Termination proof of /tmp/tmpbq2673/quicksort.itrs

Termination of the given ITRSProblem could successfully be proven:

↳ ITRS

  ↳ ITRStoQTRSProof

ITRS problem:
The following domains are used:

z

The TRS R consists of the following rules:

if_3(pair(yl, yh), x, ys) → app(qsort(yl), cons(x, qsort(yh)))
if_1(pair(zl, zh), x, y, zs) → Cond_if_1(>@z(x, y), zl, zh, x, y, zs)
qsort(ins(x, ys)) → if_3(split(x, ys), x, ys)
if_2(pair(zl, zh), x, y, zs) → Cond_if_2(>=@z(y, x), zl, zh, x, y, zs)
Cond_if_2(TRUE, zl, zh, x, y, zs) → pair(zl, ins(y, zh))
app(cons(x, ys), zs) → cons(x, app(ys, zs))
split(x, e) → pair(e, e)
Cond_if_1(TRUE, zl, zh, x, y, zs) → pair(ins(y, zl), zh)
Cond_split(TRUE, x, y, zs) → if_1(split(x, zs), x, y, zs)
split(x, ins(y, zs)) → Cond_split1(>=@z(y, x), x, y, zs)
Cond_split1(TRUE, x, y, zs) → if_2(split(x, zs), x, y, zs)
qsort(e) → nil
app(nil, zs) → zs
split(x, ins(y, zs)) → Cond_split(>@z(x, y), x, y, zs)

The set Q consists of the following terms:

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(TRUE, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(TRUE, x0, x1, x2, x3, x4)
Cond_split(TRUE, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(TRUE, x0, x1, x2)
qsort(e)
app(nil, x0)

Represented integers and predefined function symbols by Terms

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

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

if_3(pair(yl, yh), x, ys) → app(qsort(yl), cons(x, qsort(yh)))
if_1(pair(zl, zh), x, y, zs) → Cond_if_1(greater_int(x, y), zl, zh, x, y, zs)
qsort(ins(x, ys)) → if_3(split(x, ys), x, ys)
if_2(pair(zl, zh), x, y, zs) → Cond_if_2(greatereq_int(y, x), zl, zh, x, y, zs)
Cond_if_2(true, zl, zh, x, y, zs) → pair(zl, ins(y, zh))
app(cons(x, ys), zs) → cons(x, app(ys, zs))
split(x, e) → pair(e, e)
Cond_if_1(true, zl, zh, x, y, zs) → pair(ins(y, zl), zh)
Cond_split(true, x, y, zs) → if_1(split(x, zs), x, y, zs)
split(x, ins(y, zs)) → Cond_split1(greatereq_int(y, x), x, y, zs)
Cond_split1(true, x, y, zs) → if_2(split(x, zs), x, y, zs)
qsort(e) → nil
app(nil, zs) → zs
split(x, ins(y, zs)) → Cond_split(greater_int(x, y), x, y, zs)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

IF_3(pair(yl, yh), x, ys) → APP(qsort(yl), cons(x, qsort(yh)))
IF_3(pair(yl, yh), x, ys) → QSORT(yl)
IF_3(pair(yl, yh), x, ys) → QSORT(yh)
IF_1(pair(zl, zh), x, y, zs) → COND_IF_1(greater_int(x, y), zl, zh, x, y, zs)
IF_1(pair(zl, zh), x, y, zs) → GREATER_INT(x, y)
QSORT(ins(x, ys)) → IF_3(split(x, ys), x, ys)
QSORT(ins(x, ys)) → SPLIT(x, ys)
IF_2(pair(zl, zh), x, y, zs) → COND_IF_2(greatereq_int(y, x), zl, zh, x, y, zs)
IF_2(pair(zl, zh), x, y, zs) → GREATEREQ_INT(y, x)
APP(cons(x, ys), zs) → APP(ys, zs)
COND_SPLIT(true, x, y, zs) → IF_1(split(x, zs), x, y, zs)
COND_SPLIT(true, x, y, zs) → SPLIT(x, zs)
SPLIT(x, ins(y, zs)) → COND_SPLIT1(greatereq_int(y, x), x, y, zs)
SPLIT(x, ins(y, zs)) → GREATEREQ_INT(y, x)
COND_SPLIT1(true, x, y, zs) → IF_2(split(x, zs), x, y, zs)
COND_SPLIT1(true, x, y, zs) → SPLIT(x, zs)
SPLIT(x, ins(y, zs)) → COND_SPLIT(greater_int(x, y), x, y, zs)
SPLIT(x, ins(y, zs)) → GREATER_INT(x, y)
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))

The TRS R consists of the following rules:

if_3(pair(yl, yh), x, ys) → app(qsort(yl), cons(x, qsort(yh)))
if_1(pair(zl, zh), x, y, zs) → Cond_if_1(greater_int(x, y), zl, zh, x, y, zs)
qsort(ins(x, ys)) → if_3(split(x, ys), x, ys)
if_2(pair(zl, zh), x, y, zs) → Cond_if_2(greatereq_int(y, x), zl, zh, x, y, zs)
Cond_if_2(true, zl, zh, x, y, zs) → pair(zl, ins(y, zh))
app(cons(x, ys), zs) → cons(x, app(ys, zs))
split(x, e) → pair(e, e)
Cond_if_1(true, zl, zh, x, y, zs) → pair(ins(y, zl), zh)
Cond_split(true, x, y, zs) → if_1(split(x, zs), x, y, zs)
split(x, ins(y, zs)) → Cond_split1(greatereq_int(y, x), x, y, zs)
Cond_split1(true, x, y, zs) → if_2(split(x, zs), x, y, zs)
qsort(e) → nil
app(nil, zs) → zs
split(x, ins(y, zs)) → Cond_split(greater_int(x, y), x, y, zs)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

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

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

IF_3(pair(yl, yh), x, ys) → APP(qsort(yl), cons(x, qsort(yh)))
IF_3(pair(yl, yh), x, ys) → QSORT(yl)
IF_3(pair(yl, yh), x, ys) → QSORT(yh)
IF_1(pair(zl, zh), x, y, zs) → COND_IF_1(greater_int(x, y), zl, zh, x, y, zs)
IF_1(pair(zl, zh), x, y, zs) → GREATER_INT(x, y)
QSORT(ins(x, ys)) → IF_3(split(x, ys), x, ys)
QSORT(ins(x, ys)) → SPLIT(x, ys)
IF_2(pair(zl, zh), x, y, zs) → COND_IF_2(greatereq_int(y, x), zl, zh, x, y, zs)
IF_2(pair(zl, zh), x, y, zs) → GREATEREQ_INT(y, x)
APP(cons(x, ys), zs) → APP(ys, zs)
COND_SPLIT(true, x, y, zs) → IF_1(split(x, zs), x, y, zs)
COND_SPLIT(true, x, y, zs) → SPLIT(x, zs)
SPLIT(x, ins(y, zs)) → COND_SPLIT1(greatereq_int(y, x), x, y, zs)
SPLIT(x, ins(y, zs)) → GREATEREQ_INT(y, x)
COND_SPLIT1(true, x, y, zs) → IF_2(split(x, zs), x, y, zs)
COND_SPLIT1(true, x, y, zs) → SPLIT(x, zs)
SPLIT(x, ins(y, zs)) → COND_SPLIT(greater_int(x, y), x, y, zs)
SPLIT(x, ins(y, zs)) → GREATER_INT(x, y)
GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))
GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))
GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))
GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))

The TRS R consists of the following rules:

if_3(pair(yl, yh), x, ys) → app(qsort(yl), cons(x, qsort(yh)))
if_1(pair(zl, zh), x, y, zs) → Cond_if_1(greater_int(x, y), zl, zh, x, y, zs)
qsort(ins(x, ys)) → if_3(split(x, ys), x, ys)
if_2(pair(zl, zh), x, y, zs) → Cond_if_2(greatereq_int(y, x), zl, zh, x, y, zs)
Cond_if_2(true, zl, zh, x, y, zs) → pair(zl, ins(y, zh))
app(cons(x, ys), zs) → cons(x, app(ys, zs))
split(x, e) → pair(e, e)
Cond_if_1(true, zl, zh, x, y, zs) → pair(ins(y, zl), zh)
Cond_split(true, x, y, zs) → if_1(split(x, zs), x, y, zs)
split(x, ins(y, zs)) → Cond_split1(greatereq_int(y, x), x, y, zs)
Cond_split1(true, x, y, zs) → if_2(split(x, zs), x, y, zs)
qsort(e) → nil
app(nil, zs) → zs
split(x, ins(y, zs)) → Cond_split(greater_int(x, y), x, y, zs)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 7 SCCs with 10 less nodes.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))

The TRS R consists of the following rules:

if_3(pair(yl, yh), x, ys) → app(qsort(yl), cons(x, qsort(yh)))
if_1(pair(zl, zh), x, y, zs) → Cond_if_1(greater_int(x, y), zl, zh, x, y, zs)
qsort(ins(x, ys)) → if_3(split(x, ys), x, ys)
if_2(pair(zl, zh), x, y, zs) → Cond_if_2(greatereq_int(y, x), zl, zh, x, y, zs)
Cond_if_2(true, zl, zh, x, y, zs) → pair(zl, ins(y, zh))
app(cons(x, ys), zs) → cons(x, app(ys, zs))
split(x, e) → pair(e, e)
Cond_if_1(true, zl, zh, x, y, zs) → pair(ins(y, zl), zh)
Cond_split(true, x, y, zs) → if_1(split(x, zs), x, y, zs)
split(x, ins(y, zs)) → Cond_split1(greatereq_int(y, x), x, y, zs)
Cond_split1(true, x, y, zs) → if_2(split(x, zs), x, y, zs)
qsort(e) → nil
app(nil, zs) → zs
split(x, ins(y, zs)) → Cond_split(greater_int(x, y), x, y, zs)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))

R is empty.
The set Q consists of the following terms:

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

GREATEREQ_INT(neg(s(x)), neg(s(y))) → GREATEREQ_INT(neg(x), neg(y))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(GREATEREQ_INT(x₁, x₂)) = 2·x₁ + x₂
POL(neg(x₁)) = x₁
POL(s(x₁)) = 2·x₁

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
P is empty.
R is empty.
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.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))

The TRS R consists of the following rules:

if_3(pair(yl, yh), x, ys) → app(qsort(yl), cons(x, qsort(yh)))
if_1(pair(zl, zh), x, y, zs) → Cond_if_1(greater_int(x, y), zl, zh, x, y, zs)
qsort(ins(x, ys)) → if_3(split(x, ys), x, ys)
if_2(pair(zl, zh), x, y, zs) → Cond_if_2(greatereq_int(y, x), zl, zh, x, y, zs)
Cond_if_2(true, zl, zh, x, y, zs) → pair(zl, ins(y, zh))
app(cons(x, ys), zs) → cons(x, app(ys, zs))
split(x, e) → pair(e, e)
Cond_if_1(true, zl, zh, x, y, zs) → pair(ins(y, zl), zh)
Cond_split(true, x, y, zs) → if_1(split(x, zs), x, y, zs)
split(x, ins(y, zs)) → Cond_split1(greatereq_int(y, x), x, y, zs)
Cond_split1(true, x, y, zs) → if_2(split(x, zs), x, y, zs)
qsort(e) → nil
app(nil, zs) → zs
split(x, ins(y, zs)) → Cond_split(greater_int(x, y), x, y, zs)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))

R is empty.
The set Q consists of the following terms:

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATEREQ_INT(pos(s(x)), pos(s(y))) → GREATEREQ_INT(pos(x), pos(y))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(GREATEREQ_INT(x₁, x₂)) = 2·x₁ + x₂
POL(pos(x₁)) = x₁
POL(s(x₁)) = 2·x₁

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
P is empty.
R is empty.
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.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))

The TRS R consists of the following rules:

if_3(pair(yl, yh), x, ys) → app(qsort(yl), cons(x, qsort(yh)))
if_1(pair(zl, zh), x, y, zs) → Cond_if_1(greater_int(x, y), zl, zh, x, y, zs)
qsort(ins(x, ys)) → if_3(split(x, ys), x, ys)
if_2(pair(zl, zh), x, y, zs) → Cond_if_2(greatereq_int(y, x), zl, zh, x, y, zs)
Cond_if_2(true, zl, zh, x, y, zs) → pair(zl, ins(y, zh))
app(cons(x, ys), zs) → cons(x, app(ys, zs))
split(x, e) → pair(e, e)
Cond_if_1(true, zl, zh, x, y, zs) → pair(ins(y, zl), zh)
Cond_split(true, x, y, zs) → if_1(split(x, zs), x, y, zs)
split(x, ins(y, zs)) → Cond_split1(greatereq_int(y, x), x, y, zs)
Cond_split1(true, x, y, zs) → if_2(split(x, zs), x, y, zs)
qsort(e) → nil
app(nil, zs) → zs
split(x, ins(y, zs)) → Cond_split(greater_int(x, y), x, y, zs)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))

R is empty.
The set Q consists of the following terms:

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATER_INT(neg(s(x)), neg(s(y))) → GREATER_INT(neg(x), neg(y))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(GREATER_INT(x₁, x₂)) = 2·x₁ + x₂
POL(neg(x₁)) = x₁
POL(s(x₁)) = 2·x₁

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
P is empty.
R is empty.
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.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))

The TRS R consists of the following rules:

if_3(pair(yl, yh), x, ys) → app(qsort(yl), cons(x, qsort(yh)))
if_1(pair(zl, zh), x, y, zs) → Cond_if_1(greater_int(x, y), zl, zh, x, y, zs)
qsort(ins(x, ys)) → if_3(split(x, ys), x, ys)
if_2(pair(zl, zh), x, y, zs) → Cond_if_2(greatereq_int(y, x), zl, zh, x, y, zs)
Cond_if_2(true, zl, zh, x, y, zs) → pair(zl, ins(y, zh))
app(cons(x, ys), zs) → cons(x, app(ys, zs))
split(x, e) → pair(e, e)
Cond_if_1(true, zl, zh, x, y, zs) → pair(ins(y, zl), zh)
Cond_split(true, x, y, zs) → if_1(split(x, zs), x, y, zs)
split(x, ins(y, zs)) → Cond_split1(greatereq_int(y, x), x, y, zs)
Cond_split1(true, x, y, zs) → if_2(split(x, zs), x, y, zs)
qsort(e) → nil
app(nil, zs) → zs
split(x, ins(y, zs)) → Cond_split(greater_int(x, y), x, y, zs)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))

R is empty.
The set Q consists of the following terms:

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

GREATER_INT(pos(s(x)), pos(s(y))) → GREATER_INT(pos(x), pos(y))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(GREATER_INT(x₁, x₂)) = 2·x₁ + x₂
POL(pos(x₁)) = x₁
POL(s(x₁)) = 2·x₁

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ UsableRulesReductionPairsProof

                          ↳ QDP

                            ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
P is empty.
R is empty.
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.

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

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

APP(cons(x, ys), zs) → APP(ys, zs)

The TRS R consists of the following rules:

if_3(pair(yl, yh), x, ys) → app(qsort(yl), cons(x, qsort(yh)))
if_1(pair(zl, zh), x, y, zs) → Cond_if_1(greater_int(x, y), zl, zh, x, y, zs)
qsort(ins(x, ys)) → if_3(split(x, ys), x, ys)
if_2(pair(zl, zh), x, y, zs) → Cond_if_2(greatereq_int(y, x), zl, zh, x, y, zs)
Cond_if_2(true, zl, zh, x, y, zs) → pair(zl, ins(y, zh))
app(cons(x, ys), zs) → cons(x, app(ys, zs))
split(x, e) → pair(e, e)
Cond_if_1(true, zl, zh, x, y, zs) → pair(ins(y, zl), zh)
Cond_split(true, x, y, zs) → if_1(split(x, zs), x, y, zs)
split(x, ins(y, zs)) → Cond_split1(greatereq_int(y, x), x, y, zs)
Cond_split1(true, x, y, zs) → if_2(split(x, zs), x, y, zs)
qsort(e) → nil
app(nil, zs) → zs
split(x, ins(y, zs)) → Cond_split(greater_int(x, y), x, y, zs)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

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

APP(cons(x, ys), zs) → APP(ys, zs)

R is empty.
The set Q consists of the following terms:

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

APP(cons(x, ys), zs) → APP(ys, zs)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

APP(cons(x, ys), zs) → APP(ys, zs)
The graph contains the following edges 1 > 1, 2 >= 2

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

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

COND_SPLIT(true, x, y, zs) → SPLIT(x, zs)
SPLIT(x, ins(y, zs)) → COND_SPLIT1(greatereq_int(y, x), x, y, zs)
COND_SPLIT1(true, x, y, zs) → SPLIT(x, zs)
SPLIT(x, ins(y, zs)) → COND_SPLIT(greater_int(x, y), x, y, zs)

The TRS R consists of the following rules:

if_3(pair(yl, yh), x, ys) → app(qsort(yl), cons(x, qsort(yh)))
if_1(pair(zl, zh), x, y, zs) → Cond_if_1(greater_int(x, y), zl, zh, x, y, zs)
qsort(ins(x, ys)) → if_3(split(x, ys), x, ys)
if_2(pair(zl, zh), x, y, zs) → Cond_if_2(greatereq_int(y, x), zl, zh, x, y, zs)
Cond_if_2(true, zl, zh, x, y, zs) → pair(zl, ins(y, zh))
app(cons(x, ys), zs) → cons(x, app(ys, zs))
split(x, e) → pair(e, e)
Cond_if_1(true, zl, zh, x, y, zs) → pair(ins(y, zl), zh)
Cond_split(true, x, y, zs) → if_1(split(x, zs), x, y, zs)
split(x, ins(y, zs)) → Cond_split1(greatereq_int(y, x), x, y, zs)
Cond_split1(true, x, y, zs) → if_2(split(x, zs), x, y, zs)
qsort(e) → nil
app(nil, zs) → zs
split(x, ins(y, zs)) → Cond_split(greater_int(x, y), x, y, zs)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

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

COND_SPLIT(true, x, y, zs) → SPLIT(x, zs)
SPLIT(x, ins(y, zs)) → COND_SPLIT1(greatereq_int(y, x), x, y, zs)
COND_SPLIT1(true, x, y, zs) → SPLIT(x, zs)
SPLIT(x, ins(y, zs)) → COND_SPLIT(greater_int(x, y), x, y, zs)

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

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

COND_SPLIT(true, x, y, zs) → SPLIT(x, zs)
SPLIT(x, ins(y, zs)) → COND_SPLIT1(greatereq_int(y, x), x, y, zs)
COND_SPLIT1(true, x, y, zs) → SPLIT(x, zs)
SPLIT(x, ins(y, zs)) → COND_SPLIT(greater_int(x, y), x, y, zs)

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

SPLIT(x, ins(y, zs)) → COND_SPLIT(greater_int(x, y), x, y, zs)
The graph contains the following edges 1 >= 2, 2 > 3, 2 > 4
SPLIT(x, ins(y, zs)) → COND_SPLIT1(greatereq_int(y, x), x, y, zs)
The graph contains the following edges 1 >= 2, 2 > 3, 2 > 4
COND_SPLIT1(true, x, y, zs) → SPLIT(x, zs)
The graph contains the following edges 2 >= 1, 4 >= 2
COND_SPLIT(true, x, y, zs) → SPLIT(x, zs)
The graph contains the following edges 2 >= 1, 4 >= 2

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

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

IF_3(pair(yl, yh), x, ys) → QSORT(yl)
QSORT(ins(x, ys)) → IF_3(split(x, ys), x, ys)
IF_3(pair(yl, yh), x, ys) → QSORT(yh)

The TRS R consists of the following rules:

if_3(pair(yl, yh), x, ys) → app(qsort(yl), cons(x, qsort(yh)))
if_1(pair(zl, zh), x, y, zs) → Cond_if_1(greater_int(x, y), zl, zh, x, y, zs)
qsort(ins(x, ys)) → if_3(split(x, ys), x, ys)
if_2(pair(zl, zh), x, y, zs) → Cond_if_2(greatereq_int(y, x), zl, zh, x, y, zs)
Cond_if_2(true, zl, zh, x, y, zs) → pair(zl, ins(y, zh))
app(cons(x, ys), zs) → cons(x, app(ys, zs))
split(x, e) → pair(e, e)
Cond_if_1(true, zl, zh, x, y, zs) → pair(ins(y, zl), zh)
Cond_split(true, x, y, zs) → if_1(split(x, zs), x, y, zs)
split(x, ins(y, zs)) → Cond_split1(greatereq_int(y, x), x, y, zs)
Cond_split1(true, x, y, zs) → if_2(split(x, zs), x, y, zs)
qsort(e) → nil
app(nil, zs) → zs
split(x, ins(y, zs)) → Cond_split(greater_int(x, y), x, y, zs)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))

The set Q consists of the following terms:

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

IF_3(pair(yl, yh), x, ys) → QSORT(yl)
QSORT(ins(x, ys)) → IF_3(split(x, ys), x, ys)
IF_3(pair(yl, yh), x, ys) → QSORT(yh)

The TRS R consists of the following rules:

split(x, e) → pair(e, e)
split(x, ins(y, zs)) → Cond_split1(greatereq_int(y, x), x, y, zs)
split(x, ins(y, zs)) → Cond_split(greater_int(x, y), x, y, zs)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
Cond_split(true, x, y, zs) → if_1(split(x, zs), x, y, zs)
if_1(pair(zl, zh), x, y, zs) → Cond_if_1(greater_int(x, y), zl, zh, x, y, zs)
Cond_if_1(true, zl, zh, x, y, zs) → pair(ins(y, zl), zh)
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
Cond_split1(true, x, y, zs) → if_2(split(x, zs), x, y, zs)
if_2(pair(zl, zh), x, y, zs) → Cond_if_2(greatereq_int(y, x), zl, zh, x, y, zs)
Cond_if_2(true, zl, zh, x, y, zs) → pair(zl, ins(y, zh))

The set Q consists of the following terms:

if_3(pair(x0, x1), x2, x3)
if_1(pair(x0, x1), x2, x3, x4)
qsort(ins(x0, x1))
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
app(cons(x0, x1), x2)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
qsort(e)
app(nil, x0)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

if_3(pair(x0, x1), x2, x3)
qsort(ins(x0, x1))
app(cons(x0, x1), x2)
qsort(e)
app(nil, x0)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPOrderProof

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

IF_3(pair(yl, yh), x, ys) → QSORT(yl)
QSORT(ins(x, ys)) → IF_3(split(x, ys), x, ys)
IF_3(pair(yl, yh), x, ys) → QSORT(yh)

The TRS R consists of the following rules:

split(x, e) → pair(e, e)
split(x, ins(y, zs)) → Cond_split1(greatereq_int(y, x), x, y, zs)
split(x, ins(y, zs)) → Cond_split(greater_int(x, y), x, y, zs)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
Cond_split(true, x, y, zs) → if_1(split(x, zs), x, y, zs)
if_1(pair(zl, zh), x, y, zs) → Cond_if_1(greater_int(x, y), zl, zh, x, y, zs)
Cond_if_1(true, zl, zh, x, y, zs) → pair(ins(y, zl), zh)
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
Cond_split1(true, x, y, zs) → if_2(split(x, zs), x, y, zs)
if_2(pair(zl, zh), x, y, zs) → Cond_if_2(greatereq_int(y, x), zl, zh, x, y, zs)
Cond_if_2(true, zl, zh, x, y, zs) → pair(zl, ins(y, zh))

The set Q consists of the following terms:

if_1(pair(x0, x1), x2, x3, x4)
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

QSORT(ins(x, ys)) → IF_3(split(x, ys), x, ys)

The remaining pairs can at least be oriented weakly.

IF_3(pair(yl, yh), x, ys) → QSORT(yl)
IF_3(pair(yl, yh), x, ys) → QSORT(yh)

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(Cond_if_1(x₁, x₂, x₃, x₄, x₅, x₆)) = 1 + x₂ + x₃
POL(Cond_if_2(x₁, x₂, x₃, x₄, x₅, x₆)) = 1 + x₂ + x₃
POL(Cond_split(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(Cond_split1(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(IF_3(x₁, x₂, x₃)) = x₁
POL(QSORT(x₁)) = x₁
POL(e) = 0
POL(false) = 0
POL(greater_int(x₁, x₂)) = 0
POL(greatereq_int(x₁, x₂)) = x₁
POL(if_1(x₁, x₂, x₃, x₄)) = 1 + x₁
POL(if_2(x₁, x₂, x₃, x₄)) = 1 + x₁
POL(ins(x₁, x₂)) = 1 + x₂
POL(neg(x₁)) = x₁
POL(pair(x₁, x₂)) = x₁ + x₂
POL(pos(x₁)) = 1 + x₁
POL(s(x₁)) = 1 + x₁
POL(split(x₁, x₂)) = x₂
POL(true) = 0

The following usable rules [FROCOS05] were oriented:

split(x, ins(y, zs)) → Cond_split(greater_int(x, y), x, y, zs)
Cond_split1(true, x, y, zs) → if_2(split(x, zs), x, y, zs)
if_2(pair(zl, zh), x, y, zs) → Cond_if_2(greatereq_int(y, x), zl, zh, x, y, zs)
Cond_split(true, x, y, zs) → if_1(split(x, zs), x, y, zs)
if_1(pair(zl, zh), x, y, zs) → Cond_if_1(greater_int(x, y), zl, zh, x, y, zs)
Cond_if_1(true, zl, zh, x, y, zs) → pair(ins(y, zl), zh)
Cond_if_2(true, zl, zh, x, y, zs) → pair(zl, ins(y, zh))
split(x, e) → pair(e, e)
split(x, ins(y, zs)) → Cond_split1(greatereq_int(y, x), x, y, zs)

↳ ITRS

  ↳ ITRStoQTRSProof

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

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

IF_3(pair(yl, yh), x, ys) → QSORT(yl)
IF_3(pair(yl, yh), x, ys) → QSORT(yh)

The TRS R consists of the following rules:

split(x, e) → pair(e, e)
split(x, ins(y, zs)) → Cond_split1(greatereq_int(y, x), x, y, zs)
split(x, ins(y, zs)) → Cond_split(greater_int(x, y), x, y, zs)
greater_int(pos(0), pos(0)) → false
greater_int(pos(0), neg(0)) → false
greater_int(neg(0), pos(0)) → false
greater_int(neg(0), neg(0)) → false
greater_int(pos(0), pos(s(y))) → false
greater_int(neg(0), pos(s(y))) → false
greater_int(pos(0), neg(s(y))) → true
greater_int(neg(0), neg(s(y))) → true
greater_int(pos(s(x)), pos(0)) → true
greater_int(neg(s(x)), pos(0)) → false
greater_int(pos(s(x)), neg(0)) → true
greater_int(neg(s(x)), neg(0)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))
Cond_split(true, x, y, zs) → if_1(split(x, zs), x, y, zs)
if_1(pair(zl, zh), x, y, zs) → Cond_if_1(greater_int(x, y), zl, zh, x, y, zs)
Cond_if_1(true, zl, zh, x, y, zs) → pair(ins(y, zl), zh)
greatereq_int(pos(x), pos(0)) → true
greatereq_int(neg(0), pos(0)) → true
greatereq_int(neg(0), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(0), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(0)) → false
greatereq_int(neg(s(x)), neg(0)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
Cond_split1(true, x, y, zs) → if_2(split(x, zs), x, y, zs)
if_2(pair(zl, zh), x, y, zs) → Cond_if_2(greatereq_int(y, x), zl, zh, x, y, zs)
Cond_if_2(true, zl, zh, x, y, zs) → pair(zl, ins(y, zh))

The set Q consists of the following terms:

if_1(pair(x0, x1), x2, x3, x4)
if_2(pair(x0, x1), x2, x3, x4)
Cond_if_2(true, x0, x1, x2, x3, x4)
split(x0, e)
Cond_if_1(true, x0, x1, x2, x3, x4)
Cond_split(true, x0, x1, x2)
split(x0, ins(x1, x2))
Cond_split1(true, x0, x1, x2)
greater_int(pos(0), pos(0))
greater_int(pos(0), neg(0))
greater_int(neg(0), pos(0))
greater_int(neg(0), neg(0))
greater_int(pos(0), pos(s(x0)))
greater_int(neg(0), pos(s(x0)))
greater_int(pos(0), neg(s(x0)))
greater_int(neg(0), neg(s(x0)))
greater_int(pos(s(x0)), pos(0))
greater_int(neg(s(x0)), pos(0))
greater_int(pos(s(x0)), neg(0))
greater_int(neg(s(x0)), neg(0))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))
greatereq_int(pos(x0), pos(0))
greatereq_int(neg(0), pos(0))
greatereq_int(neg(0), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(0), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(0))
greatereq_int(neg(s(x0)), neg(0))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.