YES Termination w.r.t. Q proof of Applicative_first_order_05_3.13.ari

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 Overlay + Local Confluence (⇔, 0 ms)
↳2 QTRS
↳3 DependencyPairsProof (⇔, 73 ms)
↳4 QDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
    ↳7 QDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 QDP
        ↳10 ATransformationProof (⇔, 0 ms)
        ↳11 QDP
        ↳12 QReductionProof (⇔, 0 ms)
        ↳13 QDP
        ↳14 QDPSizeChangeProof (⇔, 0 ms)
        ↳15 YES
    ↳16 QDP
        ↳17 UsableRulesProof (⇔, 0 ms)
        ↳18 QDP
        ↳19 ATransformationProof (⇔, 0 ms)
        ↳20 QDP
        ↳21 QReductionProof (⇔, 0 ms)
        ↳22 QDP
        ↳23 QDPSizeChangeProof (⇔, 0 ms)
        ↳24 YES
    ↳25 QDP
        ↳26 UsableRulesProof (⇔, 0 ms)
        ↳27 QDP
        ↳28 ATransformationProof (⇔, 0 ms)
        ↳29 QDP
        ↳30 QReductionProof (⇔, 0 ms)
        ↳31 QDP
        ↳32 QDPOrderProof (⇔, 14 ms)
        ↳33 QDP
        ↳34 DependencyGraphProof (⇔, 0 ms)
        ↳35 QDP
        ↳36 UsableRulesProof (⇔, 0 ms)
        ↳37 QDP
        ↳38 QReductionProof (⇔, 0 ms)
        ↳39 QDP
        ↳40 QDPSizeChangeProof (⇔, 0 ms)
        ↳41 YES
    ↳42 QDP
        ↳43 QDPSizeChangeProof (⇔, 0 ms)
        ↳44 YES

(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) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(9) 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)

R is empty.
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) ATransformationProof (EQUIVALENT transformation)

We have applied the A-Transformation [FROCOS05] to get from an applicative problem to a standard problem.

(11) Obligation:

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

union1(edge(x, y, i), h) → union1(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.

(12) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

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)

(13) Obligation:

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

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

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • union1(edge(x, y, i), h) → union1(i, h)
    The graph contains the following edges 1 > 1, 2 >= 2

(15) YES

(16) 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.

(17) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(18) 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)

R is empty.
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) ATransformationProof (EQUIVALENT transformation)

We have applied the A-Transformation [FROCOS05] to get from an applicative problem to a standard problem.

(20) Obligation:

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

eq1(s(x), s(y)) → eq1(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.

(21) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

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)

(22) Obligation:

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

eq1(s(x), s(y)) → eq1(x, y)

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

(23) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • eq1(s(x), s(y)) → eq1(x, y)
    The graph contains the following edges 1 > 1, 2 > 2

(24) YES

(25) 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.

(26) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(27) 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(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(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)

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.

(28) ATransformationProof (EQUIVALENT transformation)

We have applied the A-Transformation [FROCOS05] to get from an applicative problem to a standard problem.

(29) Obligation:

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

reach1(x, y, edge(u, v, i), h) → if_reach_11(eq(x, u), x, y, edge(u, v, i), h)
if_reach_11(true, x, y, edge(u, v, i), h) → if_reach_21(eq(y, v), x, y, edge(u, v, i), h)
if_reach_21(false, x, y, edge(u, v, i), h) → reach1(x, y, i, h)
if_reach_21(false, x, y, edge(u, v, i), h) → reach1(v, y, union(i, h), empty)
if_reach_11(false, x, y, edge(u, v, i), h) → reach1(x, y, i, edge(u, v, h))

The TRS R consists of the following rules:

union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)

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.

(30) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

or(true, x0)
or(false, x0)
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)

(31) Obligation:

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

reach1(x, y, edge(u, v, i), h) → if_reach_11(eq(x, u), x, y, edge(u, v, i), h)
if_reach_11(true, x, y, edge(u, v, i), h) → if_reach_21(eq(y, v), x, y, edge(u, v, i), h)
if_reach_21(false, x, y, edge(u, v, i), h) → reach1(x, y, i, h)
if_reach_21(false, x, y, edge(u, v, i), h) → reach1(v, y, union(i, h), empty)
if_reach_11(false, x, y, edge(u, v, i), h) → reach1(x, y, i, edge(u, v, h))

The TRS R consists of the following rules:

union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)

The set Q consists of the following terms:

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
union(empty, x0)
union(edge(x0, x1, x2), x3)

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

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.


if_reach_11(true, x, y, edge(u, v, i), h) → if_reach_21(eq(y, v), x, y, edge(u, v, i), h)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( if_reach_11(x1, ..., x5) ) = 2x1 + 2x4 + 2x5 + 2

POL( if_reach_21(x1, ..., x5) ) = 2x4 + 2x5

POL( eq(x1, x2) ) = 0

POL( 0 ) = 2

POL( true ) = 0

POL( s(x1) ) = x1 + 2

POL( false ) = 0

POL( reach1(x1, ..., x4) ) = 2x3 + 2x4 + 2

POL( union(x1, x2) ) = x1 + x2

POL( empty ) = 0

POL( edge(x1, ..., x3) ) = x3 + 1


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))

(33) Obligation:

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

reach1(x, y, edge(u, v, i), h) → if_reach_11(eq(x, u), x, y, edge(u, v, i), h)
if_reach_21(false, x, y, edge(u, v, i), h) → reach1(x, y, i, h)
if_reach_21(false, x, y, edge(u, v, i), h) → reach1(v, y, union(i, h), empty)
if_reach_11(false, x, y, edge(u, v, i), h) → reach1(x, y, i, edge(u, v, h))

The TRS R consists of the following rules:

union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)

The set Q consists of the following terms:

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
union(empty, x0)
union(edge(x0, x1, x2), x3)

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

(34) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(35) Obligation:

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

if_reach_11(false, x, y, edge(u, v, i), h) → reach1(x, y, i, edge(u, v, h))
reach1(x, y, edge(u, v, i), h) → if_reach_11(eq(x, u), x, y, edge(u, v, i), h)

The TRS R consists of the following rules:

union(empty, h) → h
union(edge(x, y, i), h) → edge(x, y, union(i, h))
eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)

The set Q consists of the following terms:

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
union(empty, x0)
union(edge(x0, x1, x2), x3)

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

(36) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(37) Obligation:

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

if_reach_11(false, x, y, edge(u, v, i), h) → reach1(x, y, i, edge(u, v, h))
reach1(x, y, edge(u, v, i), h) → if_reach_11(eq(x, u), x, y, edge(u, v, i), h)

The TRS R consists of the following rules:

eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)

The set Q consists of the following terms:

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
union(empty, x0)
union(edge(x0, x1, x2), x3)

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

(38) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

union(empty, x0)
union(edge(x0, x1, x2), x3)

(39) Obligation:

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

if_reach_11(false, x, y, edge(u, v, i), h) → reach1(x, y, i, edge(u, v, h))
reach1(x, y, edge(u, v, i), h) → if_reach_11(eq(x, u), x, y, edge(u, v, i), h)

The TRS R consists of the following rules:

eq(0, 0) → true
eq(0, s(x)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)

The set Q consists of the following terms:

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))

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

(40) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • reach1(x, y, edge(u, v, i), h) → if_reach_11(eq(x, u), x, y, edge(u, v, i), h)
    The graph contains the following edges 1 >= 2, 2 >= 3, 3 >= 4, 4 >= 5

  • if_reach_11(false, x, y, edge(u, v, i), h) → reach1(x, y, i, edge(u, v, h))
    The graph contains the following edges 2 >= 1, 3 >= 2, 4 > 3

(41) YES

(42) 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.

(43) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • APP(app(filter, f), app(app(cons, x), xs)) → APP(f, x)
    The graph contains the following edges 1 > 1, 2 > 2

  • APP(app(map, f), app(app(cons, x), xs)) → APP(f, x)
    The graph contains the following edges 1 > 1, 2 > 2

  • APP(app(map, f), app(app(cons, x), xs)) → APP(app(map, f), xs)
    The graph contains the following edges 1 >= 1, 2 > 2

  • APP(app(filter, f), app(app(cons, x), xs)) → APP(app(app(app(filter2, app(f, x)), f), x), xs)
    The graph contains the following edges 2 > 2

  • APP(app(app(app(filter2, true), f), x), xs) → APP(app(filter, f), xs)
    The graph contains the following edges 2 >= 2

  • APP(app(app(app(filter2, false), f), x), xs) → APP(app(filter, f), xs)
    The graph contains the following edges 2 >= 2

(44) YES