true), false) -> false
app(app(and, false), true) -> false
app(app(and, false), false) -> false
app(app(or, true), true) -> true
app(app(or, true), false) -> true
app(app(or, false), true) -> true
app(app(or, false), false) -> false
app(app(forall, p), nil) -> true
app(app(forall, p), app(app(cons, x), xs)) -> app(app(and, app(p, x)), app(app(forall, p), xs))
app(app(forsome, p), nil) -> false
app(app(forsome, p), app(app(cons, x), xs)) -> app(app(or, app(p, x)), app(app(forsome, p), xs))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(forall, p), app(app(cons, x), xs)) -> APP(app(and, app(p, x)), app(app(forall, p), xs))
APP(app(forall, p), app(app(cons, x), xs)) -> APP(and, app(p, x))
APP(app(forall, p), app(app(cons, x), xs)) -> APP(p, x)
APP(app(forall, p), app(app(cons, x), xs)) -> APP(app(forall, p), xs)
APP(app(forsome, p), app(app(cons, x), xs)) -> APP(app(or, app(p, x)), app(app(forsome, p), xs))
APP(app(forsome, p), app(app(cons, x), xs)) -> APP(or, app(p, x))
APP(app(forsome, p), app(app(cons, x), xs)) -> APP(p, x)
APP(app(forsome, p), app(app(cons, x), xs)) -> APP(app(forsome, p), xs)

Furthermore, R contains one SCC. 


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




The following remains to be proven:
Dependency Pairs:

APP(app(forsome, p), app(app(cons, x), xs)) -> APP(app(forsome, p), xs)
APP(app(forsome, p), app(app(cons, x), xs)) -> APP(p, x)
APP(app(forsome, p), app(app(cons, x), xs)) -> APP(app(or, app(p, x)), app(app(forsome, p), xs))
APP(app(forall, p), app(app(cons, x), xs)) -> APP(app(forall, p), xs)
APP(app(forall, p), app(app(cons, x), xs)) -> APP(p, x)
APP(app(forall, p), app(app(cons, x), xs)) -> APP(app(and, app(p, x)), app(app(forall, p), xs))


Rules:


app(app(and, true), true) -> true
app(app(and, true), false) -> false
app(app(and, false), true) -> false
app(app(and, false), false) -> false
app(app(or, true), true) -> true
app(app(or, true), false) -> true
app(app(or, false), true) -> true
app(app(or, false), false) -> false
app(app(forall, p), nil) -> true
app(app(forall, p), app(app(cons, x), xs)) -> app(app(and, app(p, x)), app(app(forall, p), xs))
app(app(forsome, p), nil) -> false
app(app(forsome, p), app(app(cons, x), xs)) -> app(app(or, app(p, x)), app(app(forsome, p), xs))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:03 minutes