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

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

  ↳ 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))

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
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))

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

↳ 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))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 3 SCCs with 4 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ 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))

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

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(UNION₂(x₁, x₂)) = 2·x₁
POL(edge₃(x₁, x₂, x₃)) = 1 + 2·x₃

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ 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))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ 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))

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

The following pairs can be oriented strictly and are deleted.

EQ₂(s₁(x), s₁(y)) -> EQ₂(x, y)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(EQ₂(x₁, x₂)) = 2·x₁ + x₂
POL(s₁(x₁)) = 2 + 2·x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ 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))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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))

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

The following pairs can be oriented strictly and are deleted.

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 remaining pairs can at least be oriented weakly.

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))

Used ordering: Polynomial interpretation [21]:

POL(0) = 1
POL(IF_REACH_1₅(x₁, x₂, x₃, x₄, x₅)) = 1 + x₁ + x₂ + x₃ + 2·x₄ + 2·x₅
POL(IF_REACH_2₅(x₁, x₂, x₃, x₄, x₅)) = x₁ + x₂ + x₃ + 2·x₄ + 2·x₅
POL(REACH₄(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₂ + 2·x₃ + 2·x₄
POL(edge₃(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(empty) = 0
POL(eq₂(x₁, x₂)) = 0
POL(false) = 0
POL(s₁(x₁)) = 1 + x₁
POL(true) = 0
POL(union₂(x₁, x₂)) = x₁ + x₂

The following usable rules [14] were oriented:

eq₂(0, s₁(x)) -> false
eq₂(s₁(x), s₁(y)) -> eq₂(x, y)
eq₂(0, 0) -> true
eq₂(s₁(x), 0) -> false
union₂(empty, h) -> h
union₂(edge₃(x, y, i), h) -> edge₃(x, y, union₂(i, h))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

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))

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

The following pairs can be oriented strictly and are deleted.

REACH₄(x, y, edge₃(u, v, i), h) -> IF_REACH_1₅(eq₂(x, u), x, y, edge₃(u, v, i), h)

The remaining pairs can at least be oriented weakly.

IF_REACH_1₅(false, x, y, edge₃(u, v, i), h) -> REACH₄(x, y, i, edge₃(u, v, h))

Used ordering: Polynomial interpretation [21]:

POL(0) = 0
POL(IF_REACH_1₅(x₁, x₂, x₃, x₄, x₅)) = 1 + x₄
POL(REACH₄(x₁, x₂, x₃, x₄)) = 2 + 2·x₃
POL(edge₃(x₁, x₂, x₃)) = 1 + x₁ + 2·x₃
POL(eq₂(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(false) = 1
POL(s₁(x₁)) = 2 + x₁
POL(true) = 0

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

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

IF_REACH_1₅(false, x, y, edge₃(u, v, i), h) -> REACH₄(x, y, i, edge₃(u, v, 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))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 1 less node.