Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 Overlay + Local Confluence (⇔)
↳2 QTRS
↳3 DependencyPairsProof (⇔)
↳4 QDP
↳5 DependencyGraphProof (⇔)
↳6 AND
  ↳7 QDP
    ↳8 QDPOrderProof (⇔)
    ↳9 QDP
    ↳10 PisEmptyProof (⇔)
    ↳11 TRUE
  ↳12 QDP
    ↳13 QDPOrderProof (⇔)
    ↳14 QDP
    ↳15 PisEmptyProof (⇔)
    ↳16 TRUE

(0) Obligation:

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.

(1) Overlay + Local Confluence (EQUIVALENT transformation)

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

(2) Obligation:

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))

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(4) Obligation:

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

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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.

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


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.
Used ordering: Combined order from the following AFS and order.
GREATERS(x1, x2)  =  GREATERS(x1, x2)
.(x1, x2)  =  .(x1, x2)
qsort(x1)  =  qsort(x1)
nil  =  nil
++(x1, x2)  =  ++
lowers(x1, x2)  =  lowers(x2)
greaters(x1, x2)  =  greaters(x2)
if(x1, x2, x3)  =  if
<=(x1, x2)  =  <=

Recursive Path Order [RPO].
Precedence:
GREATERS2 > [nil, ++, if, <=]
[.2, qsort1] > lowers1 > [nil, ++, if, <=]
[.2, qsort1] > greaters1 > [nil, ++, if, <=]


The following usable rules [FROCOS05] were oriented:

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)))

(9) Obligation:

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.

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE

(12) Obligation:

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.

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


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.
Used ordering: Combined order from the following AFS and order.
LOWERS(x1, x2)  =  LOWERS(x1, x2)
.(x1, x2)  =  .(x1, x2)
qsort(x1)  =  qsort(x1)
nil  =  nil
++(x1, x2)  =  ++
lowers(x1, x2)  =  lowers(x2)
greaters(x1, x2)  =  greaters(x2)
if(x1, x2, x3)  =  if
<=(x1, x2)  =  <=

Recursive Path Order [RPO].
Precedence:
LOWERS2 > [nil, ++, if, <=]
[.2, qsort1] > lowers1 > [nil, ++, if, <=]
[.2, qsort1] > greaters1 > [nil, ++, if, <=]


The following usable rules [FROCOS05] were oriented:

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)))

(14) Obligation:

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.

(15) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(16) TRUE