Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar14/nonterminSLASHAG01SLASHHASH4.30.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(s₁(x), s₁(y)) -> minus₂(x, y)
le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
quot₂(x, s₁(y)) -> if_quot₃(le₂(s₁(y), x), x, s₁(y))
if_quot₃(true, x, y) -> s₁(quot₂(minus₂(x, y), y))
if_quot₃(false, x, y) -> 0

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

minus₂(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
quot₂(x, s₁(y)) -> if_quot₃(le₂(s₁(y), x), x, s₁(y))
if_quot₃(true, x, y) -> s₁(quot₂(minus₂(x, y), y))
if_quot₃(false, x, y) -> 0

Q is empty.

The TRS is non-overlapping. Hence, we can switch to innermost.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

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

minus₂(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
quot₂(x, s₁(y)) -> if_quot₃(le₂(s₁(y), x), x, s₁(y))
if_quot₃(true, x, y) -> s₁(quot₂(minus₂(x, y), y))
if_quot₃(false, x, y) -> 0

The set Q consists of the following terms:

minus₂(x0, 0)
minus₂(s₁(x0), s₁(x1))
le₂(0, x0)
le₂(s₁(x0), 0)
le₂(s₁(x0), s₁(x1))
quot₂(x0, s₁(x1))
if_quot₃(true, x0, x1)
if_quot₃(false, x0, x1)

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_QUOT₃(true, x, y) -> QUOT₂(minus₂(x, y), y)
QUOT₂(x, s₁(y)) -> IF_QUOT₃(le₂(s₁(y), x), x, s₁(y))
IF_QUOT₃(true, x, y) -> MINUS₂(x, y)
LE₂(s₁(x), s₁(y)) -> LE₂(x, y)
QUOT₂(x, s₁(y)) -> LE₂(s₁(y), x)
MINUS₂(s₁(x), s₁(y)) -> MINUS₂(x, y)

The TRS R consists of the following rules:

minus₂(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
quot₂(x, s₁(y)) -> if_quot₃(le₂(s₁(y), x), x, s₁(y))
if_quot₃(true, x, y) -> s₁(quot₂(minus₂(x, y), y))
if_quot₃(false, x, y) -> 0

The set Q consists of the following terms:

minus₂(x0, 0)
minus₂(s₁(x0), s₁(x1))
le₂(0, x0)
le₂(s₁(x0), 0)
le₂(s₁(x0), s₁(x1))
quot₂(x0, s₁(x1))
if_quot₃(true, x0, x1)
if_quot₃(false, x0, x1)

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

IF_QUOT₃(true, x, y) -> QUOT₂(minus₂(x, y), y)
QUOT₂(x, s₁(y)) -> IF_QUOT₃(le₂(s₁(y), x), x, s₁(y))
IF_QUOT₃(true, x, y) -> MINUS₂(x, y)
LE₂(s₁(x), s₁(y)) -> LE₂(x, y)
QUOT₂(x, s₁(y)) -> LE₂(s₁(y), x)
MINUS₂(s₁(x), s₁(y)) -> MINUS₂(x, y)

The TRS R consists of the following rules:

minus₂(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
quot₂(x, s₁(y)) -> if_quot₃(le₂(s₁(y), x), x, s₁(y))
if_quot₃(true, x, y) -> s₁(quot₂(minus₂(x, y), y))
if_quot₃(false, x, y) -> 0

The set Q consists of the following terms:

minus₂(x0, 0)
minus₂(s₁(x0), s₁(x1))
le₂(0, x0)
le₂(s₁(x0), 0)
le₂(s₁(x0), s₁(x1))
quot₂(x0, s₁(x1))
if_quot₃(true, x0, x1)
if_quot₃(false, x0, x1)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 3 SCCs with 2 less nodes.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPOrderProof

              ↳ QDP

              ↳ 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:

minus₂(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
quot₂(x, s₁(y)) -> if_quot₃(le₂(s₁(y), x), x, s₁(y))
if_quot₃(true, x, y) -> s₁(quot₂(minus₂(x, y), y))
if_quot₃(false, x, y) -> 0

The set Q consists of the following terms:

minus₂(x0, 0)
minus₂(s₁(x0), s₁(x1))
le₂(0, x0)
le₂(s₁(x0), 0)
le₂(s₁(x0), s₁(x1))
quot₂(x0, s₁(x1))
if_quot₃(true, x0, x1)
if_quot₃(false, x0, x1)

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be strictly oriented and are deleted.

LE₂(s₁(x), s₁(y)) -> LE₂(x, y)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
LE₂(x1, x2) = LE₁(x1)
s₁(x1) = s₁(x1)

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Non-Overlap Check

    ↳ 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(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
quot₂(x, s₁(y)) -> if_quot₃(le₂(s₁(y), x), x, s₁(y))
if_quot₃(true, x, y) -> s₁(quot₂(minus₂(x, y), y))
if_quot₃(false, x, y) -> 0

The set Q consists of the following terms:

minus₂(x0, 0)
minus₂(s₁(x0), s₁(x1))
le₂(0, x0)
le₂(s₁(x0), 0)
le₂(s₁(x0), s₁(x1))
quot₂(x0, s₁(x1))
if_quot₃(true, x0, x1)
if_quot₃(false, x0, x1)

We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ 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(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
quot₂(x, s₁(y)) -> if_quot₃(le₂(s₁(y), x), x, s₁(y))
if_quot₃(true, x, y) -> s₁(quot₂(minus₂(x, y), y))
if_quot₃(false, x, y) -> 0

The set Q consists of the following terms:

minus₂(x0, 0)
minus₂(s₁(x0), s₁(x1))
le₂(0, x0)
le₂(s₁(x0), 0)
le₂(s₁(x0), s₁(x1))
quot₂(x0, s₁(x1))
if_quot₃(true, x0, x1)
if_quot₃(false, x0, x1)

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be strictly oriented and are deleted.

MINUS₂(s₁(x), s₁(y)) -> MINUS₂(x, y)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
MINUS₂(x1, x2) = MINUS₁(x1)
s₁(x1) = s₁(x1)

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Non-Overlap Check

    ↳ 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(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
quot₂(x, s₁(y)) -> if_quot₃(le₂(s₁(y), x), x, s₁(y))
if_quot₃(true, x, y) -> s₁(quot₂(minus₂(x, y), y))
if_quot₃(false, x, y) -> 0

The set Q consists of the following terms:

minus₂(x0, 0)
minus₂(s₁(x0), s₁(x1))
le₂(0, x0)
le₂(s₁(x0), 0)
le₂(s₁(x0), s₁(x1))
quot₂(x0, s₁(x1))
if_quot₃(true, x0, x1)
if_quot₃(false, x0, x1)

We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

IF_QUOT₃(true, x, y) -> QUOT₂(minus₂(x, y), y)
QUOT₂(x, s₁(y)) -> IF_QUOT₃(le₂(s₁(y), x), x, s₁(y))

The TRS R consists of the following rules:

minus₂(x, 0) -> x
minus₂(s₁(x), s₁(y)) -> minus₂(x, y)
le₂(0, y) -> true
le₂(s₁(x), 0) -> false
le₂(s₁(x), s₁(y)) -> le₂(x, y)
quot₂(x, s₁(y)) -> if_quot₃(le₂(s₁(y), x), x, s₁(y))
if_quot₃(true, x, y) -> s₁(quot₂(minus₂(x, y), y))
if_quot₃(false, x, y) -> 0

The set Q consists of the following terms:

minus₂(x0, 0)
minus₂(s₁(x0), s₁(x1))
le₂(0, x0)
le₂(s₁(x0), 0)
le₂(s₁(x0), s₁(x1))
quot₂(x0, s₁(x1))
if_quot₃(true, x0, x1)
if_quot₃(false, x0, x1)

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