Termination w.r.t. Q proof of TRS/currying/AG01#3.13.trs

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

Q restricted rewrite system:
The TRS R consists of the following rules:

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)

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

Q restricted rewrite system:
The TRS R consists of the following rules:

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)

Q is empty.

The TRS is overlay and locally confluent. By [15] we can switch to innermost.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

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 set Q consists of the following terms:

app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(or, true), x0)
app(app(or, false), x0)
app(app(union, empty), x0)
app(app(union, app(app(app(edge, x0), x1), x2)), x3)
app(app(app(app(reach, x0), x1), empty), x2)
app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

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(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, f), xs))
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(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_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(reach, x), y)
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(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(app(eq, y), v)
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(filter, f), app(app(cons, x), xs)) → APP(filter2, app(f, 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(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → APP(reach, x)
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(eq, app(s, x)), app(s, y)) → APP(eq, x)
APP(app(eq, app(s, x)), app(s, y)) → APP(app(eq, x), y)
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(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(if_reach_1, 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(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(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(reach, v), y)
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(filter2, false), f), x), xs) → APP(app(filter, 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(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))
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_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(eq, y)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(app(eq, x), u)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(cons, x), app(app(filter, f), xs))
APP(app(union, app(app(app(edge, x), y), i)), h) → APP(union, i)
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(union, i)
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(filter2, app(f, x)), f)
APP(app(app(app(filter2, false), f), x), xs) → APP(filter, f)
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(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(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
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(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(union, app(app(app(edge, x), y), i)), h) → APP(app(union, i), h)
APP(app(app(app(filter2, true), f), x), xs) → APP(filter, f)
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(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(cons, x)
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(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(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(eq, x)

The TRS R consists of the following rules:

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 set Q consists of the following terms:

app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(or, true), x0)
app(app(or, false), x0)
app(app(union, empty), x0)
app(app(union, app(app(app(edge, x0), x1), x2)), x3)
app(app(app(app(reach, x0), x1), empty), x2)
app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

We have to consider all minimal (P,Q,R)-chains.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

Q DP problem:
The TRS P consists of the following rules:

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(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, f), xs))
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(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_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(reach, x), y)
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(app(if_reach_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(app(eq, y), v)
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(filter, f), app(app(cons, x), xs)) → APP(filter2, app(f, 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(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → APP(reach, x)
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(eq, app(s, x)), app(s, y)) → APP(eq, x)
APP(app(eq, app(s, x)), app(s, y)) → APP(app(eq, x), y)
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(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(if_reach_1, 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(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(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(reach, v), y)
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(filter2, false), f), x), xs) → APP(app(filter, 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(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))
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_1, true), x), y), app(app(app(edge, u), v), i)), h) → APP(eq, y)
APP(app(app(app(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(app(eq, x), u)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(cons, x), app(app(filter, f), xs))
APP(app(union, app(app(app(edge, x), y), i)), h) → APP(union, i)
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(union, i)
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(filter2, app(f, x)), f)
APP(app(app(app(filter2, false), f), x), xs) → APP(filter, f)
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(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(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
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(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(union, app(app(app(edge, x), y), i)), h) → APP(app(union, i), h)
APP(app(app(app(filter2, true), f), x), xs) → APP(filter, f)
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(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(cons, x)
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(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(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(eq, x)

The TRS R consists of the following rules:

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 set Q consists of the following terms:

app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(or, true), x0)
app(app(or, false), x0)
app(app(union, empty), x0)
app(app(union, app(app(app(edge, x0), x1), x2)), x3)
app(app(app(app(reach, x0), x1), empty), x2)
app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

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(map, f), app(app(cons, x), xs)) → APP(app(cons, app(f, x)), app(app(map, f), xs))
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(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_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(reach, x), y)
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(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(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(filter2, app(f, x))
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(union, app(app(app(edge, x), y), i)), h) → APP(app(app(edge, x), y), app(app(union, i), h))
APP(app(app(app(app(if_reach_1, false), x), y), app(app(app(edge, u), v), i)), h) → APP(reach, x)
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(eq, app(s, x)), app(s, y)) → APP(eq, x)
APP(app(eq, app(s, x)), app(s, y)) → APP(app(eq, x), y)
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(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(if_reach_1, 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(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(reach, v), y)
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(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(map, f), app(app(cons, x), xs)) → APP(cons, app(f, x))
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(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(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(app(eq, x), u)
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(filter2, true), f), x), xs) → APP(app(cons, x), app(app(filter, f), xs))
APP(app(union, app(app(app(edge, x), y), i)), h) → APP(union, i)
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(union, i)
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(filter2, app(f, x)), f)
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, false), 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(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(filter, 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(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(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(union, app(app(app(edge, x), y), i)), h) → APP(app(union, i), h)
APP(app(app(app(filter2, true), f), x), xs) → APP(filter, f)
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(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(cons, x)
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(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(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(eq, x)

The TRS R consists of the following rules:

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 set Q consists of the following terms:

app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(or, true), x0)
app(app(or, false), x0)
app(app(union, empty), x0)
app(app(union, app(app(app(edge, x0), x1), x2)), x3)
app(app(app(app(reach, x0), x1), empty), x2)
app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 4 SCCs with 38 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

APP(app(union, app(app(app(edge, x), y), i)), h) → APP(app(union, i), h)

The TRS R consists of the following rules:

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 set Q consists of the following terms:

app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(or, true), x0)
app(app(or, false), x0)
app(app(union, empty), x0)
app(app(union, app(app(app(edge, x0), x1), x2)), x3)
app(app(app(app(reach, x0), x1), empty), x2)
app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13]. Here, we combined the reduction pair processor with the A-transformation [14] which results in the following intermediate Q-DP Problem.
Q DP problem:
The TRS P consists of the following rules:

UNION(edge(x, y, i), h) → UNION(i, h)

R is empty.
The set Q consists of the following terms:

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
or(true, x0)
or(false, x0)
union(empty, x0)
union(edge(x0, x1, x2), x3)
reach(x0, x1, empty, x2)
reach(x0, x1, edge(x2, x3, x4), x5)
if_reach_1(true, x0, x1, edge(x2, x3, x4), x5)
if_reach_1(false, x0, x1, edge(x2, x3, x4), x5)
if_reach_2(true, x0, x1, edge(x2, x3, x4), x5)
if_reach_2(false, x0, x1, edge(x2, x3, x4), x5)
map(x0, nil)
map(x0, cons(x1, x2))
filter(x0, nil)
filter(x0, cons(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.

The following pairs can be oriented strictly and are deleted.

APP(app(union, app(app(app(edge, x), y), i)), h) → APP(app(union, i), h)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
UNION(x1, x2) = UNION(x1, x2)
edge(x1, x2, x3) = edge(x1, x3)

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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 set Q consists of the following terms:

app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(or, true), x0)
app(app(or, false), x0)
app(app(union, empty), x0)
app(app(union, app(app(app(edge, x0), x1), x2)), x3)
app(app(app(app(reach, x0), x1), empty), x2)
app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

APP(app(eq, app(s, x)), app(s, y)) → APP(app(eq, x), y)

The TRS R consists of the following rules:

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 set Q consists of the following terms:

app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(or, true), x0)
app(app(or, false), x0)
app(app(union, empty), x0)
app(app(union, app(app(app(edge, x0), x1), x2)), x3)
app(app(app(app(reach, x0), x1), empty), x2)
app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

EQ(s(x), s(y)) → EQ(x, y)

R is empty.
The set Q consists of the following terms:

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
or(true, x0)
or(false, x0)
union(empty, x0)
union(edge(x0, x1, x2), x3)
reach(x0, x1, empty, x2)
reach(x0, x1, edge(x2, x3, x4), x5)
if_reach_1(true, x0, x1, edge(x2, x3, x4), x5)
if_reach_1(false, x0, x1, edge(x2, x3, x4), x5)
if_reach_2(true, x0, x1, edge(x2, x3, x4), x5)
if_reach_2(false, x0, x1, edge(x2, x3, x4), x5)
map(x0, nil)
map(x0, cons(x1, x2))
filter(x0, nil)
filter(x0, cons(x1, x2))
filter2(true, x0, x1, x2)
filter2(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.

The following pairs can be oriented strictly and are deleted.

APP(app(eq, app(s, x)), app(s, y)) → APP(app(eq, x), y)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
EQ(x1, x2) = EQ(x2)
s(x1) = s(x1)

Lexicographic Path Order [19].
Precedence:

s₁ > EQ₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

                  ↳ QDP

                  ↳ QDP

Q DP problem:
P is empty.
The TRS R consists of the following rules:

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 set Q consists of the following terms:

app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(or, true), x0)
app(app(or, false), x0)
app(app(union, empty), x0)
app(app(union, app(app(app(edge, x0), x1), x2)), x3)
app(app(app(app(reach, x0), x1), empty), x2)
app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

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(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, 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(app(app(app(if_reach_2, app(app(eq, y), v)), x), y), app(app(app(edge, u), v), i)), h)

The TRS R consists of the following rules:

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 set Q consists of the following terms:

app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(or, true), x0)
app(app(or, false), x0)
app(app(union, empty), x0)
app(app(union, app(app(app(edge, x0), x1), x2)), x3)
app(app(app(app(reach, x0), x1), empty), x2)
app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

We have to consider all minimal (P,Q,R)-chains.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
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, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)

The TRS R consists of the following rules:

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 set Q consists of the following terms:

app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(or, true), x0)
app(app(or, false), x0)
app(app(union, empty), x0)
app(app(union, app(app(app(edge, x0), x1), x2)), x3)
app(app(app(app(reach, x0), x1), empty), x2)
app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(f, x)), f), x), xs)
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)

The remaining pairs can at least be oriented weakly.

APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)

Used ordering: Combined order from the following AFS and order.
APP(x1, x2) = APP(x2)
app(x1, x2) = app(x1, x2)
filter2 = filter2
true = true
filter = filter
map = map
cons = cons
false = false
union = union
edge = edge
nil = nil
eq = eq
0 = 0
s = s
reach = reach
if_reach_1 = if_reach_1
or = or
empty = empty
if_reach_2 = if_reach_2

Lexicographic Path Order [19].
Precedence:

filter2 > APP₁ > app₂
true > APP₁ > app₂
filter > APP₁ > app₂
cons > map > APP₁ > app₂
false > APP₁ > app₂
union > app₂
edge > app₂
eq > app₂
s > app₂
reach > app₂
if_reach₁ > app₂
if_reach₂ > app₂

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)

The TRS R consists of the following rules:

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 set Q consists of the following terms:

app(app(eq, 0), 0)
app(app(eq, 0), app(s, x0))
app(app(eq, app(s, x0)), 0)
app(app(eq, app(s, x0)), app(s, x1))
app(app(or, true), x0)
app(app(or, false), x0)
app(app(union, empty), x0)
app(app(union, app(app(app(edge, x0), x1), x2)), x3)
app(app(app(app(reach, x0), x1), empty), x2)
app(app(app(app(reach, x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_1, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, true), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(app(app(app(if_reach_2, false), x0), x1), app(app(app(edge, x2), x3), x4)), x5)
app(app(map, x0), nil)
app(app(map, x0), app(app(cons, x1), x2))
app(app(filter, x0), nil)
app(app(filter, x0), app(app(cons, x1), x2))
app(app(app(app(filter2, true), x0), x1), x2)
app(app(app(app(filter2, false), x0), x1), x2)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 2 less nodes.