R
↳Dependency Pair Analysis
APP(app(f, 0), n) -> APP(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
APP(app(f, 0), n) -> APP(hd, app(app(map, f), app(app(cons, 0), nil)))
APP(app(f, 0), n) -> APP(app(map, f), app(app(cons, 0), nil))
APP(app(f, 0), n) -> APP(map, f)
APP(app(f, 0), n) -> APP(app(cons, 0), nil)
APP(app(f, 0), n) -> APP(cons, 0)
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)
R
↳DPs
→DP Problem 1
↳Narrowing Transformation
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(f, 0), n) -> APP(app(cons, 0), nil)
APP(app(f, 0), n) -> APP(app(map, f), app(app(cons, 0), nil))
APP(app(f, 0), n) -> APP(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
app(app(f, 0), n) -> app(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
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))
innermost
three new Dependency Pairs are created:
APP(app(f, 0), n) -> APP(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
APP(app(0, 0), n) -> APP(app(hd, app(app(hd, app(app(map, map), app(app(cons, 0), nil))), app(app(cons, 0), nil))), n)
APP(app(f'', 0), n) -> APP(app(hd, app(app(cons, app(f'', 0)), app(app(map, f''), nil))), n)
APP(app(f, 0), n) -> APP(app(hd, app(app(map, f), app(app(hd, app(app(map, cons), app(app(cons, 0), nil))), nil))), n)
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Narrowing Transformation
APP(app(f, 0), n) -> APP(app(hd, app(app(map, f), app(app(hd, app(app(map, cons), app(app(cons, 0), nil))), nil))), n)
APP(app(f'', 0), n) -> APP(app(hd, app(app(cons, app(f'', 0)), app(app(map, f''), nil))), n)
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(0, 0), n) -> APP(app(hd, app(app(hd, app(app(map, map), app(app(cons, 0), nil))), app(app(cons, 0), nil))), n)
APP(app(f, 0), n) -> APP(app(cons, 0), nil)
APP(app(f, 0), n) -> APP(app(map, f), app(app(cons, 0), nil))
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(map, f), xs)
app(app(f, 0), n) -> app(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
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))
innermost
one new Dependency Pair is created:
APP(app(f, 0), n) -> APP(app(map, f), app(app(cons, 0), nil))
APP(app(f, 0), n) -> APP(app(map, f), app(app(hd, app(app(map, cons), app(app(cons, 0), nil))), nil))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 3
↳Narrowing Transformation
APP(app(f, 0), n) -> APP(app(map, f), app(app(hd, app(app(map, cons), app(app(cons, 0), nil))), nil))
APP(app(f'', 0), n) -> APP(app(hd, app(app(cons, app(f'', 0)), app(app(map, f''), nil))), n)
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(0, 0), n) -> APP(app(hd, app(app(hd, app(app(map, map), app(app(cons, 0), nil))), app(app(cons, 0), nil))), n)
APP(app(f, 0), n) -> APP(app(cons, 0), nil)
APP(app(f, 0), n) -> APP(app(hd, app(app(map, f), app(app(hd, app(app(map, cons), app(app(cons, 0), nil))), nil))), n)
app(app(f, 0), n) -> app(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
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))
innermost
six new Dependency Pairs are created:
APP(app(map, f), app(app(cons, x), xs)) -> APP(app(cons, app(f, x)), app(app(map, f), xs))
APP(app(map, app(f'', 0)), app(app(cons, x'), xs)) -> APP(app(cons, app(app(hd, app(app(map, f''), app(app(cons, 0), nil))), x')), app(app(map, app(f'', 0)), xs))
APP(app(map, app(map, f'')), app(app(cons, nil), xs)) -> APP(app(cons, nil), app(app(map, app(map, f'')), xs))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(map, f'')), xs))
APP(app(map, 0), app(app(cons, x), xs')) -> APP(app(cons, app(0, x)), app(app(hd, app(app(map, map), app(app(cons, 0), nil))), xs'))
APP(app(map, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))
R
↳DPs
→DP Problem 1
↳Nar
→DP Problem 2
↳Nar
...
→DP Problem 4
↳Remaining Obligation(s)
APP(app(map, f''), app(app(cons, x), app(app(cons, x''), xs''))) -> APP(app(cons, app(f'', x)), app(app(cons, app(f'', x'')), app(app(map, f''), xs'')))
APP(app(map, f''), app(app(cons, x), nil)) -> APP(app(cons, app(f'', x)), nil)
APP(app(map, 0), app(app(cons, x), xs')) -> APP(app(cons, app(0, x)), app(app(hd, app(app(map, map), app(app(cons, 0), nil))), xs'))
APP(app(map, app(map, f'')), app(app(cons, app(app(cons, x''), xs'')), xs)) -> APP(app(cons, app(app(cons, app(f'', x'')), app(app(map, f''), xs''))), app(app(map, app(map, f'')), xs))
APP(app(map, app(f'', 0)), app(app(cons, x'), xs)) -> APP(app(cons, app(app(hd, app(app(map, f''), app(app(cons, 0), nil))), x')), app(app(map, app(f'', 0)), xs))
APP(app(f, 0), n) -> APP(app(hd, app(app(map, f), app(app(hd, app(app(map, cons), app(app(cons, 0), nil))), nil))), n)
APP(app(f'', 0), n) -> APP(app(hd, app(app(cons, app(f'', 0)), app(app(map, f''), nil))), n)
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(0, 0), n) -> APP(app(hd, app(app(hd, app(app(map, map), app(app(cons, 0), nil))), app(app(cons, 0), nil))), n)
APP(app(f, 0), n) -> APP(app(cons, 0), nil)
APP(app(f, 0), n) -> APP(app(map, f), app(app(hd, app(app(map, cons), app(app(cons, 0), nil))), nil))
app(app(f, 0), n) -> app(app(hd, app(app(map, f), app(app(cons, 0), nil))), n)
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))
innermost