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:

qsort(nil) → nil
qsort(.(x, y)) → ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
lowers(x, nil) → nil
lowers(x, .(y, z)) → if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
greaters(x, nil) → nil
greaters(x, .(y, z)) → if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

Q is empty.


QTRS
  ↳ Overlay + Local Confluence

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

qsort(nil) → nil
qsort(.(x, y)) → ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
lowers(x, nil) → nil
lowers(x, .(y, z)) → if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
greaters(x, nil) → nil
greaters(x, .(y, z)) → if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

Q is empty.

The TRS is overlay and locally confluent. By [15] we can switch to innermost.

↳ QTRS
  ↳ Overlay + Local Confluence
QTRS
      ↳ DependencyPairsProof

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

qsort(nil) → nil
qsort(.(x, y)) → ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
lowers(x, nil) → nil
lowers(x, .(y, z)) → if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
greaters(x, nil) → nil
greaters(x, .(y, z)) → if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

The set Q consists of the following terms:

qsort(nil)
qsort(.(x0, x1))
lowers(x0, nil)
lowers(x0, .(x1, x2))
greaters(x0, nil)
greaters(x0, .(x1, x2))


Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

QSORT(.(x, y)) → LOWERS(x, y)
GREATERS(x, .(y, z)) → GREATERS(x, z)
LOWERS(x, .(y, z)) → LOWERS(x, z)
QSORT(.(x, y)) → GREATERS(x, y)
QSORT(.(x, y)) → QSORT(lowers(x, y))
QSORT(.(x, y)) → QSORT(greaters(x, y))

The TRS R consists of the following rules:

qsort(nil) → nil
qsort(.(x, y)) → ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
lowers(x, nil) → nil
lowers(x, .(y, z)) → if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
greaters(x, nil) → nil
greaters(x, .(y, z)) → if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

The set Q consists of the following terms:

qsort(nil)
qsort(.(x0, x1))
lowers(x0, nil)
lowers(x0, .(x1, x2))
greaters(x0, nil)
greaters(x0, .(x1, x2))

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

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
QDP
          ↳ EdgeDeletionProof

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

QSORT(.(x, y)) → LOWERS(x, y)
GREATERS(x, .(y, z)) → GREATERS(x, z)
LOWERS(x, .(y, z)) → LOWERS(x, z)
QSORT(.(x, y)) → GREATERS(x, y)
QSORT(.(x, y)) → QSORT(lowers(x, y))
QSORT(.(x, y)) → QSORT(greaters(x, y))

The TRS R consists of the following rules:

qsort(nil) → nil
qsort(.(x, y)) → ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
lowers(x, nil) → nil
lowers(x, .(y, z)) → if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
greaters(x, nil) → nil
greaters(x, .(y, z)) → if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

The set Q consists of the following terms:

qsort(nil)
qsort(.(x0, x1))
lowers(x0, nil)
lowers(x0, .(x1, x2))
greaters(x0, nil)
greaters(x0, .(x1, x2))

We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
QDP
              ↳ DependencyGraphProof

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

QSORT(.(x, y)) → LOWERS(x, y)
GREATERS(x, .(y, z)) → GREATERS(x, z)
LOWERS(x, .(y, z)) → LOWERS(x, z)
QSORT(.(x, y)) → QSORT(lowers(x, y))
QSORT(.(x, y)) → GREATERS(x, y)
QSORT(.(x, y)) → QSORT(greaters(x, y))

The TRS R consists of the following rules:

qsort(nil) → nil
qsort(.(x, y)) → ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
lowers(x, nil) → nil
lowers(x, .(y, z)) → if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
greaters(x, nil) → nil
greaters(x, .(y, z)) → if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

The set Q consists of the following terms:

qsort(nil)
qsort(.(x0, x1))
lowers(x0, nil)
lowers(x0, .(x1, x2))
greaters(x0, nil)
greaters(x0, .(x1, x2))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 2 SCCs with 4 less nodes.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
QDP
                    ↳ QDPOrderProof
                  ↳ QDP

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

GREATERS(x, .(y, z)) → GREATERS(x, z)

The TRS R consists of the following rules:

qsort(nil) → nil
qsort(.(x, y)) → ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
lowers(x, nil) → nil
lowers(x, .(y, z)) → if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
greaters(x, nil) → nil
greaters(x, .(y, z)) → if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

The set Q consists of the following terms:

qsort(nil)
qsort(.(x0, x1))
lowers(x0, nil)
lowers(x0, .(x1, x2))
greaters(x0, nil)
greaters(x0, .(x1, x2))

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.


GREATERS(x, .(y, z)) → GREATERS(x, z)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
GREATERS(x1, x2)  =  GREATERS(x1, x2)
.(x1, x2)  =  .(x2)

Recursive path order with status [2].
Quasi-Precedence:
trivial

Status:
.1: multiset
GREATERS2: [2,1]


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ QDPOrderProof
QDP
                        ↳ PisEmptyProof
                  ↳ QDP

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

qsort(nil) → nil
qsort(.(x, y)) → ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
lowers(x, nil) → nil
lowers(x, .(y, z)) → if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
greaters(x, nil) → nil
greaters(x, .(y, z)) → if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

The set Q consists of the following terms:

qsort(nil)
qsort(.(x0, x1))
lowers(x0, nil)
lowers(x0, .(x1, x2))
greaters(x0, nil)
greaters(x0, .(x1, x2))

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

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
QDP
                    ↳ QDPOrderProof

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

LOWERS(x, .(y, z)) → LOWERS(x, z)

The TRS R consists of the following rules:

qsort(nil) → nil
qsort(.(x, y)) → ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
lowers(x, nil) → nil
lowers(x, .(y, z)) → if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
greaters(x, nil) → nil
greaters(x, .(y, z)) → if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

The set Q consists of the following terms:

qsort(nil)
qsort(.(x0, x1))
lowers(x0, nil)
lowers(x0, .(x1, x2))
greaters(x0, nil)
greaters(x0, .(x1, x2))

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.


LOWERS(x, .(y, z)) → LOWERS(x, z)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
LOWERS(x1, x2)  =  LOWERS(x1, x2)
.(x1, x2)  =  .(x2)

Recursive path order with status [2].
Quasi-Precedence:
[LOWERS2, .1]

Status:
.1: multiset
LOWERS2: [1,2]


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ QDPOrderProof
QDP
                        ↳ PisEmptyProof

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

qsort(nil) → nil
qsort(.(x, y)) → ++(qsort(lowers(x, y)), .(x, qsort(greaters(x, y))))
lowers(x, nil) → nil
lowers(x, .(y, z)) → if(<=(y, x), .(y, lowers(x, z)), lowers(x, z))
greaters(x, nil) → nil
greaters(x, .(y, z)) → if(<=(y, x), greaters(x, z), .(y, greaters(x, z)))

The set Q consists of the following terms:

qsort(nil)
qsort(.(x0, x1))
lowers(x0, nil)
lowers(x0, .(x1, x2))
greaters(x0, nil)
greaters(x0, .(x1, x2))

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