Term Rewriting System R:
[f, x, xs, y]
app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, app(p, app(s, x))), app(p, app(s, y)))
app(p, app(s, x)) -> x
app(app(div, 0), app(s, y)) -> 0
app(app(div, app(s, x)), app(s, y)) -> app(s, app(app(div, app(app(minus, x), y)), app(s, y)))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(map, f), app(app(cons, x), xs)) -> APP(cons, app(f, x))
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(minus, app(s, x)), app(s, y)) -> APP(app(minus, app(p, app(s, x))), app(p, app(s, y)))
APP(app(minus, app(s, x)), app(s, y)) -> APP(minus, app(p, app(s, x)))
APP(app(minus, app(s, x)), app(s, y)) -> APP(p, app(s, x))
APP(app(minus, app(s, x)), app(s, y)) -> APP(p, app(s, y))
APP(app(div, app(s, x)), app(s, y)) -> APP(s, app(app(div, app(app(minus, x), y)), app(s, y)))
APP(app(div, app(s, x)), app(s, y)) -> APP(app(div, app(app(minus, x), y)), app(s, y))
APP(app(div, app(s, x)), app(s, y)) -> APP(div, app(app(minus, x), y))
APP(app(div, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(div, app(s, x)), app(s, y)) -> APP(minus, x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Rewriting Transformation


Dependency Pairs:

APP(app(div, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(div, app(s, x)), app(s, y)) -> APP(app(div, app(app(minus, x), y)), app(s, y))
APP(app(minus, app(s, x)), app(s, y)) -> APP(app(minus, app(p, app(s, x))), app(p, app(s, y)))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, app(p, app(s, x))), app(p, app(s, y)))
app(p, app(s, x)) -> x
app(app(div, 0), app(s, y)) -> 0
app(app(div, app(s, x)), app(s, y)) -> app(s, app(app(div, app(app(minus, x), y)), app(s, y)))


Strategy:

innermost




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

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

APP(app(minus, app(s, x)), app(s, y)) -> APP(app(minus, x), app(p, app(s, y)))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rewriting Transformation


Dependency Pairs:

APP(app(minus, app(s, x)), app(s, y)) -> APP(app(minus, x), app(p, app(s, y)))
APP(app(div, app(s, x)), app(s, y)) -> APP(app(div, app(app(minus, x), y)), app(s, y))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(div, app(s, x)), app(s, y)) -> APP(app(minus, x), y)


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, app(p, app(s, x))), app(p, app(s, y)))
app(p, app(s, x)) -> x
app(app(div, 0), app(s, y)) -> 0
app(app(div, app(s, x)), app(s, y)) -> app(s, app(app(div, app(app(minus, x), y)), app(s, y)))


Strategy:

innermost




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

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

APP(app(minus, app(s, x)), app(s, y)) -> APP(app(minus, x), y)

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:
Dependency Pairs:

APP(app(minus, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
APP(app(map, f), app(app(cons, x), xs)) -> APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(div, app(s, x)), app(s, y)) -> APP(app(minus, x), y)
APP(app(div, app(s, x)), app(s, y)) -> APP(app(div, app(app(minus, x), y)), app(s, y))


Rules:


app(app(map, f), nil) -> nil
app(app(map, f), app(app(cons, x), xs)) -> app(app(cons, app(f, x)), app(app(map, f), xs))
app(app(minus, x), 0) -> x
app(app(minus, app(s, x)), app(s, y)) -> app(app(minus, app(p, app(s, x))), app(p, app(s, y)))
app(p, app(s, x)) -> x
app(app(div, 0), app(s, y)) -> 0
app(app(div, app(s, x)), app(s, y)) -> app(s, app(app(div, app(app(minus, x), y)), app(s, y)))


Strategy:

innermost



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