app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, x)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(or, true), y) → true
app(app(or, false), y) → y
app(app(union, empty), h) → h
app(app(union, app(app(app(edge, x), y), i)), h) → app(app(app(edge, x), y), app(app(union, i), h))
app(app(app(app(reach, x), y), empty), h) → false
app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h))
app(app(app(app(app(if_reach_2, true), x), y), app(app(app(edge, u), v), i)), h) → true
app(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)
↳ QTRS
↳ DependencyPairsProof
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, x)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(or, true), y) → true
app(app(or, false), y) → y
app(app(union, empty), h) → h
app(app(union, app(app(app(edge, x), y), i)), h) → app(app(app(edge, x), y), app(app(union, i), h))
app(app(app(app(reach, x), y), empty), h) → false
app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h))
app(app(app(app(app(if_reach_2, true), x), y), app(app(app(edge, u), v), i)), h) → true
app(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(filter2, app(f, x)), f)
APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h)
APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i))
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(union, i), h)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(f, x)), f), x), xs)
APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(if_reach_2, app(app(eq, y), v)), x), y)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(app(if_reach_1, app(app(eq, x), u)), x)
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(reach, v), y)
APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(reach, x), y), i)
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(reach, x), y), i)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i))
APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(eq, y)
APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(reach, x), y)
APP(app(app(app(filter2, false), f), x), xs) → APP(filter, f)
APP(app(union, app(app(app(edge, x), y), i)), h) → APP(app(union, i), h)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(filter2, app(f, x)), f), x)
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(or, app(app(app(app(reach, x), y), i), h))
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(reach, x)
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(reach, x), y), i), h)
APP(app(map, f), app(app(cons, x), xs)) → APP(cons, 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(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(app(eq, y), v)
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h))
APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(edge, u), v), h)
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(union, i)
APP(app(eq, app(s, x)), app(s, y)) → APP(eq, x)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(if_reach_1, app(app(eq, x), u)), x), y)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(union, app(app(app(edge, x), y), i)), h) → APP(union, i)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(eq, x)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(if_reach_1, app(app(eq, x), u))
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(app(eq, x), u)
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(reach, x), y)
APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(if_reach_2, app(app(eq, y), v))
APP(app(eq, app(s, x)), app(s, y)) → APP(app(eq, x), y)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(reach, v), y), app(app(union, i), h))
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(reach, v)
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty))
APP(app(app(app(filter2, true), f), x), xs) → APP(cons, x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(filter2, app(f, x))
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(reach, v), y), app(app(union, i), h)), empty)
APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(app(if_reach_2, app(app(eq, y), v)), x)
APP(app(app(app(filter2, true), f), x), xs) → APP(filter, f)
APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → APP(reach, x)
APP(app(union, app(app(app(edge, x), y), i)), h) → APP(app(app(edge, x), y), app(app(union, i), h))
APP(app(app(app(filter2, true), f), x), xs) → APP(app(cons, x), app(app(filter, f), xs))
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, x)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(or, true), y) → true
app(app(or, false), y) → y
app(app(union, empty), h) → h
app(app(union, app(app(app(edge, x), y), i)), h) → app(app(app(edge, x), y), app(app(union, i), h))
app(app(app(app(reach, x), y), empty), h) → false
app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h))
app(app(app(app(app(if_reach_2, true), x), y), app(app(app(edge, u), v), i)), h) → true
app(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(filter2, app(f, x)), f)
APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h)
APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i))
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(union, i), h)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(f, x)), f), x), xs)
APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(if_reach_2, app(app(eq, y), v)), x), y)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(app(if_reach_1, app(app(eq, x), u)), x)
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(reach, v), y)
APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(reach, x), y), i)
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(reach, x), y), i)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i))
APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(eq, y)
APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(reach, x), y)
APP(app(app(app(filter2, false), f), x), xs) → APP(filter, f)
APP(app(union, app(app(app(edge, x), y), i)), h) → APP(app(union, i), h)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(filter2, app(f, x)), f), x)
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(or, app(app(app(app(reach, x), y), i), h))
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(reach, x)
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(reach, x), y), i), h)
APP(app(map, f), app(app(cons, x), xs)) → APP(cons, 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(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(app(eq, y), v)
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h))
APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(edge, u), v), h)
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(union, i)
APP(app(eq, app(s, x)), app(s, y)) → APP(eq, x)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(if_reach_1, app(app(eq, x), u)), x), y)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(union, app(app(app(edge, x), y), i)), h) → APP(union, i)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(eq, x)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(if_reach_1, app(app(eq, x), u))
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(app(eq, x), u)
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(reach, x), y)
APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(if_reach_2, app(app(eq, y), v))
APP(app(eq, app(s, x)), app(s, y)) → APP(app(eq, x), y)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(reach, v), y), app(app(union, i), h))
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(reach, v)
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty))
APP(app(app(app(filter2, true), f), x), xs) → APP(cons, x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(filter2, app(f, x))
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(reach, v), y), app(app(union, i), h)), empty)
APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(app(if_reach_2, app(app(eq, y), v)), x)
APP(app(app(app(filter2, true), f), x), xs) → APP(filter, f)
APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → APP(reach, x)
APP(app(union, app(app(app(edge, x), y), i)), h) → APP(app(app(edge, x), y), app(app(union, i), h))
APP(app(app(app(filter2, true), f), x), xs) → APP(app(cons, x), app(app(filter, f), xs))
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, x)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(or, true), y) → true
app(app(or, false), y) → y
app(app(union, empty), h) → h
app(app(union, app(app(app(edge, x), y), i)), h) → app(app(app(edge, x), y), app(app(union, i), h))
app(app(app(app(reach, x), y), empty), h) → false
app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h))
app(app(app(app(app(if_reach_2, true), x), y), app(app(app(edge, u), v), i)), h) → true
app(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
APP(app(union, app(app(app(edge, x), y), i)), h) → APP(app(union, i), h)
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, x)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(or, true), y) → true
app(app(or, false), y) → y
app(app(union, empty), h) → h
app(app(union, app(app(app(edge, x), y), i)), h) → app(app(app(edge, x), y), app(app(union, i), h))
app(app(app(app(reach, x), y), empty), h) → false
app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h))
app(app(app(app(app(if_reach_2, true), x), y), app(app(app(edge, u), v), i)), h) → true
app(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(app(union, app(app(app(edge, x), y), i)), h) → APP(app(union, i), h)
The value of delta used in the strict ordering is 8.
POL(APP(x1, x2)) = (2)x_1
POL(edge) = 0
POL(union) = 0
POL(app(x1, x2)) = 2 + (2)x_2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, x)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(or, true), y) → true
app(app(or, false), y) → y
app(app(union, empty), h) → h
app(app(union, app(app(app(edge, x), y), i)), h) → app(app(app(edge, x), y), app(app(union, i), h))
app(app(app(app(reach, x), y), empty), h) → false
app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h))
app(app(app(app(app(if_reach_2, true), x), y), app(app(app(edge, u), v), i)), h) → true
app(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
APP(app(eq, app(s, x)), app(s, y)) → APP(app(eq, x), y)
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, x)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(or, true), y) → true
app(app(or, false), y) → y
app(app(union, empty), h) → h
app(app(union, app(app(app(edge, x), y), i)), h) → app(app(app(edge, x), y), app(app(union, i), h))
app(app(app(app(reach, x), y), empty), h) → false
app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h))
app(app(app(app(app(if_reach_2, true), x), y), app(app(app(edge, u), v), i)), h) → true
app(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
APP(app(eq, app(s, x)), app(s, y)) → APP(app(eq, x), y)
The value of delta used in the strict ordering is 1/2.
POL(eq) = 0
POL(APP(x1, x2)) = (1/4)x_1
POL(app(x1, x2)) = 1 + (2)x_2
POL(s) = 0
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, x)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(or, true), y) → true
app(app(or, false), y) → y
app(app(union, empty), h) → h
app(app(union, app(app(app(edge, x), y), i)), h) → app(app(app(edge, x), y), app(app(union, i), h))
app(app(app(app(reach, x), y), empty), h) → false
app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h))
app(app(app(app(app(if_reach_2, true), x), y), app(app(app(edge, u), v), i)), h) → true
app(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h))
APP(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h)
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(reach, v), y), app(app(union, i), h)), empty)
APP(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(reach, x), y), i), h)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h)
app(app(eq, 0), 0) → true
app(app(eq, 0), app(s, x)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(or, true), y) → true
app(app(or, false), y) → y
app(app(union, empty), h) → h
app(app(union, app(app(app(edge, x), y), i)), h) → app(app(app(edge, x), y), app(app(union, i), h))
app(app(app(app(reach, x), y), empty), h) → false
app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h))
app(app(app(app(app(if_reach_2, true), x), y), app(app(app(edge, u), v), i)), h) → true
app(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(f, x)), f), x), xs)
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(eq, 0), 0) → true
app(app(eq, 0), app(s, x)) → false
app(app(eq, app(s, x)), 0) → false
app(app(eq, app(s, x)), app(s, y)) → app(app(eq, x), y)
app(app(or, true), y) → true
app(app(or, false), y) → y
app(app(union, empty), h) → h
app(app(union, app(app(app(edge, x), y), i)), h) → app(app(app(edge, x), y), app(app(union, i), h))
app(app(app(app(reach, x), y), empty), h) → false
app(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_1, app(app(eq, x), u)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h)
app(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(app(app(reach, x), y), i), app(app(app(edge, u), v), h))
app(app(app(app(app(if_reach_2, true), x), y), app(app(app(edge, u), v), i)), h) → true
app(app(app(app(app(if_reach_2, false), x), y), app(app(app(edge, u), v), i)), h) → app(app(or, app(app(app(app(reach, x), y), i), h)), app(app(app(app(reach, v), y), app(app(union, i), h)), empty))
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(filter, f), nil) → nil
app(app(filter, f), app(app(cons, x), xs)) → app(app(app(app(filter2, app(f, x)), f), x), xs)
app(app(app(app(filter2, true), f), x), xs) → app(app(cons, x), app(app(filter, f), xs))
app(app(app(app(filter2, false), f), x), xs) → app(app(filter, f), xs)