Termination w.r.t. Q proof of TRS/AG01#3.6a.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:

le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
minus₂(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
gcd₂(0, y) -> y
gcd₂(s₁(x), 0) -> s₁(x)
gcd₂(s₁(x), s₁(y)) -> if_gcd₃(le₂(y, x), s₁(x), s₁(y))
if_gcd₃(true, s₁(x), s₁(y)) -> gcd₂(minus₂(x, y), s₁(y))
if_gcd₃(false, s₁(x), s₁(y)) -> gcd₂(minus₂(y, x), s₁(x))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
minus₂(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
gcd₂(0, y) -> y
gcd₂(s₁(x), 0) -> s₁(x)
gcd₂(s₁(x), s₁(y)) -> if_gcd₃(le₂(y, x), s₁(x), s₁(y))
if_gcd₃(true, s₁(x), s₁(y)) -> gcd₂(minus₂(x, y), s₁(y))
if_gcd₃(false, s₁(x), s₁(y)) -> gcd₂(minus₂(y, x), s₁(x))

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:

IF_GCD₃(false, s₁(x), s₁(y)) -> GCD₂(minus₂(y, x), s₁(x))
IF_GCD₃(true, s₁(x), s₁(y)) -> MINUS₂(x, y)
LE₂(s₁(x), s₁(y)) -> LE₂(x, y)
GCD₂(s₁(x), s₁(y)) -> LE₂(y, x)
MINUS₂(s₁(x), s₁(y)) -> MINUS₂(x, y)
IF_GCD₃(true, s₁(x), s₁(y)) -> GCD₂(minus₂(x, y), s₁(y))
GCD₂(s₁(x), s₁(y)) -> IF_GCD₃(le₂(y, x), s₁(x), s₁(y))
IF_GCD₃(false, s₁(x), s₁(y)) -> MINUS₂(y, x)

The TRS R consists of the following rules:

le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
minus₂(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
gcd₂(0, y) -> y
gcd₂(s₁(x), 0) -> s₁(x)
gcd₂(s₁(x), s₁(y)) -> if_gcd₃(le₂(y, x), s₁(x), s₁(y))
if_gcd₃(true, s₁(x), s₁(y)) -> gcd₂(minus₂(x, y), s₁(y))
if_gcd₃(false, s₁(x), s₁(y)) -> gcd₂(minus₂(y, x), s₁(x))

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:

IF_GCD₃(false, s₁(x), s₁(y)) -> GCD₂(minus₂(y, x), s₁(x))
IF_GCD₃(true, s₁(x), s₁(y)) -> MINUS₂(x, y)
LE₂(s₁(x), s₁(y)) -> LE₂(x, y)
GCD₂(s₁(x), s₁(y)) -> LE₂(y, x)
MINUS₂(s₁(x), s₁(y)) -> MINUS₂(x, y)
IF_GCD₃(true, s₁(x), s₁(y)) -> GCD₂(minus₂(x, y), s₁(y))
GCD₂(s₁(x), s₁(y)) -> IF_GCD₃(le₂(y, x), s₁(x), s₁(y))
IF_GCD₃(false, s₁(x), s₁(y)) -> MINUS₂(y, x)

The TRS R consists of the following rules:

le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
minus₂(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
gcd₂(0, y) -> y
gcd₂(s₁(x), 0) -> s₁(x)
gcd₂(s₁(x), s₁(y)) -> if_gcd₃(le₂(y, x), s₁(x), s₁(y))
if_gcd₃(true, s₁(x), s₁(y)) -> gcd₂(minus₂(x, y), s₁(y))
if_gcd₃(false, s₁(x), s₁(y)) -> gcd₂(minus₂(y, x), s₁(x))

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 3 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

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

MINUS₂(s₁(x), s₁(y)) -> MINUS₂(x, y)

The TRS R consists of the following rules:

le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
minus₂(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
gcd₂(0, y) -> y
gcd₂(s₁(x), 0) -> s₁(x)
gcd₂(s₁(x), s₁(y)) -> if_gcd₃(le₂(y, x), s₁(x), s₁(y))
if_gcd₃(true, s₁(x), s₁(y)) -> gcd₂(minus₂(x, y), s₁(y))
if_gcd₃(false, s₁(x), s₁(y)) -> gcd₂(minus₂(y, x), s₁(x))

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(x, y)

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

POL( MINUS₂(x₁, x₂) ) = 3x₂ + 3

POL( s₁(x₁) ) = x₁ + 3

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:

le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
minus₂(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
gcd₂(0, y) -> y
gcd₂(s₁(x), 0) -> s₁(x)
gcd₂(s₁(x), s₁(y)) -> if_gcd₃(le₂(y, x), s₁(x), s₁(y))
if_gcd₃(true, s₁(x), s₁(y)) -> gcd₂(minus₂(x, y), s₁(y))
if_gcd₃(false, s₁(x), s₁(y)) -> gcd₂(minus₂(y, x), s₁(x))

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:

LE₂(s₁(x), s₁(y)) -> LE₂(x, y)

The TRS R consists of the following rules:

le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
minus₂(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
gcd₂(0, y) -> y
gcd₂(s₁(x), 0) -> s₁(x)
gcd₂(s₁(x), s₁(y)) -> if_gcd₃(le₂(y, x), s₁(x), s₁(y))
if_gcd₃(true, s₁(x), s₁(y)) -> gcd₂(minus₂(x, y), s₁(y))
if_gcd₃(false, s₁(x), s₁(y)) -> gcd₂(minus₂(y, x), s₁(x))

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.

LE₂(s₁(x), s₁(y)) -> LE₂(x, y)

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

POL( LE₂(x₁, x₂) ) = 3x₂ + 3

POL( s₁(x₁) ) = x₁ + 3

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:

le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
minus₂(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
gcd₂(0, y) -> y
gcd₂(s₁(x), 0) -> s₁(x)
gcd₂(s₁(x), s₁(y)) -> if_gcd₃(le₂(y, x), s₁(x), s₁(y))
if_gcd₃(true, s₁(x), s₁(y)) -> gcd₂(minus₂(x, y), s₁(y))
if_gcd₃(false, s₁(x), s₁(y)) -> gcd₂(minus₂(y, x), s₁(x))

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:

IF_GCD₃(false, s₁(x), s₁(y)) -> GCD₂(minus₂(y, x), s₁(x))
IF_GCD₃(true, s₁(x), s₁(y)) -> GCD₂(minus₂(x, y), s₁(y))
GCD₂(s₁(x), s₁(y)) -> IF_GCD₃(le₂(y, x), s₁(x), s₁(y))

The TRS R consists of the following rules:

le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
minus₂(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
gcd₂(0, y) -> y
gcd₂(s₁(x), 0) -> s₁(x)
gcd₂(s₁(x), s₁(y)) -> if_gcd₃(le₂(y, x), s₁(x), s₁(y))
if_gcd₃(true, s₁(x), s₁(y)) -> gcd₂(minus₂(x, y), s₁(y))
if_gcd₃(false, s₁(x), s₁(y)) -> gcd₂(minus₂(y, x), s₁(x))

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_GCD₃(false, s₁(x), s₁(y)) -> GCD₂(minus₂(y, x), s₁(x))

The remaining pairs can at least be oriented weakly.

IF_GCD₃(true, s₁(x), s₁(y)) -> GCD₂(minus₂(x, y), s₁(y))
GCD₂(s₁(x), s₁(y)) -> IF_GCD₃(le₂(y, x), s₁(x), s₁(y))

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

POL( minus₂(x₁, x₂) ) = x₁

POL( true ) = 2

POL( false ) = 3

POL( GCD₂(x₁, x₂) ) = 3x₁ + x₂

POL( IF_GCD₃(x₁, ..., x₃) ) = max{0, x₁ + 2x₂ + x₃ - 2}

POL( 0 ) = 3

POL( s₁(x₁) ) = 3x₁

POL( le₂(x₁, x₂) ) = 3x₂ + 2

The following usable rules [14] were oriented:

le₂(s₁(x), s₁(y)) -> le₂(x, y)
le₂(0, y) -> true
minus₂(x, 0) -> x
le₂(s₁(x), 0) -> false
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

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

IF_GCD₃(true, s₁(x), s₁(y)) -> GCD₂(minus₂(x, y), s₁(y))
GCD₂(s₁(x), s₁(y)) -> IF_GCD₃(le₂(y, x), s₁(x), s₁(y))

The TRS R consists of the following rules:

le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
minus₂(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
gcd₂(0, y) -> y
gcd₂(s₁(x), 0) -> s₁(x)
gcd₂(s₁(x), s₁(y)) -> if_gcd₃(le₂(y, x), s₁(x), s₁(y))
if_gcd₃(true, s₁(x), s₁(y)) -> gcd₂(minus₂(x, y), s₁(y))
if_gcd₃(false, s₁(x), s₁(y)) -> gcd₂(minus₂(y, x), s₁(x))

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.

GCD₂(s₁(x), s₁(y)) -> IF_GCD₃(le₂(y, x), s₁(x), s₁(y))

The remaining pairs can at least be oriented weakly.

IF_GCD₃(true, s₁(x), s₁(y)) -> GCD₂(minus₂(x, y), s₁(y))

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

POL( minus₂(x₁, x₂) ) = x₁

POL( true ) = 1

POL( false ) = 2

POL( GCD₂(x₁, x₂) ) = max{0, 2x₁ - 1}

POL( IF_GCD₃(x₁, ..., x₃) ) = max{0, 2x₂ - 2}

POL( 0 ) = 0

POL( s₁(x₁) ) = x₁ + 1

POL( le₂(x₁, x₂) ) = 3x₂ + 1

The following usable rules [14] were oriented:

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

↳ 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_GCD₃(true, s₁(x), s₁(y)) -> GCD₂(minus₂(x, y), s₁(y))

The TRS R consists of the following rules:

le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
minus₂(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
gcd₂(0, y) -> y
gcd₂(s₁(x), 0) -> s₁(x)
gcd₂(s₁(x), s₁(y)) -> if_gcd₃(le₂(y, x), s₁(x), s₁(y))
if_gcd₃(true, s₁(x), s₁(y)) -> gcd₂(minus₂(x, y), s₁(y))
if_gcd₃(false, s₁(x), s₁(y)) -> gcd₂(minus₂(y, x), s₁(x))

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.