Termination w.r.t. Q proof of TRS/AProVErta1.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:

plus₂(s₁(s₁(x)), y) -> s₁(plus₂(x, s₁(y)))
plus₂(x, s₁(s₁(y))) -> s₁(plus₂(s₁(x), y))
plus₂(s₁(0), y) -> s₁(y)
plus₂(0, y) -> y
ack₂(0, y) -> s₁(y)
ack₂(s₁(x), 0) -> ack₂(x, s₁(0))
ack₂(s₁(x), s₁(y)) -> ack₂(x, plus₂(y, ack₂(s₁(x), y)))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

plus₂(s₁(s₁(x)), y) -> s₁(plus₂(x, s₁(y)))
plus₂(x, s₁(s₁(y))) -> s₁(plus₂(s₁(x), y))
plus₂(s₁(0), y) -> s₁(y)
plus₂(0, y) -> y
ack₂(0, y) -> s₁(y)
ack₂(s₁(x), 0) -> ack₂(x, s₁(0))
ack₂(s₁(x), s₁(y)) -> ack₂(x, plus₂(y, ack₂(s₁(x), y)))

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:

ACK₂(s₁(x), s₁(y)) -> ACK₂(s₁(x), y)
ACK₂(s₁(x), s₁(y)) -> ACK₂(x, plus₂(y, ack₂(s₁(x), y)))
PLUS₂(x, s₁(s₁(y))) -> PLUS₂(s₁(x), y)
ACK₂(s₁(x), s₁(y)) -> PLUS₂(y, ack₂(s₁(x), y))
PLUS₂(s₁(s₁(x)), y) -> PLUS₂(x, s₁(y))
ACK₂(s₁(x), 0) -> ACK₂(x, s₁(0))

The TRS R consists of the following rules:

plus₂(s₁(s₁(x)), y) -> s₁(plus₂(x, s₁(y)))
plus₂(x, s₁(s₁(y))) -> s₁(plus₂(s₁(x), y))
plus₂(s₁(0), y) -> s₁(y)
plus₂(0, y) -> y
ack₂(0, y) -> s₁(y)
ack₂(s₁(x), 0) -> ack₂(x, s₁(0))
ack₂(s₁(x), s₁(y)) -> ack₂(x, plus₂(y, ack₂(s₁(x), y)))

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:

ACK₂(s₁(x), s₁(y)) -> ACK₂(s₁(x), y)
ACK₂(s₁(x), s₁(y)) -> ACK₂(x, plus₂(y, ack₂(s₁(x), y)))
PLUS₂(x, s₁(s₁(y))) -> PLUS₂(s₁(x), y)
ACK₂(s₁(x), s₁(y)) -> PLUS₂(y, ack₂(s₁(x), y))
PLUS₂(s₁(s₁(x)), y) -> PLUS₂(x, s₁(y))
ACK₂(s₁(x), 0) -> ACK₂(x, s₁(0))

The TRS R consists of the following rules:

plus₂(s₁(s₁(x)), y) -> s₁(plus₂(x, s₁(y)))
plus₂(x, s₁(s₁(y))) -> s₁(plus₂(s₁(x), y))
plus₂(s₁(0), y) -> s₁(y)
plus₂(0, y) -> y
ack₂(0, y) -> s₁(y)
ack₂(s₁(x), 0) -> ack₂(x, s₁(0))
ack₂(s₁(x), s₁(y)) -> ack₂(x, plus₂(y, ack₂(s₁(x), y)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 2 SCCs with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

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

PLUS₂(x, s₁(s₁(y))) -> PLUS₂(s₁(x), y)
PLUS₂(s₁(s₁(x)), y) -> PLUS₂(x, s₁(y))

The TRS R consists of the following rules:

plus₂(s₁(s₁(x)), y) -> s₁(plus₂(x, s₁(y)))
plus₂(x, s₁(s₁(y))) -> s₁(plus₂(s₁(x), y))
plus₂(s₁(0), y) -> s₁(y)
plus₂(0, y) -> y
ack₂(0, y) -> s₁(y)
ack₂(s₁(x), 0) -> ack₂(x, s₁(0))
ack₂(s₁(x), s₁(y)) -> ack₂(x, plus₂(y, ack₂(s₁(x), y)))

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.

PLUS₂(x, s₁(s₁(y))) -> PLUS₂(s₁(x), y)
PLUS₂(s₁(s₁(x)), y) -> PLUS₂(x, s₁(y))

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( s₁(x₁) ) = x₁ + 1

POL( PLUS₂(x₁, x₂) ) = max{0, x₁ + x₂ - 1}

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

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

plus₂(s₁(s₁(x)), y) -> s₁(plus₂(x, s₁(y)))
plus₂(x, s₁(s₁(y))) -> s₁(plus₂(s₁(x), y))
plus₂(s₁(0), y) -> s₁(y)
plus₂(0, y) -> y
ack₂(0, y) -> s₁(y)
ack₂(s₁(x), 0) -> ack₂(x, s₁(0))
ack₂(s₁(x), s₁(y)) -> ack₂(x, plus₂(y, ack₂(s₁(x), y)))

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

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

ACK₂(s₁(x), s₁(y)) -> ACK₂(s₁(x), y)
ACK₂(s₁(x), s₁(y)) -> ACK₂(x, plus₂(y, ack₂(s₁(x), y)))
ACK₂(s₁(x), 0) -> ACK₂(x, s₁(0))

The TRS R consists of the following rules:

plus₂(s₁(s₁(x)), y) -> s₁(plus₂(x, s₁(y)))
plus₂(x, s₁(s₁(y))) -> s₁(plus₂(s₁(x), y))
plus₂(s₁(0), y) -> s₁(y)
plus₂(0, y) -> y
ack₂(0, y) -> s₁(y)
ack₂(s₁(x), 0) -> ack₂(x, s₁(0))
ack₂(s₁(x), s₁(y)) -> ack₂(x, plus₂(y, ack₂(s₁(x), y)))

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.

ACK₂(s₁(x), s₁(y)) -> ACK₂(x, plus₂(y, ack₂(s₁(x), y)))
ACK₂(s₁(x), 0) -> ACK₂(x, s₁(0))

The remaining pairs can at least be oriented weakly.

ACK₂(s₁(x), s₁(y)) -> ACK₂(s₁(x), y)

Used ordering: Polynomial Order [17,21] with Interpretation:

POL( s₁(x₁) ) = x₁ + 1

POL( 0 ) = max{0, -1}

POL( ack₂(x₁, x₂) ) = max{0, -1}

POL( plus₂(x₁, x₂) ) = max{0, -1}

POL( ACK₂(x₁, x₂) ) = x₁ + 1

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

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

ACK₂(s₁(x), s₁(y)) -> ACK₂(s₁(x), y)

The TRS R consists of the following rules:

plus₂(s₁(s₁(x)), y) -> s₁(plus₂(x, s₁(y)))
plus₂(x, s₁(s₁(y))) -> s₁(plus₂(s₁(x), y))
plus₂(s₁(0), y) -> s₁(y)
plus₂(0, y) -> y
ack₂(0, y) -> s₁(y)
ack₂(s₁(x), 0) -> ack₂(x, s₁(0))
ack₂(s₁(x), s₁(y)) -> ack₂(x, plus₂(y, ack₂(s₁(x), y)))

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.

ACK₂(s₁(x), s₁(y)) -> ACK₂(s₁(x), y)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( s₁(x₁) ) = x₁ + 1

POL( ACK₂(x₁, x₂) ) = x₂ + 1

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

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

plus₂(s₁(s₁(x)), y) -> s₁(plus₂(x, s₁(y)))
plus₂(x, s₁(s₁(y))) -> s₁(plus₂(s₁(x), y))
plus₂(s₁(0), y) -> s₁(y)
plus₂(0, y) -> y
ack₂(0, y) -> s₁(y)
ack₂(s₁(x), 0) -> ack₂(x, s₁(0))
ack₂(s₁(x), s₁(y)) -> ack₂(x, plus₂(y, ack₂(s₁(x), y)))

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.