Termination proof of /tmp/tmpRplUMB/quicksort.itrs

Termination of the given ITRSProblem could successfully be proven:

↳ ITRS

  ↳ ITRStoIDPProof

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)

Added dependency pairs

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

The ITRS 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 integer pair graph contains the following rules and edges:

(0): QSORT(ins(x[0], ys[0])) → IF_3(split(x[0], ys[0]), x[0], ys[0])
(1): COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1])
(2): SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])
(3): IF_3(pair(yl[3], yh[3]), x[3], ys[3]) → APP(qsort(yl[3]), cons(x[3], qsort(yh[3])))
(4): IF_2(pair(zl[4], zh[4]), x[4], y[4], zs[4]) → COND_IF_2(>=@z(y[4], x[4]), zl[4], zh[4], x[4], y[4], zs[4])
(5): QSORT(ins(x[5], ys[5])) → SPLIT(x[5], ys[5])
(6): IF_1(pair(zl[6], zh[6]), x[6], y[6], zs[6]) → COND_IF_1(>@z(x[6], y[6]), zl[6], zh[6], x[6], y[6], zs[6])
(7): SPLIT(x[7], ins(y[7], zs[7])) → COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])
(8): IF_3(pair(yl[8], yh[8]), x[8], ys[8]) → QSORT(yh[8])
(9): APP(cons(x[9], ys[9]), zs[9]) → APP(ys[9], zs[9])
(10): COND_SPLIT(TRUE, x[10], y[10], zs[10]) → IF_1(split(x[10], zs[10]), x[10], y[10], zs[10])
(11): IF_3(pair(yl[11], yh[11]), x[11], ys[11]) → QSORT(yl[11])
(12): COND_SPLIT1(TRUE, x[12], y[12], zs[12]) → SPLIT(x[12], zs[12])
(13): COND_SPLIT1(TRUE, x[13], y[13], zs[13]) → IF_2(split(x[13], zs[13]), x[13], y[13], zs[13])

(0) -> (3), if ((x[0] →^* x[3])∧(ys[0] →^* ys[3])∧(split(x[0], ys[0]) →^* pair(yl[3], yh[3])))

(0) -> (8), if ((x[0] →^* x[8])∧(ys[0] →^* ys[8])∧(split(x[0], ys[0]) →^* pair(yl[8], yh[8])))

(0) -> (11), if ((x[0] →^* x[11])∧(ys[0] →^* ys[11])∧(split(x[0], ys[0]) →^* pair(yl[11], yh[11])))

(1) -> (2), if ((zs[1] →^* ins(y[2], zs[2]))∧(x[1] →^* x[2]))

(1) -> (7), if ((zs[1] →^* ins(y[7], zs[7]))∧(x[1] →^* x[7]))

(2) -> (1), if ((zs[2] →^* zs[1])∧(x[2] →^* x[1])∧(y[2] →^* y[1])∧(>@z(x[2], y[2]) →^* TRUE))

(2) -> (10), if ((zs[2] →^* zs[10])∧(x[2] →^* x[10])∧(y[2] →^* y[10])∧(>@z(x[2], y[2]) →^* TRUE))

(3) -> (9), if ((qsort(yl[3]) →^* cons(x[9], ys[9])))

(5) -> (2), if ((ys[5] →^* ins(y[2], zs[2]))∧(x[5] →^* x[2]))

(5) -> (7), if ((ys[5] →^* ins(y[7], zs[7]))∧(x[5] →^* x[7]))

(7) -> (12), if ((zs[7] →^* zs[12])∧(x[7] →^* x[12])∧(y[7] →^* y[12])∧(>=@z(y[7], x[7]) →^* TRUE))

(7) -> (13), if ((zs[7] →^* zs[13])∧(x[7] →^* x[13])∧(y[7] →^* y[13])∧(>=@z(y[7], x[7]) →^* TRUE))

(8) -> (0), if ((yh[8] →^* ins(x[0], ys[0])))

(8) -> (5), if ((yh[8] →^* ins(x[5], ys[5])))

(9) -> (9), if ((zs[9] →^* zs[9]a)∧(ys[9] →^* cons(x[9]a, ys[9]a)))

(10) -> (6), if ((zs[10] →^* zs[6])∧(x[10] →^* x[6])∧(y[10] →^* y[6])∧(split(x[10], zs[10]) →^* pair(zl[6], zh[6])))

(11) -> (0), if ((yl[11] →^* ins(x[0], ys[0])))

(11) -> (5), if ((yl[11] →^* ins(x[5], ys[5])))

(12) -> (2), if ((zs[12] →^* ins(y[2], zs[2]))∧(x[12] →^* x[2]))

(12) -> (7), if ((zs[12] →^* ins(y[7], zs[7]))∧(x[12] →^* x[7]))

(13) -> (4), if ((zs[13] →^* zs[4])∧(x[13] →^* x[4])∧(y[13] →^* y[4])∧(split(x[13], zs[13]) →^* pair(zl[4], zh[4])))

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)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 6 less nodes.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ IDependencyGraphProof

        ↳ AND

          ↳ IDP

            ↳ UsableRulesProof

          ↳ IDP

          ↳ IDP

I DP problem:
The following domains are used:

z

The ITRS 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 integer pair graph contains the following rules and edges:

(1): COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1])
(2): SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])
(12): COND_SPLIT1(TRUE, x[12], y[12], zs[12]) → SPLIT(x[12], zs[12])
(7): SPLIT(x[7], ins(y[7], zs[7])) → COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])

(2) -> (1), if ((zs[2] →^* zs[1])∧(x[2] →^* x[1])∧(y[2] →^* y[1])∧(>@z(x[2], y[2]) →^* TRUE))

(12) -> (2), if ((zs[12] →^* ins(y[2], zs[2]))∧(x[12] →^* x[2]))

(12) -> (7), if ((zs[12] →^* ins(y[7], zs[7]))∧(x[12] →^* x[7]))

(1) -> (2), if ((zs[1] →^* ins(y[2], zs[2]))∧(x[1] →^* x[2]))

(1) -> (7), if ((zs[1] →^* ins(y[7], zs[7]))∧(x[1] →^* x[7]))

(7) -> (12), if ((zs[7] →^* zs[12])∧(x[7] →^* x[12])∧(y[7] →^* y[12])∧(>=@z(y[7], x[7]) →^* TRUE))

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)

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

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ IDependencyGraphProof

        ↳ AND

          ↳ IDP

            ↳ UsableRulesProof

              ↳ IDP

                ↳ IDPNonInfProof

          ↳ IDP

          ↳ IDP

I DP problem:
The following domains are used:

z

R is empty.
The integer pair graph contains the following rules and edges:

(1): COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1])
(2): SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])
(12): COND_SPLIT1(TRUE, x[12], y[12], zs[12]) → SPLIT(x[12], zs[12])
(7): SPLIT(x[7], ins(y[7], zs[7])) → COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])

(2) -> (1), if ((zs[2] →^* zs[1])∧(x[2] →^* x[1])∧(y[2] →^* y[1])∧(>@z(x[2], y[2]) →^* TRUE))

(12) -> (2), if ((zs[12] →^* ins(y[2], zs[2]))∧(x[12] →^* x[2]))

(12) -> (7), if ((zs[12] →^* ins(y[7], zs[7]))∧(x[12] →^* x[7]))

(1) -> (2), if ((zs[1] →^* ins(y[2], zs[2]))∧(x[1] →^* x[2]))

(1) -> (7), if ((zs[1] →^* ins(y[7], zs[7]))∧(x[1] →^* x[7]))

(7) -> (12), if ((zs[7] →^* zs[12])∧(x[7] →^* x[12])∧(y[7] →^* y[12])∧(>=@z(y[7], x[7]) →^* TRUE))

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)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1]) the following chains were created:

We consider the chain SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2]), COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1]), SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2]) which results in the following constraint:
(1)    (y[2]=y[1]∧>@z(x[2], y[2])=TRUE∧x[2]=x[1]∧x[1]=x[2]1∧zs[2]=zs[1]∧zs[1]=ins(y[2]1, zs[2]1) ⇒ COND_SPLIT(TRUE, x[1], y[1], zs[1])≥SPLIT(x[1], zs[1])∧(U^Increasing(SPLIT(x[1], zs[1])), ≥))

We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2)    (>@z(x[2], y[2])=TRUE ⇒ COND_SPLIT(TRUE, x[2], y[2], ins(y[2]1, zs[2]1))≥SPLIT(x[2], ins(y[2]1, zs[2]1))∧(U^Increasing(SPLIT(x[1], zs[1])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (0 ≥ 0 ⇒ (U^Increasing(SPLIT(x[1], zs[1])), ≥)∧0 ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (0 ≥ 0 ⇒ (U^Increasing(SPLIT(x[1], zs[1])), ≥)∧0 ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (0 ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(SPLIT(x[1], zs[1])), ≥))

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (0 ≥ 0 ⇒ (U^Increasing(SPLIT(x[1], zs[1])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0)
We consider the chain SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2]), COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1]), SPLIT(x[7], ins(y[7], zs[7])) → COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7]) which results in the following constraint:
(7)    (y[2]=y[1]∧>@z(x[2], y[2])=TRUE∧zs[1]=ins(y[7], zs[7])∧x[2]=x[1]∧zs[2]=zs[1]∧x[1]=x[7] ⇒ COND_SPLIT(TRUE, x[1], y[1], zs[1])≥SPLIT(x[1], zs[1])∧(U^Increasing(SPLIT(x[1], zs[1])), ≥))

We simplified constraint (7) using rules (III), (IV) which results in the following new constraint:
(8)    (>@z(x[2], y[2])=TRUE ⇒ COND_SPLIT(TRUE, x[2], y[2], ins(y[7], zs[7]))≥SPLIT(x[2], ins(y[7], zs[7]))∧(U^Increasing(SPLIT(x[1], zs[1])), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    (0 ≥ 0 ⇒ (U^Increasing(SPLIT(x[1], zs[1])), ≥)∧0 ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    (0 ≥ 0 ⇒ (U^Increasing(SPLIT(x[1], zs[1])), ≥)∧0 ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    (0 ≥ 0 ⇒ (U^Increasing(SPLIT(x[1], zs[1])), ≥)∧0 ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    (0 ≥ 0 ⇒ 0 = 0∧(U^Increasing(SPLIT(x[1], zs[1])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)

For Pair SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2]) the following chains were created:

We consider the chain COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1]), SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2]), COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1]) which results in the following constraint:
(13)    (>@z(x[2], y[2])=TRUE∧x[1]=x[2]∧x[2]=x[1]1∧zs[1]=ins(y[2], zs[2])∧y[2]=y[1]1∧zs[2]=zs[1]1 ⇒ SPLIT(x[2], ins(y[2], zs[2]))≥COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])∧(U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥))

We simplified constraint (13) using rules (III), (IV) which results in the following new constraint:
(14)    (>@z(x[2], y[2])=TRUE ⇒ SPLIT(x[2], ins(y[2], zs[2]))≥COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])∧(U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥))

We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(15)    (0 ≥ 0 ⇒ (U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥)∧1 ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(16)    (0 ≥ 0 ⇒ (U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥)∧1 ≥ 0)

We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(17)    (0 ≥ 0 ⇒ (U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥)∧1 ≥ 0)

We simplified constraint (17) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(18)    (0 ≥ 0 ⇒ 1 ≥ 0∧0 = 0∧(U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥)∧0 = 0∧0 = 0)
We consider the chain COND_SPLIT1(TRUE, x[12], y[12], zs[12]) → SPLIT(x[12], zs[12]), SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2]), COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1]) which results in the following constraint:
(19)    (y[2]=y[1]∧>@z(x[2], y[2])=TRUE∧x[12]=x[2]∧x[2]=x[1]∧zs[2]=zs[1]∧zs[12]=ins(y[2], zs[2]) ⇒ SPLIT(x[2], ins(y[2], zs[2]))≥COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])∧(U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥))

We simplified constraint (19) using rules (III), (IV) which results in the following new constraint:
(20)    (>@z(x[2], y[2])=TRUE ⇒ SPLIT(x[2], ins(y[2], zs[2]))≥COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])∧(U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥))

We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21)    (0 ≥ 0 ⇒ (U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥)∧1 ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22)    (0 ≥ 0 ⇒ (U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥)∧1 ≥ 0)

We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23)    (0 ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥))

We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(24)    (0 ≥ 0 ⇒ 0 = 0∧1 ≥ 0∧0 = 0∧(U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥)∧0 = 0)

For Pair COND_SPLIT1(TRUE, x[12], y[12], zs[12]) → SPLIT(x[12], zs[12]) the following chains were created:

We consider the chain SPLIT(x[7], ins(y[7], zs[7])) → COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7]), COND_SPLIT1(TRUE, x[12], y[12], zs[12]) → SPLIT(x[12], zs[12]), SPLIT(x[7], ins(y[7], zs[7])) → COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7]) which results in the following constraint:
(25)    (>=@z(y[7], x[7])=TRUE∧zs[7]=zs[12]∧zs[12]=ins(y[7]1, zs[7]1)∧x[7]=x[12]∧y[7]=y[12]∧x[12]=x[7]1 ⇒ COND_SPLIT1(TRUE, x[12], y[12], zs[12])≥SPLIT(x[12], zs[12])∧(U^Increasing(SPLIT(x[12], zs[12])), ≥))

We simplified constraint (25) using rules (III), (IV) which results in the following new constraint:
(26)    (>=@z(y[7], x[7])=TRUE ⇒ COND_SPLIT1(TRUE, x[7], y[7], ins(y[7]1, zs[7]1))≥SPLIT(x[7], ins(y[7]1, zs[7]1))∧(U^Increasing(SPLIT(x[12], zs[12])), ≥))

We simplified constraint (26) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(27)    (0 ≥ 0 ⇒ (U^Increasing(SPLIT(x[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(28)    (0 ≥ 0 ⇒ (U^Increasing(SPLIT(x[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(29)    (0 ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(SPLIT(x[12], zs[12])), ≥))

We simplified constraint (29) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(30)    (0 ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(SPLIT(x[12], zs[12])), ≥)∧0 = 0)
We consider the chain SPLIT(x[7], ins(y[7], zs[7])) → COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7]), COND_SPLIT1(TRUE, x[12], y[12], zs[12]) → SPLIT(x[12], zs[12]), SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2]) which results in the following constraint:
(31)    (>=@z(y[7], x[7])=TRUE∧zs[7]=zs[12]∧x[12]=x[2]∧x[7]=x[12]∧y[7]=y[12]∧zs[12]=ins(y[2], zs[2]) ⇒ COND_SPLIT1(TRUE, x[12], y[12], zs[12])≥SPLIT(x[12], zs[12])∧(U^Increasing(SPLIT(x[12], zs[12])), ≥))

We simplified constraint (31) using rules (III), (IV) which results in the following new constraint:
(32)    (>=@z(y[7], x[7])=TRUE ⇒ COND_SPLIT1(TRUE, x[7], y[7], ins(y[2], zs[2]))≥SPLIT(x[7], ins(y[2], zs[2]))∧(U^Increasing(SPLIT(x[12], zs[12])), ≥))

We simplified constraint (32) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(33)    (0 ≥ 0 ⇒ (U^Increasing(SPLIT(x[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (33) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(34)    (0 ≥ 0 ⇒ (U^Increasing(SPLIT(x[12], zs[12])), ≥)∧0 ≥ 0)

We simplified constraint (34) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(35)    (0 ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(SPLIT(x[12], zs[12])), ≥))

We simplified constraint (35) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(36)    (0 ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(SPLIT(x[12], zs[12])), ≥)∧0 = 0)

For Pair SPLIT(x[7], ins(y[7], zs[7])) → COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7]) the following chains were created:

We consider the chain COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1]), SPLIT(x[7], ins(y[7], zs[7])) → COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7]), COND_SPLIT1(TRUE, x[12], y[12], zs[12]) → SPLIT(x[12], zs[12]) which results in the following constraint:
(37)    (>=@z(y[7], x[7])=TRUE∧zs[1]=ins(y[7], zs[7])∧zs[7]=zs[12]∧x[7]=x[12]∧y[7]=y[12]∧x[1]=x[7] ⇒ SPLIT(x[7], ins(y[7], zs[7]))≥COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])∧(U^Increasing(COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])), ≥))

We simplified constraint (37) using rules (III), (IV) which results in the following new constraint:
(38)    (>=@z(y[7], x[7])=TRUE ⇒ SPLIT(x[7], ins(y[7], zs[7]))≥COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])∧(U^Increasing(COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])), ≥))

We simplified constraint (38) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(39)    (0 ≥ 0 ⇒ (U^Increasing(COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])), ≥)∧0 ≥ 0)

We simplified constraint (39) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(40)    (0 ≥ 0 ⇒ (U^Increasing(COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])), ≥)∧0 ≥ 0)

We simplified constraint (40) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(41)    (0 ≥ 0 ⇒ (U^Increasing(COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])), ≥)∧0 ≥ 0)

We simplified constraint (41) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(42)    (0 ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])), ≥)∧0 = 0)
We consider the chain COND_SPLIT1(TRUE, x[12], y[12], zs[12]) → SPLIT(x[12], zs[12]), SPLIT(x[7], ins(y[7], zs[7])) → COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7]), COND_SPLIT1(TRUE, x[12], y[12], zs[12]) → SPLIT(x[12], zs[12]) which results in the following constraint:
(43)    (>=@z(y[7], x[7])=TRUE∧zs[7]=zs[12]1∧x[7]=x[12]1∧x[12]=x[7]∧y[7]=y[12]1∧zs[12]=ins(y[7], zs[7]) ⇒ SPLIT(x[7], ins(y[7], zs[7]))≥COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])∧(U^Increasing(COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])), ≥))

We simplified constraint (43) using rules (III), (IV) which results in the following new constraint:
(44)    (>=@z(y[7], x[7])=TRUE ⇒ SPLIT(x[7], ins(y[7], zs[7]))≥COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])∧(U^Increasing(COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])), ≥))

We simplified constraint (44) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(45)    (0 ≥ 0 ⇒ (U^Increasing(COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])), ≥)∧0 ≥ 0)

We simplified constraint (45) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(46)    (0 ≥ 0 ⇒ (U^Increasing(COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])), ≥)∧0 ≥ 0)

We simplified constraint (46) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(47)    (0 ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])), ≥))

We simplified constraint (47) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(48)    (0 ≥ 0 ⇒ 0 = 0∧0 = 0∧(U^Increasing(COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])), ≥)∧0 ≥ 0∧0 = 0)

To summarize, we get the following constraints P_≥ for the following pairs.

COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1])
- (0 ≥ 0 ⇒ (U^Increasing(SPLIT(x[1], zs[1])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0)
- (0 ≥ 0 ⇒ 0 = 0∧(U^Increasing(SPLIT(x[1], zs[1])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])
- (0 ≥ 0 ⇒ 1 ≥ 0∧0 = 0∧(U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥)∧0 = 0∧0 = 0)
- (0 ≥ 0 ⇒ 0 = 0∧1 ≥ 0∧0 = 0∧(U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥)∧0 = 0)
COND_SPLIT1(TRUE, x[12], y[12], zs[12]) → SPLIT(x[12], zs[12])
- (0 ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(SPLIT(x[12], zs[12])), ≥)∧0 = 0)
- (0 ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(SPLIT(x[12], zs[12])), ≥)∧0 = 0)
SPLIT(x[7], ins(y[7], zs[7])) → COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])
- (0 ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])), ≥)∧0 = 0)
- (0 ≥ 0 ⇒ 0 = 0∧0 = 0∧(U^Increasing(COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])), ≥)∧0 ≥ 0∧0 = 0)

The constraints for P_> respective P_bound are constructed from P_≥ where we just replace every occurence of "t ≥ s" in P_≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers with natural coefficients for non-tuple symbols [NONINF][POLO]:

POL(>=@z(x₁, x₂)) = 0
POL(ins(x₁, x₂)) = 1 + x₂
POL(COND_SPLIT(x₁, x₂, x₃, x₄)) = 1 + x₄ + (2)x₂
POL(TRUE) = 0
POL(COND_SPLIT1(x₁, x₂, x₃, x₄)) = 1 + x₄ + (2)x₂
POL(FALSE) = 0
POL(SPLIT(x₁, x₂)) = 1 + x₂ + (2)x₁
POL(undefined) = 0
POL(>@z(x₁, x₂)) = 0

The following pairs are in P_>:

SPLIT(x[7], ins(y[7], zs[7])) → COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])

The following pairs are in P_bound:

COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1])
SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])
COND_SPLIT1(TRUE, x[12], y[12], zs[12]) → SPLIT(x[12], zs[12])
SPLIT(x[7], ins(y[7], zs[7])) → COND_SPLIT1(>=@z(y[7], x[7]), x[7], y[7], zs[7])

The following pairs are in P_≥:

COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1])
SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])
COND_SPLIT1(TRUE, x[12], y[12], zs[12]) → SPLIT(x[12], zs[12])

There are no usable rules.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ IDependencyGraphProof

        ↳ AND

          ↳ IDP

            ↳ UsableRulesProof

              ↳ IDP

                ↳ IDPNonInfProof

                  ↳ AND

                    ↳ IDP

                      ↳ IDependencyGraphProof

                    ↳ IDP

          ↳ IDP

          ↳ IDP

I DP problem:
The following domains are used:none

R is empty.
The integer pair graph 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)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ IDependencyGraphProof

        ↳ AND

          ↳ IDP

            ↳ UsableRulesProof

              ↳ IDP

                ↳ IDPNonInfProof

                  ↳ AND

                    ↳ IDP

                    ↳ IDP

                      ↳ IDependencyGraphProof

          ↳ IDP

          ↳ IDP

I DP problem:
The following domains are used:

z

(2) -> (1), if ((zs[2] →^* zs[1])∧(x[2] →^* x[1])∧(y[2] →^* y[1])∧(>@z(x[2], y[2]) →^* TRUE))

(12) -> (2), if ((zs[12] →^* ins(y[2], zs[2]))∧(x[12] →^* x[2]))

(1) -> (2), if ((zs[1] →^* ins(y[2], zs[2]))∧(x[1] →^* x[2]))

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)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ IDependencyGraphProof

        ↳ AND

          ↳ IDP

            ↳ UsableRulesProof

              ↳ IDP

                ↳ IDPNonInfProof

                  ↳ AND

                    ↳ IDP

                    ↳ IDP

                      ↳ IDependencyGraphProof

                        ↳ IDP

                          ↳ IDPNonInfProof

          ↳ IDP

          ↳ IDP

I DP problem:
The following domains are used:

z

(2) -> (1), if ((zs[2] →^* zs[1])∧(x[2] →^* x[1])∧(y[2] →^* y[1])∧(>@z(x[2], y[2]) →^* TRUE))

(1) -> (2), if ((zs[1] →^* ins(y[2], zs[2]))∧(x[1] →^* x[2]))

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)

We consider the chain SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2]), COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1]), SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2]) which results in the following constraint:
(1)    (y[2]=y[1]∧>@z(x[2], y[2])=TRUE∧x[2]=x[1]∧x[1]=x[2]1∧zs[2]=zs[1]∧zs[1]=ins(y[2]1, zs[2]1) ⇒ COND_SPLIT(TRUE, x[1], y[1], zs[1])≥SPLIT(x[1], zs[1])∧(U^Increasing(SPLIT(x[1], zs[1])), ≥))

We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2)    (>@z(x[2], y[2])=TRUE ⇒ COND_SPLIT(TRUE, x[2], y[2], ins(y[2]1, zs[2]1))≥SPLIT(x[2], ins(y[2]1, zs[2]1))∧(U^Increasing(SPLIT(x[1], zs[1])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (0 ≥ 0 ⇒ (U^Increasing(SPLIT(x[1], zs[1])), ≥)∧0 ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (0 ≥ 0 ⇒ (U^Increasing(SPLIT(x[1], zs[1])), ≥)∧0 ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (0 ≥ 0 ⇒ (U^Increasing(SPLIT(x[1], zs[1])), ≥)∧0 ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (0 ≥ 0 ⇒ (U^Increasing(SPLIT(x[1], zs[1])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)

For Pair SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2]) the following chains were created:

We consider the chain COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1]), SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2]), COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1]) which results in the following constraint:
(7)    (>@z(x[2], y[2])=TRUE∧x[1]=x[2]∧x[2]=x[1]1∧zs[1]=ins(y[2], zs[2])∧y[2]=y[1]1∧zs[2]=zs[1]1 ⇒ SPLIT(x[2], ins(y[2], zs[2]))≥COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])∧(U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥))

We simplified constraint (7) using rules (III), (IV) which results in the following new constraint:
(8)    (>@z(x[2], y[2])=TRUE ⇒ SPLIT(x[2], ins(y[2], zs[2]))≥COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])∧(U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    (0 ≥ 0 ⇒ (U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥)∧1 ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    (0 ≥ 0 ⇒ (U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥)∧1 ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    (0 ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥))

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    (0 ≥ 0 ⇒ 0 = 0∧1 ≥ 0∧0 = 0∧(U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥)∧0 = 0)

To summarize, we get the following constraints P_≥ for the following pairs.

COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1])
- (0 ≥ 0 ⇒ (U^Increasing(SPLIT(x[1], zs[1])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])
- (0 ≥ 0 ⇒ 0 = 0∧1 ≥ 0∧0 = 0∧(U^Increasing(COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])), ≥)∧0 = 0)

POL(ins(x₁, x₂)) = 2 + x₂
POL(COND_SPLIT(x₁, x₂, x₃, x₄)) = -1 + x₄ + x₂
POL(TRUE) = 0
POL(FALSE) = 0
POL(SPLIT(x₁, x₂)) = -1 + x₂ + x₁
POL(undefined) = 0
POL(>@z(x₁, x₂)) = 0

The following pairs are in P_>:

SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])

The following pairs are in P_bound:

COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1])
SPLIT(x[2], ins(y[2], zs[2])) → COND_SPLIT(>@z(x[2], y[2]), x[2], y[2], zs[2])

The following pairs are in P_≥:

COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1])

There are no usable rules.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ IDependencyGraphProof

        ↳ AND

          ↳ IDP

            ↳ UsableRulesProof

              ↳ IDP

                ↳ IDPNonInfProof

                  ↳ AND

                    ↳ IDP

                    ↳ IDP

                      ↳ IDependencyGraphProof

                        ↳ IDP

                          ↳ IDPNonInfProof

                            ↳ AND

                              ↳ IDP

                                ↳ IDependencyGraphProof

                              ↳ IDP

          ↳ IDP

          ↳ IDP

I DP problem:
The following domains are used:none

R is empty.
The integer pair graph 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)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ IDependencyGraphProof

        ↳ AND

          ↳ IDP

            ↳ UsableRulesProof

              ↳ IDP

                ↳ IDPNonInfProof

                  ↳ AND

                    ↳ IDP

                    ↳ IDP

                      ↳ IDependencyGraphProof

                        ↳ IDP

                          ↳ IDPNonInfProof

                            ↳ AND

                              ↳ IDP

                              ↳ IDP

                                ↳ IDependencyGraphProof

          ↳ IDP

          ↳ IDP

I DP problem:
The following domains are used:none

R is empty.
The integer pair graph contains the following rules and edges:

(1): COND_SPLIT(TRUE, x[1], y[1], zs[1]) → SPLIT(x[1], zs[1])

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)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ IDependencyGraphProof

        ↳ AND

          ↳ IDP

          ↳ IDP

            ↳ UsableRulesProof

          ↳ IDP

I DP problem:
The following domains are used:

z

The ITRS 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 integer pair graph contains the following rules and edges:

(9): APP(cons(x[9], ys[9]), zs[9]) → APP(ys[9], zs[9])

(9) -> (9), if ((zs[9] →^* zs[9]a)∧(ys[9] →^* cons(x[9]a, ys[9]a)))

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)

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

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ IDependencyGraphProof

        ↳ AND

          ↳ IDP

          ↳ IDP

            ↳ UsableRulesProof

              ↳ IDP

                ↳ IDPNonInfProof

          ↳ IDP

I DP problem:
The following domains are used:none

R is empty.
The integer pair graph contains the following rules and edges:

(9): APP(cons(x[9], ys[9]), zs[9]) → APP(ys[9], zs[9])

(9) -> (9), if ((zs[9] →^* zs[9]a)∧(ys[9] →^* cons(x[9]a, ys[9]a)))

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)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair APP(cons(x[9], ys[9]), zs[9]) → APP(ys[9], zs[9]) the following chains were created:

We consider the chain APP(cons(x[9], ys[9]), zs[9]) → APP(ys[9], zs[9]), APP(cons(x[9], ys[9]), zs[9]) → APP(ys[9], zs[9]), APP(cons(x[9], ys[9]), zs[9]) → APP(ys[9], zs[9]) which results in the following constraint:
(1)    (zs[9]=zs[9]1∧zs[9]1=zs[9]2∧ys[9]=cons(x[9]1, ys[9]1)∧ys[9]1=cons(x[9]2, ys[9]2) ⇒ APP(cons(x[9]1, ys[9]1), zs[9]1)≥APP(ys[9]1, zs[9]1)∧(U^Increasing(APP(ys[9]1, zs[9]1)), ≥))

We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2)    (APP(cons(x[9]1, cons(x[9]2, ys[9]2)), zs[9])≥APP(cons(x[9]2, ys[9]2), zs[9])∧(U^Increasing(APP(ys[9]1, zs[9]1)), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    ((U^Increasing(APP(ys[9]1, zs[9]1)), ≥)∧1 ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    ((U^Increasing(APP(ys[9]1, zs[9]1)), ≥)∧1 ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (1 ≥ 0∧(U^Increasing(APP(ys[9]1, zs[9]1)), ≥))

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    ((U^Increasing(APP(ys[9]1, zs[9]1)), ≥)∧0 = 0∧0 = 0∧1 ≥ 0∧0 = 0∧0 = 0)

To summarize, we get the following constraints P_≥ for the following pairs.

APP(cons(x[9], ys[9]), zs[9]) → APP(ys[9], zs[9])
- ((U^Increasing(APP(ys[9]1, zs[9]1)), ≥)∧0 = 0∧0 = 0∧1 ≥ 0∧0 = 0∧0 = 0)

POL(cons(x₁, x₂)) = 1 + x₂
POL(APP(x₁, x₂)) = -1 + (-1)x₂ + (2)x₁
POL(TRUE) = 0
POL(FALSE) = 0
POL(undefined) = 0

The following pairs are in P_>:

APP(cons(x[9], ys[9]), zs[9]) → APP(ys[9], zs[9])

The following pairs are in P_bound:

APP(cons(x[9], ys[9]), zs[9]) → APP(ys[9], zs[9])

The following pairs are in P_≥:
none

There are no usable rules.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ IDependencyGraphProof

        ↳ AND

          ↳ IDP

          ↳ IDP

            ↳ UsableRulesProof

              ↳ IDP

                ↳ IDPNonInfProof

                  ↳ IDP

                    ↳ IDependencyGraphProof

          ↳ IDP

I DP problem:
The following domains are used:none

R is empty.
The integer pair graph 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)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ IDependencyGraphProof

        ↳ AND

          ↳ IDP

          ↳ IDP

          ↳ IDP

            ↳ UsableRulesProof

I DP problem:
The following domains are used:

z

The ITRS 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 integer pair graph contains the following rules and edges:

(8): IF_3(pair(yl[8], yh[8]), x[8], ys[8]) → QSORT(yh[8])
(11): IF_3(pair(yl[11], yh[11]), x[11], ys[11]) → QSORT(yl[11])
(0): QSORT(ins(x[0], ys[0])) → IF_3(split(x[0], ys[0]), x[0], ys[0])

(11) -> (0), if ((yl[11] →^* ins(x[0], ys[0])))

(0) -> (11), if ((x[0] →^* x[11])∧(ys[0] →^* ys[11])∧(split(x[0], ys[0]) →^* pair(yl[11], yh[11])))

(0) -> (8), if ((x[0] →^* x[8])∧(ys[0] →^* ys[8])∧(split(x[0], ys[0]) →^* pair(yl[8], yh[8])))

(8) -> (0), if ((yh[8] →^* ins(x[0], ys[0])))

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)

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

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ IDependencyGraphProof

        ↳ AND

          ↳ IDP

          ↳ IDP

          ↳ IDP

            ↳ UsableRulesProof

              ↳ IDP

                ↳ IDPtoQDPProof

I DP problem:
The following domains are used:

z

The ITRS R consists of the following rules:

if_1(pair(zl, zh), x, y, zs) → Cond_if_1(>@z(x, y), zl, zh, x, y, zs)
if_2(pair(zl, zh), x, y, zs) → Cond_if_2(>=@z(y, x), zl, zh, x, y, zs)
split(x, ins(y, zs)) → Cond_split(>@z(x, y), x, y, zs)
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)
Cond_if_2(TRUE, zl, zh, x, y, zs) → pair(zl, ins(y, zh))
split(x, e) → pair(e, e)
Cond_if_1(TRUE, zl, zh, x, y, zs) → pair(ins(y, zl), zh)

(11) -> (0), if ((yl[11] →^* ins(x[0], ys[0])))

(0) -> (11), if ((x[0] →^* x[11])∧(ys[0] →^* ys[11])∧(split(x[0], ys[0]) →^* pair(yl[11], yh[11])))

(0) -> (8), if ((x[0] →^* x[8])∧(ys[0] →^* ys[8])∧(split(x[0], ys[0]) →^* pair(yl[8], yh[8])))

(8) -> (0), if ((yh[8] →^* ins(x[0], ys[0])))

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

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ IDependencyGraphProof

        ↳ AND

          ↳ IDP

          ↳ IDP

          ↳ IDP

            ↳ UsableRulesProof

              ↳ IDP

                ↳ IDPtoQDPProof

                  ↳ QDP

                    ↳ QReductionProof

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

IF_3(pair(yl[8], yh[8]), x[8], ys[8]) → QSORT(yh[8])
IF_3(pair(yl[11], yh[11]), x[11], ys[11]) → QSORT(yl[11])
QSORT(ins(x[0], ys[0])) → IF_3(split(x[0], ys[0]), x[0], ys[0])

The TRS R consists of the following rules:

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

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ IDependencyGraphProof

        ↳ AND

          ↳ IDP

          ↳ IDP

          ↳ IDP

            ↳ UsableRulesProof

              ↳ IDP

                ↳ IDPtoQDPProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPOrderProof

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

IF_3(pair(yl[8], yh[8]), x[8], ys[8]) → QSORT(yh[8])
IF_3(pair(yl[11], yh[11]), x[11], ys[11]) → QSORT(yl[11])
QSORT(ins(x[0], ys[0])) → IF_3(split(x[0], ys[0]), x[0], ys[0])

The TRS R consists of the following rules:

if_1(pair(zl, zh), x, y, zs) → Cond_if_1(greater_int(x, y), zl, zh, x, y, zs)
if_2(pair(zl, zh), x, y, zs) → Cond_if_2(greatereq_int(y, x), zl, zh, x, y, zs)
split(x, ins(y, zs)) → Cond_split(greater_int(x, y), x, y, zs)
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)
Cond_if_2(true, zl, zh, x, y, zs) → pair(zl, ins(y, zh))
split(x, e) → pair(e, e)
Cond_if_1(true, zl, zh, x, y, zs) → pair(ins(y, zl), zh)
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_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[0], ys[0])) → IF_3(split(x[0], ys[0]), x[0], ys[0])

The remaining pairs can at least be oriented weakly.

IF_3(pair(yl[8], yh[8]), x[8], ys[8]) → QSORT(yh[8])
IF_3(pair(yl[11], yh[11]), x[11], ys[11]) → QSORT(yl[11])

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)) = x₂
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)) = x₁
POL(s(x₁)) = 1 + x₁
POL(split(x₁, x₂)) = x₂
POL(true) = 0

The following usable rules [FROCOS05] were oriented:

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ IDependencyGraphProof

        ↳ AND

          ↳ IDP

          ↳ IDP

          ↳ IDP

            ↳ UsableRulesProof

              ↳ IDP

                ↳ IDPtoQDPProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

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

IF_3(pair(yl[8], yh[8]), x[8], ys[8]) → QSORT(yh[8])
IF_3(pair(yl[11], yh[11]), x[11], ys[11]) → QSORT(yl[11])

The TRS R consists of the following rules:

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