Termination w.r.t. Q proof of TRS/secret05aprove5.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:

minus₂(x, 0) -> x
minus₂(0, y) -> 0
minus₂(s₁(x), s₁(y)) -> minus₂(p₁(s₁(x)), p₁(s₁(y)))
minus₂(x, plus₂(y, z)) -> minus₂(minus₂(x, y), z)
p₁(s₁(s₁(x))) -> s₁(p₁(s₁(x)))
p₁(0) -> s₁(s₁(0))
div₂(s₁(x), s₁(y)) -> s₁(div₂(minus₂(x, y), s₁(y)))
div₂(plus₂(x, y), z) -> plus₂(div₂(x, z), div₂(y, z))
plus₂(0, y) -> y
plus₂(s₁(x), y) -> s₁(plus₂(y, minus₂(s₁(x), s₁(0))))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

minus₂(x, 0) -> x
minus₂(0, y) -> 0
minus₂(s₁(x), s₁(y)) -> minus₂(p₁(s₁(x)), p₁(s₁(y)))
minus₂(x, plus₂(y, z)) -> minus₂(minus₂(x, y), z)
p₁(s₁(s₁(x))) -> s₁(p₁(s₁(x)))
p₁(0) -> s₁(s₁(0))
div₂(s₁(x), s₁(y)) -> s₁(div₂(minus₂(x, y), s₁(y)))
div₂(plus₂(x, y), z) -> plus₂(div₂(x, z), div₂(y, z))
plus₂(0, y) -> y
plus₂(s₁(x), y) -> s₁(plus₂(y, minus₂(s₁(x), s₁(0))))

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:

DIV₂(plus₂(x, y), z) -> PLUS₂(div₂(x, z), div₂(y, z))
DIV₂(s₁(x), s₁(y)) -> MINUS₂(x, y)
MINUS₂(s₁(x), s₁(y)) -> P₁(s₁(x))
MINUS₂(s₁(x), s₁(y)) -> P₁(s₁(y))
MINUS₂(x, plus₂(y, z)) -> MINUS₂(minus₂(x, y), z)
PLUS₂(s₁(x), y) -> PLUS₂(y, minus₂(s₁(x), s₁(0)))
MINUS₂(s₁(x), s₁(y)) -> MINUS₂(p₁(s₁(x)), p₁(s₁(y)))
P₁(s₁(s₁(x))) -> P₁(s₁(x))
DIV₂(s₁(x), s₁(y)) -> DIV₂(minus₂(x, y), s₁(y))
MINUS₂(x, plus₂(y, z)) -> MINUS₂(x, y)
PLUS₂(s₁(x), y) -> MINUS₂(s₁(x), s₁(0))
DIV₂(plus₂(x, y), z) -> DIV₂(x, z)
DIV₂(plus₂(x, y), z) -> DIV₂(y, z)

The TRS R consists of the following rules:

minus₂(x, 0) -> x
minus₂(0, y) -> 0
minus₂(s₁(x), s₁(y)) -> minus₂(p₁(s₁(x)), p₁(s₁(y)))
minus₂(x, plus₂(y, z)) -> minus₂(minus₂(x, y), z)
p₁(s₁(s₁(x))) -> s₁(p₁(s₁(x)))
p₁(0) -> s₁(s₁(0))
div₂(s₁(x), s₁(y)) -> s₁(div₂(minus₂(x, y), s₁(y)))
div₂(plus₂(x, y), z) -> plus₂(div₂(x, z), div₂(y, z))
plus₂(0, y) -> y
plus₂(s₁(x), y) -> s₁(plus₂(y, minus₂(s₁(x), s₁(0))))

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:

DIV₂(plus₂(x, y), z) -> PLUS₂(div₂(x, z), div₂(y, z))
DIV₂(s₁(x), s₁(y)) -> MINUS₂(x, y)
MINUS₂(s₁(x), s₁(y)) -> P₁(s₁(x))
MINUS₂(s₁(x), s₁(y)) -> P₁(s₁(y))
MINUS₂(x, plus₂(y, z)) -> MINUS₂(minus₂(x, y), z)
PLUS₂(s₁(x), y) -> PLUS₂(y, minus₂(s₁(x), s₁(0)))
MINUS₂(s₁(x), s₁(y)) -> MINUS₂(p₁(s₁(x)), p₁(s₁(y)))
P₁(s₁(s₁(x))) -> P₁(s₁(x))
DIV₂(s₁(x), s₁(y)) -> DIV₂(minus₂(x, y), s₁(y))
MINUS₂(x, plus₂(y, z)) -> MINUS₂(x, y)
PLUS₂(s₁(x), y) -> MINUS₂(s₁(x), s₁(0))
DIV₂(plus₂(x, y), z) -> DIV₂(x, z)
DIV₂(plus₂(x, y), z) -> DIV₂(y, z)

The TRS R consists of the following rules:

minus₂(x, 0) -> x
minus₂(0, y) -> 0
minus₂(s₁(x), s₁(y)) -> minus₂(p₁(s₁(x)), p₁(s₁(y)))
minus₂(x, plus₂(y, z)) -> minus₂(minus₂(x, y), z)
p₁(s₁(s₁(x))) -> s₁(p₁(s₁(x)))
p₁(0) -> s₁(s₁(0))
div₂(s₁(x), s₁(y)) -> s₁(div₂(minus₂(x, y), s₁(y)))
div₂(plus₂(x, y), z) -> plus₂(div₂(x, z), div₂(y, z))
plus₂(0, y) -> y
plus₂(s₁(x), y) -> s₁(plus₂(y, minus₂(s₁(x), s₁(0))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 4 SCCs with 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

P₁(s₁(s₁(x))) -> P₁(s₁(x))

The TRS R consists of the following rules:

minus₂(x, 0) -> x
minus₂(0, y) -> 0
minus₂(s₁(x), s₁(y)) -> minus₂(p₁(s₁(x)), p₁(s₁(y)))
minus₂(x, plus₂(y, z)) -> minus₂(minus₂(x, y), z)
p₁(s₁(s₁(x))) -> s₁(p₁(s₁(x)))
p₁(0) -> s₁(s₁(0))
div₂(s₁(x), s₁(y)) -> s₁(div₂(minus₂(x, y), s₁(y)))
div₂(plus₂(x, y), z) -> plus₂(div₂(x, z), div₂(y, z))
plus₂(0, y) -> y
plus₂(s₁(x), y) -> s₁(plus₂(y, minus₂(s₁(x), s₁(0))))

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.

P₁(s₁(s₁(x))) -> P₁(s₁(x))

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

POL( P₁(x₁) ) = max{0, x₁ - 1}

POL( s₁(x₁) ) = x₁ + 1

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

minus₂(x, 0) -> x
minus₂(0, y) -> 0
minus₂(s₁(x), s₁(y)) -> minus₂(p₁(s₁(x)), p₁(s₁(y)))
minus₂(x, plus₂(y, z)) -> minus₂(minus₂(x, y), z)
p₁(s₁(s₁(x))) -> s₁(p₁(s₁(x)))
p₁(0) -> s₁(s₁(0))
div₂(s₁(x), s₁(y)) -> s₁(div₂(minus₂(x, y), s₁(y)))
div₂(plus₂(x, y), z) -> plus₂(div₂(x, z), div₂(y, z))
plus₂(0, y) -> y
plus₂(s₁(x), y) -> s₁(plus₂(y, minus₂(s₁(x), s₁(0))))

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

          ↳ QDP

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

MINUS₂(s₁(x), s₁(y)) -> MINUS₂(p₁(s₁(x)), p₁(s₁(y)))
MINUS₂(x, plus₂(y, z)) -> MINUS₂(x, y)
MINUS₂(x, plus₂(y, z)) -> MINUS₂(minus₂(x, y), z)

The TRS R consists of the following rules:

minus₂(x, 0) -> x
minus₂(0, y) -> 0
minus₂(s₁(x), s₁(y)) -> minus₂(p₁(s₁(x)), p₁(s₁(y)))
minus₂(x, plus₂(y, z)) -> minus₂(minus₂(x, y), z)
p₁(s₁(s₁(x))) -> s₁(p₁(s₁(x)))
p₁(0) -> s₁(s₁(0))
div₂(s₁(x), s₁(y)) -> s₁(div₂(minus₂(x, y), s₁(y)))
div₂(plus₂(x, y), z) -> plus₂(div₂(x, z), div₂(y, z))
plus₂(0, y) -> y
plus₂(s₁(x), y) -> s₁(plus₂(y, minus₂(s₁(x), s₁(0))))

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.

MINUS₂(s₁(x), s₁(y)) -> MINUS₂(p₁(s₁(x)), p₁(s₁(y)))

The remaining pairs can at least be oriented weakly.

MINUS₂(x, plus₂(y, z)) -> MINUS₂(x, y)
MINUS₂(x, plus₂(y, z)) -> MINUS₂(minus₂(x, y), z)

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

POL( 0 ) = 0

POL( s₁(x₁) ) = x₁ + 1

POL( minus₂(x₁, x₂) ) = 1

POL( MINUS₂(x₁, x₂) ) = x₂

POL( plus₂(x₁, x₂) ) = x₁ + x₂

POL( p₁(x₁) ) = max{0, x₁ - 1}

The following usable rules [14] were oriented:

p₁(s₁(s₁(x))) -> s₁(p₁(s₁(x)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

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

MINUS₂(x, plus₂(y, z)) -> MINUS₂(x, y)
MINUS₂(x, plus₂(y, z)) -> MINUS₂(minus₂(x, y), z)

The TRS R consists of the following rules:

minus₂(x, 0) -> x
minus₂(0, y) -> 0
minus₂(s₁(x), s₁(y)) -> minus₂(p₁(s₁(x)), p₁(s₁(y)))
minus₂(x, plus₂(y, z)) -> minus₂(minus₂(x, y), z)
p₁(s₁(s₁(x))) -> s₁(p₁(s₁(x)))
p₁(0) -> s₁(s₁(0))
div₂(s₁(x), s₁(y)) -> s₁(div₂(minus₂(x, y), s₁(y)))
div₂(plus₂(x, y), z) -> plus₂(div₂(x, z), div₂(y, z))
plus₂(0, y) -> y
plus₂(s₁(x), y) -> s₁(plus₂(y, minus₂(s₁(x), s₁(0))))

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.

MINUS₂(x, plus₂(y, z)) -> MINUS₂(x, y)
MINUS₂(x, plus₂(y, z)) -> MINUS₂(minus₂(x, y), z)

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

POL( 0 ) = 0

POL( s₁(x₁) ) = max{0, -1}

POL( minus₂(x₁, x₂) ) = max{0, -1}

POL( MINUS₂(x₁, x₂) ) = x₂ + 1

POL( plus₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( p₁(x₁) ) = 1

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

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

minus₂(x, 0) -> x
minus₂(0, y) -> 0
minus₂(s₁(x), s₁(y)) -> minus₂(p₁(s₁(x)), p₁(s₁(y)))
minus₂(x, plus₂(y, z)) -> minus₂(minus₂(x, y), z)
p₁(s₁(s₁(x))) -> s₁(p₁(s₁(x)))
p₁(0) -> s₁(s₁(0))
div₂(s₁(x), s₁(y)) -> s₁(div₂(minus₂(x, y), s₁(y)))
div₂(plus₂(x, y), z) -> plus₂(div₂(x, z), div₂(y, z))
plus₂(0, y) -> y
plus₂(s₁(x), y) -> s₁(plus₂(y, minus₂(s₁(x), s₁(0))))

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

          ↳ QDP

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

PLUS₂(s₁(x), y) -> PLUS₂(y, minus₂(s₁(x), s₁(0)))

The TRS R consists of the following rules:

minus₂(x, 0) -> x
minus₂(0, y) -> 0
minus₂(s₁(x), s₁(y)) -> minus₂(p₁(s₁(x)), p₁(s₁(y)))
minus₂(x, plus₂(y, z)) -> minus₂(minus₂(x, y), z)
p₁(s₁(s₁(x))) -> s₁(p₁(s₁(x)))
p₁(0) -> s₁(s₁(0))
div₂(s₁(x), s₁(y)) -> s₁(div₂(minus₂(x, y), s₁(y)))
div₂(plus₂(x, y), z) -> plus₂(div₂(x, z), div₂(y, z))
plus₂(0, y) -> y
plus₂(s₁(x), y) -> s₁(plus₂(y, minus₂(s₁(x), s₁(0))))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

DIV₂(s₁(x), s₁(y)) -> DIV₂(minus₂(x, y), s₁(y))
DIV₂(plus₂(x, y), z) -> DIV₂(x, z)
DIV₂(plus₂(x, y), z) -> DIV₂(y, z)

The TRS R consists of the following rules:

minus₂(x, 0) -> x
minus₂(0, y) -> 0
minus₂(s₁(x), s₁(y)) -> minus₂(p₁(s₁(x)), p₁(s₁(y)))
minus₂(x, plus₂(y, z)) -> minus₂(minus₂(x, y), z)
p₁(s₁(s₁(x))) -> s₁(p₁(s₁(x)))
p₁(0) -> s₁(s₁(0))
div₂(s₁(x), s₁(y)) -> s₁(div₂(minus₂(x, y), s₁(y)))
div₂(plus₂(x, y), z) -> plus₂(div₂(x, z), div₂(y, z))
plus₂(0, y) -> y
plus₂(s₁(x), y) -> s₁(plus₂(y, minus₂(s₁(x), s₁(0))))

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.

DIV₂(plus₂(x, y), z) -> DIV₂(x, z)
DIV₂(plus₂(x, y), z) -> DIV₂(y, z)

The remaining pairs can at least be oriented weakly.

DIV₂(s₁(x), s₁(y)) -> DIV₂(minus₂(x, y), s₁(y))

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

POL( 0 ) = 1

POL( s₁(x₁) ) = x₁

POL( minus₂(x₁, x₂) ) = x₁

POL( DIV₂(x₁, x₂) ) = x₁ + 1

POL( plus₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( p₁(x₁) ) = 0

The following usable rules [14] were oriented:

minus₂(x, 0) -> x
minus₂(0, y) -> 0
minus₂(x, plus₂(y, z)) -> minus₂(minus₂(x, y), z)
minus₂(s₁(x), s₁(y)) -> minus₂(p₁(s₁(x)), p₁(s₁(y)))
p₁(s₁(s₁(x))) -> s₁(p₁(s₁(x)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

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

DIV₂(s₁(x), s₁(y)) -> DIV₂(minus₂(x, y), s₁(y))

The TRS R consists of the following rules:

minus₂(x, 0) -> x
minus₂(0, y) -> 0
minus₂(s₁(x), s₁(y)) -> minus₂(p₁(s₁(x)), p₁(s₁(y)))
minus₂(x, plus₂(y, z)) -> minus₂(minus₂(x, y), z)
p₁(s₁(s₁(x))) -> s₁(p₁(s₁(x)))
p₁(0) -> s₁(s₁(0))
div₂(s₁(x), s₁(y)) -> s₁(div₂(minus₂(x, y), s₁(y)))
div₂(plus₂(x, y), z) -> plus₂(div₂(x, z), div₂(y, z))
plus₂(0, y) -> y
plus₂(s₁(x), y) -> s₁(plus₂(y, minus₂(s₁(x), s₁(0))))

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.

DIV₂(s₁(x), s₁(y)) -> DIV₂(minus₂(x, y), s₁(y))

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

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

POL( s₁(x₁) ) = x₁ + 1

POL( minus₂(x₁, x₂) ) = x₁

POL( DIV₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( plus₂(x₁, x₂) ) = max{0, -1}

POL( p₁(x₁) ) = max{0, x₁ - 1}

The following usable rules [14] were oriented:

minus₂(x, 0) -> x
minus₂(0, y) -> 0
minus₂(x, plus₂(y, z)) -> minus₂(minus₂(x, y), z)
minus₂(s₁(x), s₁(y)) -> minus₂(p₁(s₁(x)), p₁(s₁(y)))
p₁(s₁(s₁(x))) -> s₁(p₁(s₁(x)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

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

minus₂(x, 0) -> x
minus₂(0, y) -> 0
minus₂(s₁(x), s₁(y)) -> minus₂(p₁(s₁(x)), p₁(s₁(y)))
minus₂(x, plus₂(y, z)) -> minus₂(minus₂(x, y), z)
p₁(s₁(s₁(x))) -> s₁(p₁(s₁(x)))
p₁(0) -> s₁(s₁(0))
div₂(s₁(x), s₁(y)) -> s₁(div₂(minus₂(x, y), s₁(y)))
div₂(plus₂(x, y), z) -> plus₂(div₂(x, z), div₂(y, z))
plus₂(0, y) -> y
plus₂(s₁(x), y) -> s₁(plus₂(y, minus₂(s₁(x), s₁(0))))

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.