Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS
↳1 Overlay + Local Confluence (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
  ↳7 QDP
    ↳8 QDPOrderProof (⇔)
    ↳9 QDP
    ↳10 PisEmptyProof (⇔)
    ↳11 TRUE
  ↳12 QDP
    ↳13 QDPOrderProof (⇔)
    ↳14 QDP
    ↳15 PisEmptyProof (⇔)
    ↳16 TRUE
  ↳17 QDP
  ↳18 QDP
    ↳19 QDPOrderProof (⇔)
    ↳20 QDP
    ↳21 DependencyGraphProof (⇔)
    ↳22 TRUE

(0) Obligation:

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.

(1) Overlay + Local Confluence (EQUIVALENT transformation)

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

(2) Obligation:

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)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(4) Obligation:

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)
APP(app(eq, app(s, x)), app(s, y)) → APP(eq, 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(union, app(app(app(edge, x), y), i)), h) → APP(app(union, i), h)
APP(app(union, app(app(app(edge, x), y), i)), h) → APP(union, i)
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(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(app(app(if_reach_1, app(app(eq, x), u)), 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(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(reach, x), y), app(app(app(edge, u), v), i)), h) → APP(eq, 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(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(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(if_reach_2, app(app(eq, y), v)), 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_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(eq, y)
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(reach, x), y), i)
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(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(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(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_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(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(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(reach, x), y)
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, 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(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(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(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_2, false), x), y), app(app(app(edge, u), v), i)), h) → APP(union, i)
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(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(app(app(filter2, app(f, x)), f), x)
APP(app(filter, f), app(app(cons, x), xs)) → APP(app(filter2, app(f, x)), f)
APP(app(filter, f), app(app(cons, x), xs)) → APP(filter2, app(f, x))
APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(cons, x), app(app(filter, f), xs))
APP(app(app(app(filter2, true), f), x), xs) → APP(cons, x)
APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, true), f), x), xs) → APP(filter, f)
APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
APP(app(app(app(filter2, false), f), x), xs) → APP(filter, f)

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.

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 38 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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.

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04]. Here, we combined the reduction pair processor with the A-transformation [FROCOS05] which results in the following intermediate Q-DP Problem.
The a-transformed P is

union1(edge(x, y, i), h) → union1(i, h)

The a-transformed usable rules are
none


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.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
[union12, edge3]

Status:
union12: [2,1]
edge3: multiset


The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

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.

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE

(12) Obligation:

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)

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04]. Here, we combined the reduction pair processor with the A-transformation [FROCOS05] which results in the following intermediate Q-DP Problem.
The a-transformed P is

eq1(s(x), s(y)) → eq1(x, y)

The a-transformed usable rules are
none


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.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
[eq12, s1]

Status:
s1: [1]
eq12: [1,2]


The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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.

(15) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(16) TRUE

(17) Obligation:

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

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_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_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, 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))

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.

(18) Obligation:

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

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(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(filter, f), xs)
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)

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.

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Recursive path order with status [RPO].
Quasi-Precedence:
APP1 > [filter, false] > [true, or] > edge > app2 > filter2 > map
cons > app2 > filter2 > map
0 > map
s > eq > [filter, false] > [true, or] > edge > app2 > filter2 > map
union > edge > app2 > filter2 > map
empty > [filter, false] > [true, or] > edge > app2 > filter2 > map
reach > [ifreach1, ifreach2] > eq > [filter, false] > [true, or] > edge > app2 > filter2 > map
nil > map

Status:
APP1: [1]
eq: multiset
edge: multiset
or: multiset
union: multiset
true: multiset
empty: multiset
s: multiset
0: multiset
reach: multiset
ifreach2: multiset
filter: multiset
cons: multiset
map: multiset
ifreach1: multiset
false: multiset
app2: multiset
filter2: multiset
nil: multiset


The following usable rules [FROCOS05] were oriented: none

(20) Obligation:

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.

(21) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(22) TRUE