Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar4/AG01SLASHHASH3.13.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:

eq₂(0, 0) -> true
eq₂(0, s₁(x)) -> false
eq₂(s₁(x), 0) -> false
eq₂(s₁(x), s₁(y)) -> eq₂(x, y)
or₂(true, y) -> true
or₂(false, y) -> y
union₂(empty, h) -> h
union₂(edge₃(x, y, i), h) -> edge₃(x, y, union₂(i, h))
reach₄(x, y, empty, h) -> false
reach₄(x, y, edge₃(u, v, i), h) -> if_reach_1₅(eq₂(x, u), x, y, edge₃(u, v, i), h)
if_reach_1₅(true, x, y, edge₃(u, v, i), h) -> if_reach_2₅(eq₂(y, v), x, y, edge₃(u, v, i), h)
if_reach_2₅(true, x, y, edge₃(u, v, i), h) -> true
if_reach_2₅(false, x, y, edge₃(u, v, i), h) -> or₂(reach₄(x, y, i, h), reach₄(v, y, union₂(i, h), empty))
if_reach_1₅(false, x, y, edge₃(u, v, i), h) -> reach₄(x, y, i, edge₃(u, v, h))

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

eq₂(0, 0) -> true
eq₂(0, s₁(x)) -> false
eq₂(s₁(x), 0) -> false
eq₂(s₁(x), s₁(y)) -> eq₂(x, y)
or₂(true, y) -> true
or₂(false, y) -> y
union₂(empty, h) -> h
union₂(edge₃(x, y, i), h) -> edge₃(x, y, union₂(i, h))
reach₄(x, y, empty, h) -> false
reach₄(x, y, edge₃(u, v, i), h) -> if_reach_1₅(eq₂(x, u), x, y, edge₃(u, v, i), h)
if_reach_1₅(true, x, y, edge₃(u, v, i), h) -> if_reach_2₅(eq₂(y, v), x, y, edge₃(u, v, i), h)
if_reach_2₅(true, x, y, edge₃(u, v, i), h) -> true
if_reach_2₅(false, x, y, edge₃(u, v, i), h) -> or₂(reach₄(x, y, i, h), reach₄(v, y, union₂(i, h), empty))
if_reach_1₅(false, x, y, edge₃(u, v, i), h) -> reach₄(x, y, i, edge₃(u, v, h))

Q is empty.

The TRS is non-overlapping. Hence, we can switch to innermost.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

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

eq₂(0, 0) -> true
eq₂(0, s₁(x)) -> false
eq₂(s₁(x), 0) -> false
eq₂(s₁(x), s₁(y)) -> eq₂(x, y)
or₂(true, y) -> true
or₂(false, y) -> y
union₂(empty, h) -> h
union₂(edge₃(x, y, i), h) -> edge₃(x, y, union₂(i, h))
reach₄(x, y, empty, h) -> false
reach₄(x, y, edge₃(u, v, i), h) -> if_reach_1₅(eq₂(x, u), x, y, edge₃(u, v, i), h)
if_reach_1₅(true, x, y, edge₃(u, v, i), h) -> if_reach_2₅(eq₂(y, v), x, y, edge₃(u, v, i), h)
if_reach_2₅(true, x, y, edge₃(u, v, i), h) -> true
if_reach_2₅(false, x, y, edge₃(u, v, i), h) -> or₂(reach₄(x, y, i, h), reach₄(v, y, union₂(i, h), empty))
if_reach_1₅(false, x, y, edge₃(u, v, i), h) -> reach₄(x, y, i, edge₃(u, v, h))

The set Q consists of the following terms:

eq₂(0, 0)
eq₂(0, s₁(x0))
eq₂(s₁(x0), 0)
eq₂(s₁(x0), s₁(x1))
or₂(true, x0)
or₂(false, x0)
union₂(empty, x0)
union₂(edge₃(x0, x1, x2), x3)
reach₄(x0, x1, empty, x2)
reach₄(x0, x1, edge₃(x2, x3, x4), x5)
if_reach_1₅(true, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_2₅(true, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_2₅(false, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_1₅(false, x0, x1, edge₃(x2, x3, x4), x5)

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

REACH₄(x, y, edge₃(u, v, i), h) -> IF_REACH_1₅(eq₂(x, u), x, y, edge₃(u, v, i), h)
IF_REACH_1₅(false, x, y, edge₃(u, v, i), h) -> REACH₄(x, y, i, edge₃(u, v, h))
UNION₂(edge₃(x, y, i), h) -> UNION₂(i, h)
EQ₂(s₁(x), s₁(y)) -> EQ₂(x, y)
IF_REACH_1₅(true, x, y, edge₃(u, v, i), h) -> EQ₂(y, v)
IF_REACH_1₅(true, x, y, edge₃(u, v, i), h) -> IF_REACH_2₅(eq₂(y, v), x, y, edge₃(u, v, i), h)
IF_REACH_2₅(false, x, y, edge₃(u, v, i), h) -> REACH₄(x, y, i, h)
IF_REACH_2₅(false, x, y, edge₃(u, v, i), h) -> UNION₂(i, h)
IF_REACH_2₅(false, x, y, edge₃(u, v, i), h) -> OR₂(reach₄(x, y, i, h), reach₄(v, y, union₂(i, h), empty))
IF_REACH_2₅(false, x, y, edge₃(u, v, i), h) -> REACH₄(v, y, union₂(i, h), empty)
REACH₄(x, y, edge₃(u, v, i), h) -> EQ₂(x, u)

The TRS R consists of the following rules:

eq₂(0, 0) -> true
eq₂(0, s₁(x)) -> false
eq₂(s₁(x), 0) -> false
eq₂(s₁(x), s₁(y)) -> eq₂(x, y)
or₂(true, y) -> true
or₂(false, y) -> y
union₂(empty, h) -> h
union₂(edge₃(x, y, i), h) -> edge₃(x, y, union₂(i, h))
reach₄(x, y, empty, h) -> false
reach₄(x, y, edge₃(u, v, i), h) -> if_reach_1₅(eq₂(x, u), x, y, edge₃(u, v, i), h)
if_reach_1₅(true, x, y, edge₃(u, v, i), h) -> if_reach_2₅(eq₂(y, v), x, y, edge₃(u, v, i), h)
if_reach_2₅(true, x, y, edge₃(u, v, i), h) -> true
if_reach_2₅(false, x, y, edge₃(u, v, i), h) -> or₂(reach₄(x, y, i, h), reach₄(v, y, union₂(i, h), empty))
if_reach_1₅(false, x, y, edge₃(u, v, i), h) -> reach₄(x, y, i, edge₃(u, v, h))

The set Q consists of the following terms:

eq₂(0, 0)
eq₂(0, s₁(x0))
eq₂(s₁(x0), 0)
eq₂(s₁(x0), s₁(x1))
or₂(true, x0)
or₂(false, x0)
union₂(empty, x0)
union₂(edge₃(x0, x1, x2), x3)
reach₄(x0, x1, empty, x2)
reach₄(x0, x1, edge₃(x2, x3, x4), x5)
if_reach_1₅(true, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_2₅(true, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_2₅(false, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_1₅(false, x0, x1, edge₃(x2, x3, x4), x5)

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

REACH₄(x, y, edge₃(u, v, i), h) -> IF_REACH_1₅(eq₂(x, u), x, y, edge₃(u, v, i), h)
IF_REACH_1₅(false, x, y, edge₃(u, v, i), h) -> REACH₄(x, y, i, edge₃(u, v, h))
UNION₂(edge₃(x, y, i), h) -> UNION₂(i, h)
EQ₂(s₁(x), s₁(y)) -> EQ₂(x, y)
IF_REACH_1₅(true, x, y, edge₃(u, v, i), h) -> EQ₂(y, v)
IF_REACH_1₅(true, x, y, edge₃(u, v, i), h) -> IF_REACH_2₅(eq₂(y, v), x, y, edge₃(u, v, i), h)
IF_REACH_2₅(false, x, y, edge₃(u, v, i), h) -> REACH₄(x, y, i, h)
IF_REACH_2₅(false, x, y, edge₃(u, v, i), h) -> UNION₂(i, h)
IF_REACH_2₅(false, x, y, edge₃(u, v, i), h) -> OR₂(reach₄(x, y, i, h), reach₄(v, y, union₂(i, h), empty))
IF_REACH_2₅(false, x, y, edge₃(u, v, i), h) -> REACH₄(v, y, union₂(i, h), empty)
REACH₄(x, y, edge₃(u, v, i), h) -> EQ₂(x, u)

The TRS R consists of the following rules:

eq₂(0, 0) -> true
eq₂(0, s₁(x)) -> false
eq₂(s₁(x), 0) -> false
eq₂(s₁(x), s₁(y)) -> eq₂(x, y)
or₂(true, y) -> true
or₂(false, y) -> y
union₂(empty, h) -> h
union₂(edge₃(x, y, i), h) -> edge₃(x, y, union₂(i, h))
reach₄(x, y, empty, h) -> false
reach₄(x, y, edge₃(u, v, i), h) -> if_reach_1₅(eq₂(x, u), x, y, edge₃(u, v, i), h)
if_reach_1₅(true, x, y, edge₃(u, v, i), h) -> if_reach_2₅(eq₂(y, v), x, y, edge₃(u, v, i), h)
if_reach_2₅(true, x, y, edge₃(u, v, i), h) -> true
if_reach_2₅(false, x, y, edge₃(u, v, i), h) -> or₂(reach₄(x, y, i, h), reach₄(v, y, union₂(i, h), empty))
if_reach_1₅(false, x, y, edge₃(u, v, i), h) -> reach₄(x, y, i, edge₃(u, v, h))

The set Q consists of the following terms:

eq₂(0, 0)
eq₂(0, s₁(x0))
eq₂(s₁(x0), 0)
eq₂(s₁(x0), s₁(x1))
or₂(true, x0)
or₂(false, x0)
union₂(empty, x0)
union₂(edge₃(x0, x1, x2), x3)
reach₄(x0, x1, empty, x2)
reach₄(x0, x1, edge₃(x2, x3, x4), x5)
if_reach_1₅(true, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_2₅(true, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_2₅(false, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_1₅(false, x0, x1, edge₃(x2, x3, x4), x5)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 3 SCCs with 4 less nodes.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPAfsSolverProof

              ↳ QDP

              ↳ QDP

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

UNION₂(edge₃(x, y, i), h) -> UNION₂(i, h)

The TRS R consists of the following rules:

eq₂(0, 0) -> true
eq₂(0, s₁(x)) -> false
eq₂(s₁(x), 0) -> false
eq₂(s₁(x), s₁(y)) -> eq₂(x, y)
or₂(true, y) -> true
or₂(false, y) -> y
union₂(empty, h) -> h
union₂(edge₃(x, y, i), h) -> edge₃(x, y, union₂(i, h))
reach₄(x, y, empty, h) -> false
reach₄(x, y, edge₃(u, v, i), h) -> if_reach_1₅(eq₂(x, u), x, y, edge₃(u, v, i), h)
if_reach_1₅(true, x, y, edge₃(u, v, i), h) -> if_reach_2₅(eq₂(y, v), x, y, edge₃(u, v, i), h)
if_reach_2₅(true, x, y, edge₃(u, v, i), h) -> true
if_reach_2₅(false, x, y, edge₃(u, v, i), h) -> or₂(reach₄(x, y, i, h), reach₄(v, y, union₂(i, h), empty))
if_reach_1₅(false, x, y, edge₃(u, v, i), h) -> reach₄(x, y, i, edge₃(u, v, h))

The set Q consists of the following terms:

eq₂(0, 0)
eq₂(0, s₁(x0))
eq₂(s₁(x0), 0)
eq₂(s₁(x0), s₁(x1))
or₂(true, x0)
or₂(false, x0)
union₂(empty, x0)
union₂(edge₃(x0, x1, x2), x3)
reach₄(x0, x1, empty, x2)
reach₄(x0, x1, edge₃(x2, x3, x4), x5)
if_reach_1₅(true, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_2₅(true, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_2₅(false, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_1₅(false, x0, x1, edge₃(x2, x3, x4), x5)

We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

UNION₂(edge₃(x, y, i), h) -> UNION₂(i, h)

Used argument filtering: UNION₂(x1, x2) = x1
edge₃(x1, x2, x3) = edge₁(x3)
Used ordering: Quasi Precedence: trivial

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPAfsSolverProof

                  ↳ QDP

                    ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

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

eq₂(0, 0) -> true
eq₂(0, s₁(x)) -> false
eq₂(s₁(x), 0) -> false
eq₂(s₁(x), s₁(y)) -> eq₂(x, y)
or₂(true, y) -> true
or₂(false, y) -> y
union₂(empty, h) -> h
union₂(edge₃(x, y, i), h) -> edge₃(x, y, union₂(i, h))
reach₄(x, y, empty, h) -> false
reach₄(x, y, edge₃(u, v, i), h) -> if_reach_1₅(eq₂(x, u), x, y, edge₃(u, v, i), h)
if_reach_1₅(true, x, y, edge₃(u, v, i), h) -> if_reach_2₅(eq₂(y, v), x, y, edge₃(u, v, i), h)
if_reach_2₅(true, x, y, edge₃(u, v, i), h) -> true
if_reach_2₅(false, x, y, edge₃(u, v, i), h) -> or₂(reach₄(x, y, i, h), reach₄(v, y, union₂(i, h), empty))
if_reach_1₅(false, x, y, edge₃(u, v, i), h) -> reach₄(x, y, i, edge₃(u, v, h))

The set Q consists of the following terms:

eq₂(0, 0)
eq₂(0, s₁(x0))
eq₂(s₁(x0), 0)
eq₂(s₁(x0), s₁(x1))
or₂(true, x0)
or₂(false, x0)
union₂(empty, x0)
union₂(edge₃(x0, x1, x2), x3)
reach₄(x0, x1, empty, x2)
reach₄(x0, x1, edge₃(x2, x3, x4), x5)
if_reach_1₅(true, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_2₅(true, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_2₅(false, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_1₅(false, x0, x1, edge₃(x2, x3, x4), x5)

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPAfsSolverProof

              ↳ QDP

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

EQ₂(s₁(x), s₁(y)) -> EQ₂(x, y)

The TRS R consists of the following rules:

eq₂(0, 0) -> true
eq₂(0, s₁(x)) -> false
eq₂(s₁(x), 0) -> false
eq₂(s₁(x), s₁(y)) -> eq₂(x, y)
or₂(true, y) -> true
or₂(false, y) -> y
union₂(empty, h) -> h
union₂(edge₃(x, y, i), h) -> edge₃(x, y, union₂(i, h))
reach₄(x, y, empty, h) -> false
reach₄(x, y, edge₃(u, v, i), h) -> if_reach_1₅(eq₂(x, u), x, y, edge₃(u, v, i), h)
if_reach_1₅(true, x, y, edge₃(u, v, i), h) -> if_reach_2₅(eq₂(y, v), x, y, edge₃(u, v, i), h)
if_reach_2₅(true, x, y, edge₃(u, v, i), h) -> true
if_reach_2₅(false, x, y, edge₃(u, v, i), h) -> or₂(reach₄(x, y, i, h), reach₄(v, y, union₂(i, h), empty))
if_reach_1₅(false, x, y, edge₃(u, v, i), h) -> reach₄(x, y, i, edge₃(u, v, h))

The set Q consists of the following terms:

eq₂(0, 0)
eq₂(0, s₁(x0))
eq₂(s₁(x0), 0)
eq₂(s₁(x0), s₁(x1))
or₂(true, x0)
or₂(false, x0)
union₂(empty, x0)
union₂(edge₃(x0, x1, x2), x3)
reach₄(x0, x1, empty, x2)
reach₄(x0, x1, edge₃(x2, x3, x4), x5)
if_reach_1₅(true, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_2₅(true, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_2₅(false, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_1₅(false, x0, x1, edge₃(x2, x3, x4), x5)

We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

EQ₂(s₁(x), s₁(y)) -> EQ₂(x, y)

Used argument filtering: EQ₂(x1, x2) = x2
s₁(x1) = s₁(x1)
Used ordering: Quasi Precedence: trivial

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPAfsSolverProof

                  ↳ QDP

                    ↳ PisEmptyProof

              ↳ QDP

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

eq₂(0, 0) -> true
eq₂(0, s₁(x)) -> false
eq₂(s₁(x), 0) -> false
eq₂(s₁(x), s₁(y)) -> eq₂(x, y)
or₂(true, y) -> true
or₂(false, y) -> y
union₂(empty, h) -> h
union₂(edge₃(x, y, i), h) -> edge₃(x, y, union₂(i, h))
reach₄(x, y, empty, h) -> false
reach₄(x, y, edge₃(u, v, i), h) -> if_reach_1₅(eq₂(x, u), x, y, edge₃(u, v, i), h)
if_reach_1₅(true, x, y, edge₃(u, v, i), h) -> if_reach_2₅(eq₂(y, v), x, y, edge₃(u, v, i), h)
if_reach_2₅(true, x, y, edge₃(u, v, i), h) -> true
if_reach_2₅(false, x, y, edge₃(u, v, i), h) -> or₂(reach₄(x, y, i, h), reach₄(v, y, union₂(i, h), empty))
if_reach_1₅(false, x, y, edge₃(u, v, i), h) -> reach₄(x, y, i, edge₃(u, v, h))

The set Q consists of the following terms:

eq₂(0, 0)
eq₂(0, s₁(x0))
eq₂(s₁(x0), 0)
eq₂(s₁(x0), s₁(x1))
or₂(true, x0)
or₂(false, x0)
union₂(empty, x0)
union₂(edge₃(x0, x1, x2), x3)
reach₄(x0, x1, empty, x2)
reach₄(x0, x1, edge₃(x2, x3, x4), x5)
if_reach_1₅(true, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_2₅(true, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_2₅(false, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_1₅(false, x0, x1, edge₃(x2, x3, x4), x5)

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

REACH₄(x, y, edge₃(u, v, i), h) -> IF_REACH_1₅(eq₂(x, u), x, y, edge₃(u, v, i), h)
IF_REACH_1₅(false, x, y, edge₃(u, v, i), h) -> REACH₄(x, y, i, edge₃(u, v, h))
IF_REACH_1₅(true, x, y, edge₃(u, v, i), h) -> IF_REACH_2₅(eq₂(y, v), x, y, edge₃(u, v, i), h)
IF_REACH_2₅(false, x, y, edge₃(u, v, i), h) -> REACH₄(x, y, i, h)
IF_REACH_2₅(false, x, y, edge₃(u, v, i), h) -> REACH₄(v, y, union₂(i, h), empty)

The TRS R consists of the following rules:

eq₂(0, 0) -> true
eq₂(0, s₁(x)) -> false
eq₂(s₁(x), 0) -> false
eq₂(s₁(x), s₁(y)) -> eq₂(x, y)
or₂(true, y) -> true
or₂(false, y) -> y
union₂(empty, h) -> h
union₂(edge₃(x, y, i), h) -> edge₃(x, y, union₂(i, h))
reach₄(x, y, empty, h) -> false
reach₄(x, y, edge₃(u, v, i), h) -> if_reach_1₅(eq₂(x, u), x, y, edge₃(u, v, i), h)
if_reach_1₅(true, x, y, edge₃(u, v, i), h) -> if_reach_2₅(eq₂(y, v), x, y, edge₃(u, v, i), h)
if_reach_2₅(true, x, y, edge₃(u, v, i), h) -> true
if_reach_2₅(false, x, y, edge₃(u, v, i), h) -> or₂(reach₄(x, y, i, h), reach₄(v, y, union₂(i, h), empty))
if_reach_1₅(false, x, y, edge₃(u, v, i), h) -> reach₄(x, y, i, edge₃(u, v, h))

The set Q consists of the following terms:

eq₂(0, 0)
eq₂(0, s₁(x0))
eq₂(s₁(x0), 0)
eq₂(s₁(x0), s₁(x1))
or₂(true, x0)
or₂(false, x0)
union₂(empty, x0)
union₂(edge₃(x0, x1, x2), x3)
reach₄(x0, x1, empty, x2)
reach₄(x0, x1, edge₃(x2, x3, x4), x5)
if_reach_1₅(true, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_2₅(true, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_2₅(false, x0, x1, edge₃(x2, x3, x4), x5)
if_reach_1₅(false, x0, x1, edge₃(x2, x3, x4), x5)

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