Termination w.r.t. Q proof of TRS/HMt002.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:

leq₂(0, y) -> true
leq₂(s₁(x), 0) -> false
leq₂(s₁(x), s₁(y)) -> leq₂(x, y)
if₃(true, x, y) -> x
if₃(false, x, y) -> y
-₂(x, 0) -> x
-₂(s₁(x), s₁(y)) -> -₂(x, y)
mod₂(0, y) -> 0
mod₂(s₁(x), 0) -> 0
mod₂(s₁(x), s₁(y)) -> if₃(leq₂(y, x), mod₂(-₂(s₁(x), s₁(y)), s₁(y)), s₁(x))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

leq₂(0, y) -> true
leq₂(s₁(x), 0) -> false
leq₂(s₁(x), s₁(y)) -> leq₂(x, y)
if₃(true, x, y) -> x
if₃(false, x, y) -> y
-₂(x, 0) -> x
-₂(s₁(x), s₁(y)) -> -₂(x, y)
mod₂(0, y) -> 0
mod₂(s₁(x), 0) -> 0
mod₂(s₁(x), s₁(y)) -> if₃(leq₂(y, x), mod₂(-₂(s₁(x), s₁(y)), s₁(y)), 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:

MOD₂(s₁(x), s₁(y)) -> LEQ₂(y, x)
-¹₂(s₁(x), s₁(y)) -> -¹₂(x, y)
MOD₂(s₁(x), s₁(y)) -> MOD₂(-₂(s₁(x), s₁(y)), s₁(y))
LEQ₂(s₁(x), s₁(y)) -> LEQ₂(x, y)
MOD₂(s₁(x), s₁(y)) -> IF₃(leq₂(y, x), mod₂(-₂(s₁(x), s₁(y)), s₁(y)), s₁(x))
MOD₂(s₁(x), s₁(y)) -> -¹₂(s₁(x), s₁(y))

The TRS R consists of the following rules:

leq₂(0, y) -> true
leq₂(s₁(x), 0) -> false
leq₂(s₁(x), s₁(y)) -> leq₂(x, y)
if₃(true, x, y) -> x
if₃(false, x, y) -> y
-₂(x, 0) -> x
-₂(s₁(x), s₁(y)) -> -₂(x, y)
mod₂(0, y) -> 0
mod₂(s₁(x), 0) -> 0
mod₂(s₁(x), s₁(y)) -> if₃(leq₂(y, x), mod₂(-₂(s₁(x), s₁(y)), s₁(y)), 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:

MOD₂(s₁(x), s₁(y)) -> LEQ₂(y, x)
-¹₂(s₁(x), s₁(y)) -> -¹₂(x, y)
MOD₂(s₁(x), s₁(y)) -> MOD₂(-₂(s₁(x), s₁(y)), s₁(y))
LEQ₂(s₁(x), s₁(y)) -> LEQ₂(x, y)
MOD₂(s₁(x), s₁(y)) -> IF₃(leq₂(y, x), mod₂(-₂(s₁(x), s₁(y)), s₁(y)), s₁(x))
MOD₂(s₁(x), s₁(y)) -> -¹₂(s₁(x), s₁(y))

The TRS R consists of the following rules:

leq₂(0, y) -> true
leq₂(s₁(x), 0) -> false
leq₂(s₁(x), s₁(y)) -> leq₂(x, y)
if₃(true, x, y) -> x
if₃(false, x, y) -> y
-₂(x, 0) -> x
-₂(s₁(x), s₁(y)) -> -₂(x, y)
mod₂(0, y) -> 0
mod₂(s₁(x), 0) -> 0
mod₂(s₁(x), s₁(y)) -> if₃(leq₂(y, x), mod₂(-₂(s₁(x), s₁(y)), s₁(y)), 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:

-¹₂(s₁(x), s₁(y)) -> -¹₂(x, y)

The TRS R consists of the following rules:

leq₂(0, y) -> true
leq₂(s₁(x), 0) -> false
leq₂(s₁(x), s₁(y)) -> leq₂(x, y)
if₃(true, x, y) -> x
if₃(false, x, y) -> y
-₂(x, 0) -> x
-₂(s₁(x), s₁(y)) -> -₂(x, y)
mod₂(0, y) -> 0
mod₂(s₁(x), 0) -> 0
mod₂(s₁(x), s₁(y)) -> if₃(leq₂(y, x), mod₂(-₂(s₁(x), s₁(y)), s₁(y)), 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.

-¹₂(s₁(x), s₁(y)) -> -¹₂(x, y)

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

POL(-¹₂(x₁, x₂)) = x₂
POL(s₁(x₁)) = 1 + 2·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:

leq₂(0, y) -> true
leq₂(s₁(x), 0) -> false
leq₂(s₁(x), s₁(y)) -> leq₂(x, y)
if₃(true, x, y) -> x
if₃(false, x, y) -> y
-₂(x, 0) -> x
-₂(s₁(x), s₁(y)) -> -₂(x, y)
mod₂(0, y) -> 0
mod₂(s₁(x), 0) -> 0
mod₂(s₁(x), s₁(y)) -> if₃(leq₂(y, x), mod₂(-₂(s₁(x), s₁(y)), s₁(y)), 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:

LEQ₂(s₁(x), s₁(y)) -> LEQ₂(x, y)

The TRS R consists of the following rules:

leq₂(0, y) -> true
leq₂(s₁(x), 0) -> false
leq₂(s₁(x), s₁(y)) -> leq₂(x, y)
if₃(true, x, y) -> x
if₃(false, x, y) -> y
-₂(x, 0) -> x
-₂(s₁(x), s₁(y)) -> -₂(x, y)
mod₂(0, y) -> 0
mod₂(s₁(x), 0) -> 0
mod₂(s₁(x), s₁(y)) -> if₃(leq₂(y, x), mod₂(-₂(s₁(x), s₁(y)), s₁(y)), 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.

LEQ₂(s₁(x), s₁(y)) -> LEQ₂(x, y)

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

POL(LEQ₂(x₁, x₂)) = x₂
POL(s₁(x₁)) = 1 + 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:

leq₂(0, y) -> true
leq₂(s₁(x), 0) -> false
leq₂(s₁(x), s₁(y)) -> leq₂(x, y)
if₃(true, x, y) -> x
if₃(false, x, y) -> y
-₂(x, 0) -> x
-₂(s₁(x), s₁(y)) -> -₂(x, y)
mod₂(0, y) -> 0
mod₂(s₁(x), 0) -> 0
mod₂(s₁(x), s₁(y)) -> if₃(leq₂(y, x), mod₂(-₂(s₁(x), s₁(y)), s₁(y)), 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

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

MOD₂(s₁(x), s₁(y)) -> MOD₂(-₂(s₁(x), s₁(y)), s₁(y))

The TRS R consists of the following rules:

leq₂(0, y) -> true
leq₂(s₁(x), 0) -> false
leq₂(s₁(x), s₁(y)) -> leq₂(x, y)
if₃(true, x, y) -> x
if₃(false, x, y) -> y
-₂(x, 0) -> x
-₂(s₁(x), s₁(y)) -> -₂(x, y)
mod₂(0, y) -> 0
mod₂(s₁(x), 0) -> 0
mod₂(s₁(x), s₁(y)) -> if₃(leq₂(y, x), mod₂(-₂(s₁(x), s₁(y)), s₁(y)), s₁(x))

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