Termination w.r.t. Q proof of TRS/nontermin/CSREx49

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:

minus₂(0, Y) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
geq₂(X, 0) -> true
geq₂(0, s₁(Y)) -> false
geq₂(s₁(X), s₁(Y)) -> geq₂(X, Y)
div₂(0, s₁(Y)) -> 0
div₂(s₁(X), s₁(Y)) -> if₃(geq₂(X, Y), s₁(div₂(minus₂(X, Y), s₁(Y))), 0)
if₃(true, X, Y) -> X
if₃(false, X, Y) -> Y

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

minus₂(0, Y) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
geq₂(X, 0) -> true
geq₂(0, s₁(Y)) -> false
geq₂(s₁(X), s₁(Y)) -> geq₂(X, Y)
div₂(0, s₁(Y)) -> 0
div₂(s₁(X), s₁(Y)) -> if₃(geq₂(X, Y), s₁(div₂(minus₂(X, Y), s₁(Y))), 0)
if₃(true, X, Y) -> X
if₃(false, X, Y) -> 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:

DIV₂(s₁(X), s₁(Y)) -> IF₃(geq₂(X, Y), s₁(div₂(minus₂(X, Y), s₁(Y))), 0)
DIV₂(s₁(X), s₁(Y)) -> MINUS₂(X, Y)
DIV₂(s₁(X), s₁(Y)) -> DIV₂(minus₂(X, Y), s₁(Y))
DIV₂(s₁(X), s₁(Y)) -> GEQ₂(X, Y)
GEQ₂(s₁(X), s₁(Y)) -> GEQ₂(X, Y)
MINUS₂(s₁(X), s₁(Y)) -> MINUS₂(X, Y)

The TRS R consists of the following rules:

minus₂(0, Y) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
geq₂(X, 0) -> true
geq₂(0, s₁(Y)) -> false
geq₂(s₁(X), s₁(Y)) -> geq₂(X, Y)
div₂(0, s₁(Y)) -> 0
div₂(s₁(X), s₁(Y)) -> if₃(geq₂(X, Y), s₁(div₂(minus₂(X, Y), s₁(Y))), 0)
if₃(true, X, Y) -> X
if₃(false, X, Y) -> 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:

DIV₂(s₁(X), s₁(Y)) -> IF₃(geq₂(X, Y), s₁(div₂(minus₂(X, Y), s₁(Y))), 0)
DIV₂(s₁(X), s₁(Y)) -> MINUS₂(X, Y)
DIV₂(s₁(X), s₁(Y)) -> DIV₂(minus₂(X, Y), s₁(Y))
DIV₂(s₁(X), s₁(Y)) -> GEQ₂(X, Y)
GEQ₂(s₁(X), s₁(Y)) -> GEQ₂(X, Y)
MINUS₂(s₁(X), s₁(Y)) -> MINUS₂(X, Y)

The TRS R consists of the following rules:

minus₂(0, Y) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
geq₂(X, 0) -> true
geq₂(0, s₁(Y)) -> false
geq₂(s₁(X), s₁(Y)) -> geq₂(X, Y)
div₂(0, s₁(Y)) -> 0
div₂(s₁(X), s₁(Y)) -> if₃(geq₂(X, Y), s₁(div₂(minus₂(X, Y), s₁(Y))), 0)
if₃(true, X, Y) -> X
if₃(false, X, Y) -> Y

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:

GEQ₂(s₁(X), s₁(Y)) -> GEQ₂(X, Y)

The TRS R consists of the following rules:

minus₂(0, Y) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
geq₂(X, 0) -> true
geq₂(0, s₁(Y)) -> false
geq₂(s₁(X), s₁(Y)) -> geq₂(X, Y)
div₂(0, s₁(Y)) -> 0
div₂(s₁(X), s₁(Y)) -> if₃(geq₂(X, Y), s₁(div₂(minus₂(X, Y), s₁(Y))), 0)
if₃(true, X, Y) -> X
if₃(false, X, Y) -> 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.

GEQ₂(s₁(X), s₁(Y)) -> GEQ₂(X, Y)

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

POL(GEQ₂(x₁, x₂)) = x₁
POL(s₁(x₁)) = 1 + x₁

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:

minus₂(0, Y) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
geq₂(X, 0) -> true
geq₂(0, s₁(Y)) -> false
geq₂(s₁(X), s₁(Y)) -> geq₂(X, Y)
div₂(0, s₁(Y)) -> 0
div₂(s₁(X), s₁(Y)) -> if₃(geq₂(X, Y), s₁(div₂(minus₂(X, Y), s₁(Y))), 0)
if₃(true, X, Y) -> X
if₃(false, X, Y) -> 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

          ↳ 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:

minus₂(0, Y) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
geq₂(X, 0) -> true
geq₂(0, s₁(Y)) -> false
geq₂(s₁(X), s₁(Y)) -> geq₂(X, Y)
div₂(0, s₁(Y)) -> 0
div₂(s₁(X), s₁(Y)) -> if₃(geq₂(X, Y), s₁(div₂(minus₂(X, Y), s₁(Y))), 0)
if₃(true, X, Y) -> X
if₃(false, X, Y) -> 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.

MINUS₂(s₁(X), s₁(Y)) -> MINUS₂(X, Y)

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

POL(MINUS₂(x₁, x₂)) = x₁
POL(s₁(x₁)) = 1 + 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:

minus₂(0, Y) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
geq₂(X, 0) -> true
geq₂(0, s₁(Y)) -> false
geq₂(s₁(X), s₁(Y)) -> geq₂(X, Y)
div₂(0, s₁(Y)) -> 0
div₂(s₁(X), s₁(Y)) -> if₃(geq₂(X, Y), s₁(div₂(minus₂(X, Y), s₁(Y))), 0)
if₃(true, X, Y) -> X
if₃(false, X, Y) -> 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

          ↳ 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(0, Y) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
geq₂(X, 0) -> true
geq₂(0, s₁(Y)) -> false
geq₂(s₁(X), s₁(Y)) -> geq₂(X, Y)
div₂(0, s₁(Y)) -> 0
div₂(s₁(X), s₁(Y)) -> if₃(geq₂(X, Y), s₁(div₂(minus₂(X, Y), s₁(Y))), 0)
if₃(true, X, Y) -> X
if₃(false, X, Y) -> 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.

DIV₂(s₁(X), s₁(Y)) -> DIV₂(minus₂(X, Y), s₁(Y))

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

POL(0) = 0
POL(DIV₂(x₁, x₂)) = x₁
POL(minus₂(x₁, x₂)) = 0
POL(s₁(x₁)) = 1

The following usable rules [14] were oriented:

minus₂(0, Y) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

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

minus₂(0, Y) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
geq₂(X, 0) -> true
geq₂(0, s₁(Y)) -> false
geq₂(s₁(X), s₁(Y)) -> geq₂(X, Y)
div₂(0, s₁(Y)) -> 0
div₂(s₁(X), s₁(Y)) -> if₃(geq₂(X, Y), s₁(div₂(minus₂(X, Y), s₁(Y))), 0)
if₃(true, X, Y) -> X
if₃(false, X, Y) -> 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.