Termination proof of /tmp/tmp0SaaxE/a.10.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:

eval(x, y) → Cond_eval3(&&(>@z(y, x), >@z(x, y)), x, y)
eval(x, y) → Cond_eval1(&&(>@z(y, x), >=@z(y, x)), x, y)
eval(x, y) → Cond_eval(&&(>@z(x, y), >=@z(y, x)), x, y)
Cond_eval1(TRUE, x, y) → eval(+@z(x, 1@z), y)
Cond_eval(TRUE, x, y) → eval(+@z(x, 1@z), y)
eval(x, y) → Cond_eval2(>@z(x, y), x, y)
Cond_eval3(TRUE, x, y) → eval(x, +@z(y, 1@z))
Cond_eval2(TRUE, x, y) → eval(x, +@z(y, 1@z))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

Added dependency pairs

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

I DP problem:
The following domains are used:

z

The ITRS R consists of the following rules:

eval(x, y) → Cond_eval3(&&(>@z(y, x), >@z(x, y)), x, y)
eval(x, y) → Cond_eval1(&&(>@z(y, x), >=@z(y, x)), x, y)
eval(x, y) → Cond_eval(&&(>@z(x, y), >=@z(y, x)), x, y)
Cond_eval1(TRUE, x, y) → eval(+@z(x, 1@z), y)
Cond_eval(TRUE, x, y) → eval(+@z(x, 1@z), y)
eval(x, y) → Cond_eval2(>@z(x, y), x, y)
Cond_eval3(TRUE, x, y) → eval(x, +@z(y, 1@z))
Cond_eval2(TRUE, x, y) → eval(x, +@z(y, 1@z))

The integer pair graph contains the following rules and edges:

(0): COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
(1): COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1])
(2): EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
(4): EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])
(5): COND_EVAL3(TRUE, x[5], y[5]) → EVAL(x[5], +@z(y[5], 1@z))
(6): COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z))
(7): EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])

(0) -> (2), if ((y[0] →^* y[2])∧(+@z(x[0], 1@z) →^* x[2]))

(0) -> (3), if ((y[0] →^* y[3])∧(+@z(x[0], 1@z) →^* x[3]))

(0) -> (4), if ((y[0] →^* y[4])∧(+@z(x[0], 1@z) →^* x[4]))

(0) -> (7), if ((y[0] →^* y[7])∧(+@z(x[0], 1@z) →^* x[7]))

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

(1) -> (3), if ((y[1] →^* y[3])∧(+@z(x[1], 1@z) →^* x[3]))

(1) -> (4), if ((y[1] →^* y[4])∧(+@z(x[1], 1@z) →^* x[4]))

(1) -> (7), if ((y[1] →^* y[7])∧(+@z(x[1], 1@z) →^* x[7]))

(2) -> (6), if ((x[2] →^* x[6])∧(y[2] →^* y[6])∧(>@z(x[2], y[2]) →^* TRUE))

(3) -> (0), if ((x[3] →^* x[0])∧(y[3] →^* y[0])∧(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])) →^* TRUE))

(4) -> (5), if ((x[4] →^* x[5])∧(y[4] →^* y[5])∧(&&(>@z(y[4], x[4]), >@z(x[4], y[4])) →^* TRUE))

(5) -> (2), if ((+@z(y[5], 1@z) →^* y[2])∧(x[5] →^* x[2]))

(5) -> (3), if ((+@z(y[5], 1@z) →^* y[3])∧(x[5] →^* x[3]))

(5) -> (4), if ((+@z(y[5], 1@z) →^* y[4])∧(x[5] →^* x[4]))

(5) -> (7), if ((+@z(y[5], 1@z) →^* y[7])∧(x[5] →^* x[7]))

(6) -> (2), if ((+@z(y[6], 1@z) →^* y[2])∧(x[6] →^* x[2]))

(6) -> (3), if ((+@z(y[6], 1@z) →^* y[3])∧(x[6] →^* x[3]))

(6) -> (4), if ((+@z(y[6], 1@z) →^* y[4])∧(x[6] →^* x[4]))

(6) -> (7), if ((+@z(y[6], 1@z) →^* y[7])∧(x[6] →^* x[7]))

(7) -> (1), if ((x[7] →^* x[1])∧(y[7] →^* y[1])∧(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])) →^* TRUE))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

I DP problem:
The following domains are used:

z

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

(0): COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
(1): COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1])
(2): EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
(4): EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])
(5): COND_EVAL3(TRUE, x[5], y[5]) → EVAL(x[5], +@z(y[5], 1@z))
(6): COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z))
(7): EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])

(0) -> (2), if ((y[0] →^* y[2])∧(+@z(x[0], 1@z) →^* x[2]))

(0) -> (3), if ((y[0] →^* y[3])∧(+@z(x[0], 1@z) →^* x[3]))

(0) -> (4), if ((y[0] →^* y[4])∧(+@z(x[0], 1@z) →^* x[4]))

(0) -> (7), if ((y[0] →^* y[7])∧(+@z(x[0], 1@z) →^* x[7]))

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

(1) -> (3), if ((y[1] →^* y[3])∧(+@z(x[1], 1@z) →^* x[3]))

(1) -> (4), if ((y[1] →^* y[4])∧(+@z(x[1], 1@z) →^* x[4]))

(1) -> (7), if ((y[1] →^* y[7])∧(+@z(x[1], 1@z) →^* x[7]))

(2) -> (6), if ((x[2] →^* x[6])∧(y[2] →^* y[6])∧(>@z(x[2], y[2]) →^* TRUE))

(3) -> (0), if ((x[3] →^* x[0])∧(y[3] →^* y[0])∧(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])) →^* TRUE))

(4) -> (5), if ((x[4] →^* x[5])∧(y[4] →^* y[5])∧(&&(>@z(y[4], x[4]), >@z(x[4], y[4])) →^* TRUE))

(5) -> (2), if ((+@z(y[5], 1@z) →^* y[2])∧(x[5] →^* x[2]))

(5) -> (3), if ((+@z(y[5], 1@z) →^* y[3])∧(x[5] →^* x[3]))

(5) -> (4), if ((+@z(y[5], 1@z) →^* y[4])∧(x[5] →^* x[4]))

(5) -> (7), if ((+@z(y[5], 1@z) →^* y[7])∧(x[5] →^* x[7]))

(6) -> (2), if ((+@z(y[6], 1@z) →^* y[2])∧(x[6] →^* x[2]))

(6) -> (3), if ((+@z(y[6], 1@z) →^* y[3])∧(x[6] →^* x[3]))

(6) -> (4), if ((+@z(y[6], 1@z) →^* y[4])∧(x[6] →^* x[4]))

(6) -> (7), if ((+@z(y[6], 1@z) →^* y[7])∧(x[6] →^* x[7]))

(7) -> (1), if ((x[7] →^* x[1])∧(y[7] →^* y[1])∧(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])) →^* TRUE))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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_EVAL1(TRUE, x, y) → EVAL(+@z(x, 1@z), y) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]) which results in the following constraint:
(1)    (x[3]=x[0]∧y[3]=y[0]∧&&(>@z(y[3], x[3]), >=@z(y[3], x[3]))=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(+@z(x[0], 1@z), y[0])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>@z(y[3], x[3])=TRUE∧>=@z(y[3], x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(+@z(x[3], 1@z), y[3])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (1 + x[3] ≥ 0∧x[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(7)    (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

(8)    (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

For Pair COND_EVAL(TRUE, x, y) → EVAL(+@z(x, 1@z), y) the following chains were created:

We consider the chain EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]), COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1]) which results in the following constraint:
(9)    (y[7]=y[1]∧x[7]=x[1]∧&&(>@z(x[7], y[7]), >=@z(y[7], x[7]))=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(+@z(x[1], 1@z), y[1])∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (9) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(10)    (>@z(x[7], y[7])=TRUE∧>=@z(y[7], x[7])=TRUE ⇒ COND_EVAL(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL(TRUE, x[7], y[7])≥EVAL(+@z(x[7], 1@z), y[7])∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (10) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(11)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧0 ≥ 0∧-2 ≥ 0)

We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(12)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧0 ≥ 0∧-2 ≥ 0)

We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(13)    (y[7] + (-1)x[7] ≥ 0∧-1 + x[7] + (-1)y[7] ≥ 0 ⇒ -2 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We solved constraint (13) using rule (IDP_SMT_SPLIT).

For Pair EVAL(x, y) → COND_EVAL2(>@z(x, y), x, y) the following chains were created:

We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2]) which results in the following constraint:
(14)    (EVAL(x[2], y[2])≥NonInfC∧EVAL(x[2], y[2])≥COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])∧(U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥))

We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(15)    ((U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(16)    ((U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(17)    ((U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (17) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(18)    (0 ≥ 0∧(U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)

For Pair EVAL(x, y) → COND_EVAL1(&&(>@z(y, x), >=@z(y, x)), x, y) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(19)    (EVAL(x[3], y[3])≥NonInfC∧EVAL(x[3], y[3])≥COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥))

We simplified constraint (19) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(20)    ((U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (20) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(21)    ((U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (21) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(22)    (1 ≥ 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0)

We simplified constraint (22) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(23)    (0 = 0∧0 = 0∧1 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥))

For Pair EVAL(x, y) → COND_EVAL3(&&(>@z(y, x), >@z(x, y)), x, y) the following chains were created:

We consider the chain EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4]) which results in the following constraint:
(24)    (EVAL(x[4], y[4])≥NonInfC∧EVAL(x[4], y[4])≥COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])∧(U^Increasing(COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])), ≥))

We simplified constraint (24) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(25)    ((U^Increasing(COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (25) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(26)    ((U^Increasing(COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (26) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(27)    (0 ≥ 0∧(U^Increasing(COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])), ≥)∧1 ≥ 0)

We simplified constraint (27) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(28)    (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])), ≥)∧0 = 0∧1 ≥ 0)

For Pair COND_EVAL3(TRUE, x, y) → EVAL(x, +@z(y, 1@z)) the following chains were created:

We consider the chain EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4]), COND_EVAL3(TRUE, x[5], y[5]) → EVAL(x[5], +@z(y[5], 1@z)) which results in the following constraint:
(29)    (x[4]=x[5]∧y[4]=y[5]∧&&(>@z(y[4], x[4]), >@z(x[4], y[4]))=TRUE ⇒ COND_EVAL3(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL3(TRUE, x[5], y[5])≥EVAL(x[5], +@z(y[5], 1@z))∧(U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥))

We simplified constraint (29) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(30)    (>@z(y[4], x[4])=TRUE∧>@z(x[4], y[4])=TRUE ⇒ COND_EVAL3(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL3(TRUE, x[4], y[4])≥EVAL(x[4], +@z(y[4], 1@z))∧(U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥))

We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(31)    (-1 + y[4] + (-1)x[4] ≥ 0∧-1 + x[4] + (-1)y[4] ≥ 0 ⇒ (U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥)∧-1 + (-1)Bound + (-1)y[4] + (-1)x[4] ≥ 0∧0 ≥ 0)

We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(32)    (-1 + y[4] + (-1)x[4] ≥ 0∧-1 + x[4] + (-1)y[4] ≥ 0 ⇒ (U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥)∧-1 + (-1)Bound + (-1)y[4] + (-1)x[4] ≥ 0∧0 ≥ 0)

We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(33)    (-1 + y[4] + (-1)x[4] ≥ 0∧-1 + x[4] + (-1)y[4] ≥ 0 ⇒ -1 + (-1)Bound + (-1)y[4] + (-1)x[4] ≥ 0∧(U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥)∧0 ≥ 0)

We solved constraint (33) using rule (IDP_SMT_SPLIT).

For Pair COND_EVAL2(TRUE, x, y) → EVAL(x, +@z(y, 1@z)) the following chains were created:

We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2]), COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z)) which results in the following constraint:
(34)    (>@z(x[2], y[2])=TRUE∧x[2]=x[6]∧y[2]=y[6] ⇒ COND_EVAL2(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL2(TRUE, x[6], y[6])≥EVAL(x[6], +@z(y[6], 1@z))∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (34) using rule (III) which results in the following new constraint:
(35)    (>@z(x[2], y[2])=TRUE ⇒ COND_EVAL2(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL2(TRUE, x[2], y[2])≥EVAL(x[2], +@z(y[2], 1@z))∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (35) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(36)    (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (36) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(37)    (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (37) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(38)    (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (38) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(39)    (x[2] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (39) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(40)    (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

(41)    (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

For Pair EVAL(x, y) → COND_EVAL(&&(>@z(x, y), >=@z(y, x)), x, y) the following chains were created:

We consider the chain EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]) which results in the following constraint:
(42)    (EVAL(x[7], y[7])≥NonInfC∧EVAL(x[7], y[7])≥COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])∧(U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥))

We simplified constraint (42) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(43)    ((U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧3 ≥ 0)

We simplified constraint (43) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(44)    ((U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧3 ≥ 0)

We simplified constraint (44) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(45)    (3 ≥ 0∧(U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥)∧0 ≥ 0)

We simplified constraint (45) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(46)    (0 ≥ 0∧0 = 0∧3 ≥ 0∧0 = 0∧(U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥)∧0 = 0∧0 = 0)

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

COND_EVAL1(TRUE, x, y) → EVAL(+@z(x, 1@z), y)
- (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)
- (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)
COND_EVAL(TRUE, x, y) → EVAL(+@z(x, 1@z), y)
EVAL(x, y) → COND_EVAL2(>@z(x, y), x, y)
- (0 ≥ 0∧(U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
EVAL(x, y) → COND_EVAL1(&&(>@z(y, x), >=@z(y, x)), x, y)
- (0 = 0∧0 = 0∧1 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥))
EVAL(x, y) → COND_EVAL3(&&(>@z(y, x), >@z(x, y)), x, y)
- (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])), ≥)∧0 = 0∧1 ≥ 0)
COND_EVAL3(TRUE, x, y) → EVAL(x, +@z(y, 1@z))
COND_EVAL2(TRUE, x, y) → EVAL(x, +@z(y, 1@z))
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))
EVAL(x, y) → COND_EVAL(&&(>@z(x, y), >=@z(y, x)), x, y)
- (0 ≥ 0∧0 = 0∧3 ≥ 0∧0 = 0∧(U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[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[POLO]:

POL(COND_EVAL1(x₁, x₂, x₃)) = 1 + (-1)x₃ + (-1)x₂ + (2)x₁
POL(TRUE) = -1
POL(&&(x₁, x₂)) = -1
POL(COND_EVAL(x₁, x₂, x₃)) = -1 + (-1)x₃ + (-1)x₂ + (2)x₁
POL(FALSE) = -1
POL(>@z(x₁, x₂)) = -1
POL(>=@z(x₁, x₂)) = -1
POL(COND_EVAL3(x₁, x₂, x₃)) = -1 + (-1)x₃ + (-1)x₂
POL(COND_EVAL2(x₁, x₂, x₃)) = -1 + (-1)x₃ + (-1)x₂
POL(EVAL(x₁, x₂)) = (-1)x₂ + (-1)x₁
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(1@z) = 1
POL(undefined) = -1

The following pairs are in P_>:

EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])

The following pairs are in P_bound:

COND_EVAL3(TRUE, x[5], y[5]) → EVAL(x[5], +@z(y[5], 1@z))

The following pairs are in P_≥:

COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1])
EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])
COND_EVAL3(TRUE, x[5], y[5]) → EVAL(x[5], +@z(y[5], 1@z))
COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z))
EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])

At least the following rules have been oriented under context sensitive arithmetic replacement:

&&(FALSE, FALSE)¹ ↔ FALSE¹
&&(TRUE, TRUE)¹ ↔ TRUE¹
+@z¹ ↔
&&(TRUE, FALSE)¹ ↔ FALSE¹
&&(FALSE, TRUE)¹ ↔ FALSE¹

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

              ↳ IDP

I DP problem:
The following domains are used:

z

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

(0): COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
(1): COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1])
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
(4): EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])
(5): COND_EVAL3(TRUE, x[5], y[5]) → EVAL(x[5], +@z(y[5], 1@z))
(6): COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z))
(7): EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])

(1) -> (4), if ((y[1] →^* y[4])∧(+@z(x[1], 1@z) →^* x[4]))

(6) -> (7), if ((+@z(y[6], 1@z) →^* y[7])∧(x[6] →^* x[7]))

(0) -> (4), if ((y[0] →^* y[4])∧(+@z(x[0], 1@z) →^* x[4]))

(1) -> (7), if ((y[1] →^* y[7])∧(+@z(x[1], 1@z) →^* x[7]))

(6) -> (3), if ((+@z(y[6], 1@z) →^* y[3])∧(x[6] →^* x[3]))

(1) -> (3), if ((y[1] →^* y[3])∧(+@z(x[1], 1@z) →^* x[3]))

(7) -> (1), if ((x[7] →^* x[1])∧(y[7] →^* y[1])∧(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])) →^* TRUE))

(6) -> (4), if ((+@z(y[6], 1@z) →^* y[4])∧(x[6] →^* x[4]))

(0) -> (3), if ((y[0] →^* y[3])∧(+@z(x[0], 1@z) →^* x[3]))

(0) -> (7), if ((y[0] →^* y[7])∧(+@z(x[0], 1@z) →^* x[7]))

(5) -> (3), if ((+@z(y[5], 1@z) →^* y[3])∧(x[5] →^* x[3]))

(3) -> (0), if ((x[3] →^* x[0])∧(y[3] →^* y[0])∧(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])) →^* TRUE))

(4) -> (5), if ((x[4] →^* x[5])∧(y[4] →^* y[5])∧(&&(>@z(y[4], x[4]), >@z(x[4], y[4])) →^* TRUE))

(5) -> (4), if ((+@z(y[5], 1@z) →^* y[4])∧(x[5] →^* x[4]))

(5) -> (7), if ((+@z(y[5], 1@z) →^* y[7])∧(x[5] →^* x[7]))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

              ↳ IDP

I DP problem:
The following domains are used:

z

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

(5): COND_EVAL3(TRUE, x[5], y[5]) → EVAL(x[5], +@z(y[5], 1@z))
(4): EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])
(0): COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
(1): COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1])
(7): EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])

(0) -> (3), if ((y[0] →^* y[3])∧(+@z(x[0], 1@z) →^* x[3]))

(1) -> (4), if ((y[1] →^* y[4])∧(+@z(x[1], 1@z) →^* x[4]))

(5) -> (3), if ((+@z(y[5], 1@z) →^* y[3])∧(x[5] →^* x[3]))

(0) -> (7), if ((y[0] →^* y[7])∧(+@z(x[0], 1@z) →^* x[7]))

(0) -> (4), if ((y[0] →^* y[4])∧(+@z(x[0], 1@z) →^* x[4]))

(5) -> (4), if ((+@z(y[5], 1@z) →^* y[4])∧(x[5] →^* x[4]))

(1) -> (3), if ((y[1] →^* y[3])∧(+@z(x[1], 1@z) →^* x[3]))

(1) -> (7), if ((y[1] →^* y[7])∧(+@z(x[1], 1@z) →^* x[7]))

(4) -> (5), if ((x[4] →^* x[5])∧(y[4] →^* y[5])∧(&&(>@z(y[4], x[4]), >@z(x[4], y[4])) →^* TRUE))

(3) -> (0), if ((x[3] →^* x[0])∧(y[3] →^* y[0])∧(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])) →^* TRUE))

(5) -> (7), if ((+@z(y[5], 1@z) →^* y[7])∧(x[5] →^* x[7]))

(7) -> (1), if ((x[7] →^* x[1])∧(y[7] →^* y[1])∧(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])) →^* TRUE))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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_EVAL3(TRUE, x[5], y[5]) → EVAL(x[5], +@z(y[5], 1@z)) the following chains were created:

We consider the chain EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4]), COND_EVAL3(TRUE, x[5], y[5]) → EVAL(x[5], +@z(y[5], 1@z)), EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(1)    (x[5]=x[3]∧x[4]=x[5]∧y[4]=y[5]∧&&(>@z(y[4], x[4]), >@z(x[4], y[4]))=TRUE∧+@z(y[5], 1@z)=y[3] ⇒ COND_EVAL3(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL3(TRUE, x[5], y[5])≥EVAL(x[5], +@z(y[5], 1@z))∧(U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥))

We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>@z(y[4], x[4])=TRUE∧>@z(x[4], y[4])=TRUE ⇒ COND_EVAL3(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL3(TRUE, x[4], y[4])≥EVAL(x[4], +@z(y[4], 1@z))∧(U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (-1 + y[4] + (-1)x[4] ≥ 0∧-1 + x[4] + (-1)y[4] ≥ 0 ⇒ (U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥)∧-1 + (-1)Bound + (-1)y[4] + (-1)x[4] ≥ 0∧0 ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (-1 + y[4] + (-1)x[4] ≥ 0∧-1 + x[4] + (-1)y[4] ≥ 0 ⇒ (U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥)∧-1 + (-1)Bound + (-1)y[4] + (-1)x[4] ≥ 0∧0 ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (-1 + x[4] + (-1)y[4] ≥ 0∧-1 + y[4] + (-1)x[4] ≥ 0 ⇒ 0 ≥ 0∧-1 + (-1)Bound + (-1)y[4] + (-1)x[4] ≥ 0∧(U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥))

We solved constraint (5) using rule (IDP_SMT_SPLIT).
We consider the chain EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4]), COND_EVAL3(TRUE, x[5], y[5]) → EVAL(x[5], +@z(y[5], 1@z)), EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]) which results in the following constraint:
(6)    (x[4]=x[5]∧y[4]=y[5]∧x[5]=x[7]∧&&(>@z(y[4], x[4]), >@z(x[4], y[4]))=TRUE∧+@z(y[5], 1@z)=y[7] ⇒ COND_EVAL3(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL3(TRUE, x[5], y[5])≥EVAL(x[5], +@z(y[5], 1@z))∧(U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥))

We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7)    (>@z(y[4], x[4])=TRUE∧>@z(x[4], y[4])=TRUE ⇒ COND_EVAL3(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL3(TRUE, x[4], y[4])≥EVAL(x[4], +@z(y[4], 1@z))∧(U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    (-1 + y[4] + (-1)x[4] ≥ 0∧-1 + x[4] + (-1)y[4] ≥ 0 ⇒ (U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥)∧-1 + (-1)Bound + (-1)y[4] + (-1)x[4] ≥ 0∧0 ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    (-1 + y[4] + (-1)x[4] ≥ 0∧-1 + x[4] + (-1)y[4] ≥ 0 ⇒ (U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥)∧-1 + (-1)Bound + (-1)y[4] + (-1)x[4] ≥ 0∧0 ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    (-1 + y[4] + (-1)x[4] ≥ 0∧-1 + x[4] + (-1)y[4] ≥ 0 ⇒ -1 + (-1)Bound + (-1)y[4] + (-1)x[4] ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥))

We solved constraint (10) using rule (IDP_SMT_SPLIT).
We consider the chain EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4]), COND_EVAL3(TRUE, x[5], y[5]) → EVAL(x[5], +@z(y[5], 1@z)), EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4]) which results in the following constraint:
(11)    (x[4]=x[5]∧y[4]=y[5]∧x[5]=x[4]1∧&&(>@z(y[4], x[4]), >@z(x[4], y[4]))=TRUE∧+@z(y[5], 1@z)=y[4]1 ⇒ COND_EVAL3(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL3(TRUE, x[5], y[5])≥EVAL(x[5], +@z(y[5], 1@z))∧(U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥))

We simplified constraint (11) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(12)    (>@z(y[4], x[4])=TRUE∧>@z(x[4], y[4])=TRUE ⇒ COND_EVAL3(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL3(TRUE, x[4], y[4])≥EVAL(x[4], +@z(y[4], 1@z))∧(U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥))

We simplified constraint (12) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(13)    (-1 + y[4] + (-1)x[4] ≥ 0∧-1 + x[4] + (-1)y[4] ≥ 0 ⇒ (U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥)∧-1 + (-1)Bound + (-1)y[4] + (-1)x[4] ≥ 0∧0 ≥ 0)

We simplified constraint (13) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(14)    (-1 + y[4] + (-1)x[4] ≥ 0∧-1 + x[4] + (-1)y[4] ≥ 0 ⇒ (U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥)∧-1 + (-1)Bound + (-1)y[4] + (-1)x[4] ≥ 0∧0 ≥ 0)

We simplified constraint (14) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(15)    (-1 + x[4] + (-1)y[4] ≥ 0∧-1 + y[4] + (-1)x[4] ≥ 0 ⇒ 0 ≥ 0∧-1 + (-1)Bound + (-1)y[4] + (-1)x[4] ≥ 0∧(U^Increasing(EVAL(x[5], +@z(y[5], 1@z))), ≥))

We solved constraint (15) using rule (IDP_SMT_SPLIT).

For Pair EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4]) the following chains were created:

We consider the chain EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4]) which results in the following constraint:
(16)    (EVAL(x[4], y[4])≥NonInfC∧EVAL(x[4], y[4])≥COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])∧(U^Increasing(COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])), ≥))

We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17)    ((U^Increasing(COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18)    ((U^Increasing(COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])), ≥))

We simplified constraint (19) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(20)    (0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)

For Pair COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]), EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]) which results in the following constraint:
(21)    (y[0]=y[7]∧x[3]=x[0]∧y[3]=y[0]∧&&(>@z(y[3], x[3]), >=@z(y[3], x[3]))=TRUE∧+@z(x[0], 1@z)=x[7] ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(+@z(x[0], 1@z), y[0])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (21) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(22)    (>@z(y[3], x[3])=TRUE∧>=@z(y[3], x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(+@z(x[3], 1@z), y[3])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (22) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(23)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (23) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(24)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (24) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(25)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (25) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(26)    (x[3] ≥ 0∧1 + x[3] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(27)    (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

(28)    (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))
We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]), EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4]) which results in the following constraint:
(29)    (+@z(x[0], 1@z)=x[4]∧y[0]=y[4]∧x[3]=x[0]∧y[3]=y[0]∧&&(>@z(y[3], x[3]), >=@z(y[3], x[3]))=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(+@z(x[0], 1@z), y[0])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (29) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(30)    (>@z(y[3], x[3])=TRUE∧>=@z(y[3], x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(+@z(x[3], 1@z), y[3])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(31)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(32)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(33)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0∧0 ≥ 0)

We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(34)    (1 + x[3] ≥ 0∧x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0∧0 ≥ 0)

We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(35)    (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0∧0 ≥ 0)

(36)    (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0∧0 ≥ 0)
We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]), EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(37)    (x[3]=x[0]∧y[3]=y[0]∧&&(>@z(y[3], x[3]), >=@z(y[3], x[3]))=TRUE∧+@z(x[0], 1@z)=x[3]1∧y[0]=y[3]1 ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(+@z(x[0], 1@z), y[0])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (37) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(38)    (>@z(y[3], x[3])=TRUE∧>=@z(y[3], x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(+@z(x[3], 1@z), y[3])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (38) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(39)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (39) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(40)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (40) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(41)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

We simplified constraint (41) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(42)    (x[3] ≥ 0∧1 + x[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

We simplified constraint (42) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(43)    (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

(44)    (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

For Pair EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(45)    (EVAL(x[3], y[3])≥NonInfC∧EVAL(x[3], y[3])≥COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥))

We simplified constraint (45) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(46)    ((U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (46) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(47)    ((U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (47) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(48)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥))

We simplified constraint (48) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(49)    (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 = 0)

For Pair COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1]) the following chains were created:

We consider the chain EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]), COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1]), EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(50)    (y[1]=y[3]∧+@z(x[1], 1@z)=x[3]∧y[7]=y[1]∧x[7]=x[1]∧&&(>@z(x[7], y[7]), >=@z(y[7], x[7]))=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(+@z(x[1], 1@z), y[1])∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (50) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(51)    (>@z(x[7], y[7])=TRUE∧>=@z(y[7], x[7])=TRUE ⇒ COND_EVAL(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL(TRUE, x[7], y[7])≥EVAL(+@z(x[7], 1@z), y[7])∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (51) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(52)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧(-1)Bound + (-1)y[7] + (-1)x[7] ≥ 0∧1 ≥ 0)

We simplified constraint (52) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(53)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧(-1)Bound + (-1)y[7] + (-1)x[7] ≥ 0∧1 ≥ 0)

We simplified constraint (53) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(54)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧(-1)Bound + (-1)y[7] + (-1)x[7] ≥ 0∧1 ≥ 0)

We solved constraint (54) using rule (IDP_SMT_SPLIT).
We consider the chain EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]), COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1]), EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4]) which results in the following constraint:
(55)    (y[7]=y[1]∧x[7]=x[1]∧+@z(x[1], 1@z)=x[4]∧y[1]=y[4]∧&&(>@z(x[7], y[7]), >=@z(y[7], x[7]))=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(+@z(x[1], 1@z), y[1])∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (55) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(56)    (>@z(x[7], y[7])=TRUE∧>=@z(y[7], x[7])=TRUE ⇒ COND_EVAL(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL(TRUE, x[7], y[7])≥EVAL(+@z(x[7], 1@z), y[7])∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (56) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(57)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧(-1)Bound + (-1)y[7] + (-1)x[7] ≥ 0∧1 ≥ 0)

We simplified constraint (57) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(58)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧(-1)Bound + (-1)y[7] + (-1)x[7] ≥ 0∧1 ≥ 0)

We simplified constraint (58) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(59)    (y[7] + (-1)x[7] ≥ 0∧-1 + x[7] + (-1)y[7] ≥ 0 ⇒ (-1)Bound + (-1)y[7] + (-1)x[7] ≥ 0∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧1 ≥ 0)

We solved constraint (59) using rule (IDP_SMT_SPLIT).
We consider the chain EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]), COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1]), EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]) which results in the following constraint:
(60)    (+@z(x[1], 1@z)=x[7]1∧y[7]=y[1]∧x[7]=x[1]∧y[1]=y[7]1∧&&(>@z(x[7], y[7]), >=@z(y[7], x[7]))=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(+@z(x[1], 1@z), y[1])∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (60) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(61)    (>@z(x[7], y[7])=TRUE∧>=@z(y[7], x[7])=TRUE ⇒ COND_EVAL(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL(TRUE, x[7], y[7])≥EVAL(+@z(x[7], 1@z), y[7])∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (61) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(62)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧(-1)Bound + (-1)y[7] + (-1)x[7] ≥ 0∧1 ≥ 0)

We simplified constraint (62) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(63)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧(-1)Bound + (-1)y[7] + (-1)x[7] ≥ 0∧1 ≥ 0)

We simplified constraint (63) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(64)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧1 ≥ 0∧(-1)Bound + (-1)y[7] + (-1)x[7] ≥ 0)

We solved constraint (64) using rule (IDP_SMT_SPLIT).

For Pair EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]) the following chains were created:

We consider the chain EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]) which results in the following constraint:
(65)    (EVAL(x[7], y[7])≥NonInfC∧EVAL(x[7], y[7])≥COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])∧(U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥))

We simplified constraint (65) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(66)    ((U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (66) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(67)    ((U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (67) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(68)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥))

We simplified constraint (68) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(69)    (0 ≥ 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥)∧0 = 0)

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

COND_EVAL3(TRUE, x[5], y[5]) → EVAL(x[5], +@z(y[5], 1@z))
EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])
- (0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)
COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
- (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))
- (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))
- (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0∧0 ≥ 0)
- (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0∧0 ≥ 0)
- (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)
- (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)
EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
- (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 = 0)
COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1])
EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])
- (0 ≥ 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥)∧0 = 0)

POL(>=@z(x₁, x₂)) = -1
POL(COND_EVAL3(x₁, x₂, x₃)) = -1 + (-1)x₃ + (-1)x₂
POL(COND_EVAL1(x₁, x₂, x₃)) = -1 + (-1)x₃ + (-1)x₂ + (-1)x₁
POL(TRUE) = -1
POL(&&(x₁, x₂)) = -1
POL(COND_EVAL(x₁, x₂, x₃)) = -1 + (-1)x₃ + (-1)x₂ + (-1)x₁
POL(EVAL(x₁, x₂)) = (-1)x₂ + (-1)x₁
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(FALSE) = 0
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x₁, x₂)) = -1

The following pairs are in P_>:

EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])

The following pairs are in P_bound:

COND_EVAL3(TRUE, x[5], y[5]) → EVAL(x[5], +@z(y[5], 1@z))
COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1])

The following pairs are in P_≥:

COND_EVAL3(TRUE, x[5], y[5]) → EVAL(x[5], +@z(y[5], 1@z))
COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1])
EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])

At least the following rules have been oriented under context sensitive arithmetic replacement:

FALSE¹ → &&(FALSE, FALSE)¹
TRUE¹ → &&(TRUE, TRUE)¹
+@z¹ ↔
FALSE¹ → &&(TRUE, FALSE)¹
FALSE¹ → &&(FALSE, TRUE)¹

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                          ↳ IDependencyGraphProof

                        ↳ IDP

              ↳ IDP

I DP problem:
The following domains are used:

z

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

(5): COND_EVAL3(TRUE, x[5], y[5]) → EVAL(x[5], +@z(y[5], 1@z))
(0): COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
(1): COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1])
(7): EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])

(0) -> (3), if ((y[0] →^* y[3])∧(+@z(x[0], 1@z) →^* x[3]))

(5) -> (3), if ((+@z(y[5], 1@z) →^* y[3])∧(x[5] →^* x[3]))

(0) -> (7), if ((y[0] →^* y[7])∧(+@z(x[0], 1@z) →^* x[7]))

(1) -> (3), if ((y[1] →^* y[3])∧(+@z(x[1], 1@z) →^* x[3]))

(1) -> (7), if ((y[1] →^* y[7])∧(+@z(x[1], 1@z) →^* x[7]))

(3) -> (0), if ((x[3] →^* x[0])∧(y[3] →^* y[0])∧(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])) →^* TRUE))

(5) -> (7), if ((+@z(y[5], 1@z) →^* y[7])∧(x[5] →^* x[7]))

(7) -> (1), if ((x[7] →^* x[1])∧(y[7] →^* y[1])∧(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])) →^* TRUE))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                          ↳ IDependencyGraphProof

                            ↳ 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:

(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
(1): COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1])
(7): EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])
(0): COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])

(0) -> (3), if ((y[0] →^* y[3])∧(+@z(x[0], 1@z) →^* x[3]))

(0) -> (7), if ((y[0] →^* y[7])∧(+@z(x[0], 1@z) →^* x[7]))

(1) -> (3), if ((y[1] →^* y[3])∧(+@z(x[1], 1@z) →^* x[3]))

(1) -> (7), if ((y[1] →^* y[7])∧(+@z(x[1], 1@z) →^* x[7]))

(3) -> (0), if ((x[3] →^* x[0])∧(y[3] →^* y[0])∧(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])) →^* TRUE))

(7) -> (1), if ((x[7] →^* x[1])∧(y[7] →^* y[1])∧(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])) →^* TRUE))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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 EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(1)    (EVAL(x[3], y[3])≥NonInfC∧EVAL(x[3], y[3])≥COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2)    ((U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3)    ((U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥))

We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5)    (0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 = 0)

For Pair COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1]) the following chains were created:

We consider the chain EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]), COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1]), EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(6)    (y[1]=y[3]∧+@z(x[1], 1@z)=x[3]∧y[7]=y[1]∧x[7]=x[1]∧&&(>@z(x[7], y[7]), >=@z(y[7], x[7]))=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(+@z(x[1], 1@z), y[1])∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7)    (>@z(x[7], y[7])=TRUE∧>=@z(y[7], x[7])=TRUE ⇒ COND_EVAL(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL(TRUE, x[7], y[7])≥EVAL(+@z(x[7], 1@z), y[7])∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧-2 + (-1)Bound + y[7] + (-1)x[7] ≥ 0∧-2 ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧-2 + (-1)Bound + y[7] + (-1)x[7] ≥ 0∧-2 ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    (y[7] + (-1)x[7] ≥ 0∧-1 + x[7] + (-1)y[7] ≥ 0 ⇒ -2 ≥ 0∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧-2 + (-1)Bound + y[7] + (-1)x[7] ≥ 0)

We solved constraint (10) using rule (IDP_SMT_SPLIT).
We consider the chain EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]), COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1]), EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]) which results in the following constraint:
(11)    (+@z(x[1], 1@z)=x[7]1∧y[7]=y[1]∧x[7]=x[1]∧y[1]=y[7]1∧&&(>@z(x[7], y[7]), >=@z(y[7], x[7]))=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(+@z(x[1], 1@z), y[1])∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (11) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(12)    (>@z(x[7], y[7])=TRUE∧>=@z(y[7], x[7])=TRUE ⇒ COND_EVAL(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL(TRUE, x[7], y[7])≥EVAL(+@z(x[7], 1@z), y[7])∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (12) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(13)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧-2 + (-1)Bound + y[7] + (-1)x[7] ≥ 0∧-2 ≥ 0)

We simplified constraint (13) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(14)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧-2 + (-1)Bound + y[7] + (-1)x[7] ≥ 0∧-2 ≥ 0)

We simplified constraint (14) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(15)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ -2 ≥ 0∧-2 + (-1)Bound + y[7] + (-1)x[7] ≥ 0∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We solved constraint (15) using rule (IDP_SMT_SPLIT).

For Pair EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]) the following chains were created:

We consider the chain EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]) which results in the following constraint:
(16)    (EVAL(x[7], y[7])≥NonInfC∧EVAL(x[7], y[7])≥COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])∧(U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥))

We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17)    ((U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18)    ((U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19)    (0 ≥ 0∧1 ≥ 0∧(U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥))

We simplified constraint (19) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(20)    (1 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥))

For Pair COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]), EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]) which results in the following constraint:
(21)    (y[0]=y[7]∧x[3]=x[0]∧y[3]=y[0]∧&&(>@z(y[3], x[3]), >=@z(y[3], x[3]))=TRUE∧+@z(x[0], 1@z)=x[7] ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(+@z(x[0], 1@z), y[0])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (21) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(22)    (>@z(y[3], x[3])=TRUE∧>=@z(y[3], x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(+@z(x[3], 1@z), y[3])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (22) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(23)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (23) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(24)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (24) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(25)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

We simplified constraint (25) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(26)    (1 + x[3] ≥ 0∧x[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(27)    (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

(28)    (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)
We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]), EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(29)    (x[3]=x[0]∧y[3]=y[0]∧&&(>@z(y[3], x[3]), >=@z(y[3], x[3]))=TRUE∧+@z(x[0], 1@z)=x[3]1∧y[0]=y[3]1 ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(+@z(x[0], 1@z), y[0])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (29) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(30)    (>@z(y[3], x[3])=TRUE∧>=@z(y[3], x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(+@z(x[3], 1@z), y[3])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(31)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(32)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(33)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(34)    (-1 + x[3] ≥ 0∧x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(35)    (x[3] ≥ 0∧1 + x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (35) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(36)    (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

(37)    (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

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

EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
- (0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 = 0)
COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1])
EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])
- (1 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥))
COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
- (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)
- (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)
- (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)
- (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

POL(>=@z(x₁, x₂)) = -1
POL(COND_EVAL1(x₁, x₂, x₃)) = -1 + x₃ + (-1)x₂ + (-1)x₁
POL(TRUE) = -1
POL(&&(x₁, x₂)) = -1
POL(COND_EVAL(x₁, x₂, x₃)) = x₃ + (-1)x₂ + (2)x₁
POL(EVAL(x₁, x₂)) = x₂ + (-1)x₁
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x₁, x₂)) = -1

The following pairs are in P_>:

COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1])
EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])

The following pairs are in P_bound:

COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1])

The following pairs are in P_≥:

EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])

At least the following rules have been oriented under context sensitive arithmetic replacement:

&&(FALSE, FALSE)¹ ↔ FALSE¹
&&(TRUE, TRUE)¹ ↔ TRUE¹
+@z¹ ↔
&&(TRUE, FALSE)¹ ↔ FALSE¹
&&(FALSE, TRUE)¹ ↔ FALSE¹

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                          ↳ IDependencyGraphProof

                            ↳ IDP

                              ↳ IDPNonInfProof

                                ↳ AND

                                  ↳ IDP

                                    ↳ IDependencyGraphProof

                                  ↳ IDP

                        ↳ IDP

              ↳ IDP

I DP problem:
The following domains are used:

z

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

(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
(7): EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])
(0): COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])

(0) -> (3), if ((y[0] →^* y[3])∧(+@z(x[0], 1@z) →^* x[3]))

(0) -> (7), if ((y[0] →^* y[7])∧(+@z(x[0], 1@z) →^* x[7]))

(3) -> (0), if ((x[3] →^* x[0])∧(y[3] →^* y[0])∧(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])) →^* TRUE))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                          ↳ IDependencyGraphProof

                            ↳ IDP

                              ↳ IDPNonInfProof

                                ↳ AND

                                  ↳ IDP

                                    ↳ IDependencyGraphProof

                                      ↳ IDP

                                        ↳ IDPNonInfProof

                                  ↳ IDP

                        ↳ IDP

              ↳ IDP

I DP problem:
The following domains are used:

z

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

(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
(0): COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])

(0) -> (3), if ((y[0] →^* y[3])∧(+@z(x[0], 1@z) →^* x[3]))

(3) -> (0), if ((x[3] →^* x[0])∧(y[3] →^* y[0])∧(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])) →^* TRUE))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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 EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(1)    (EVAL(x[3], y[3])≥NonInfC∧EVAL(x[3], y[3])≥COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2)    ((U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3)    ((U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4)    (0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0)

We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5)    (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0)

For Pair COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]), EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(6)    (x[3]=x[0]∧y[3]=y[0]∧&&(>@z(y[3], x[3]), >=@z(y[3], x[3]))=TRUE∧+@z(x[0], 1@z)=x[3]1∧y[0]=y[3]1 ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(+@z(x[0], 1@z), y[0])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7)    (>@z(y[3], x[3])=TRUE∧>=@z(y[3], x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(+@z(x[3], 1@z), y[3])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧2 + (-1)Bound + y[3] + (-1)x[3] ≥ 0∧0 ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧2 + (-1)Bound + y[3] + (-1)x[3] ≥ 0∧0 ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0 ⇒ 2 + (-1)Bound + y[3] + (-1)x[3] ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11)    (x[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ 2 + (-1)Bound + x[3] ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12)    (1 + x[3] ≥ 0∧x[3] ≥ 0 ⇒ 3 + (-1)Bound + x[3] ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(13)    (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 3 + (-1)Bound + x[3] ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

(14)    (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 3 + (-1)Bound + x[3] ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

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

EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
- (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0)
COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
- (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 3 + (-1)Bound + x[3] ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)
- (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 3 + (-1)Bound + x[3] ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

POL(>=@z(x₁, x₂)) = -1
POL(COND_EVAL1(x₁, x₂, x₃)) = 2 + x₃ + (-1)x₂
POL(TRUE) = 0
POL(&&(x₁, x₂)) = -1
POL(EVAL(x₁, x₂)) = 2 + x₂ + (-1)x₁
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x₁, x₂)) = -1

The following pairs are in P_>:

COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])

The following pairs are in P_bound:

COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])

The following pairs are in P_≥:

EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])

At least the following rules have been oriented under context sensitive arithmetic replacement:

&&(FALSE, FALSE)¹ ↔ FALSE¹
TRUE¹ → &&(TRUE, TRUE)¹
+@z¹ ↔
&&(TRUE, FALSE)¹ → FALSE¹
&&(FALSE, TRUE)¹ ↔ FALSE¹

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                          ↳ IDependencyGraphProof

                            ↳ IDP

                              ↳ IDPNonInfProof

                                ↳ AND

                                  ↳ IDP

                                    ↳ IDependencyGraphProof

                                      ↳ IDP

                                        ↳ IDPNonInfProof

                                          ↳ IDP

                                            ↳ IDependencyGraphProof

                                  ↳ IDP

                        ↳ IDP

              ↳ IDP

I DP problem:
The following domains are used:

z

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

(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                          ↳ IDependencyGraphProof

                            ↳ IDP

                              ↳ IDPNonInfProof

                                ↳ AND

                                  ↳ IDP

                                  ↳ IDP

                                    ↳ IDPNonInfProof

                        ↳ IDP

              ↳ IDP

I DP problem:
The following domains are used:

z

(0) -> (3), if ((y[0] →^* y[3])∧(+@z(x[0], 1@z) →^* x[3]))

(3) -> (0), if ((x[3] →^* x[0])∧(y[3] →^* y[0])∧(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])) →^* TRUE))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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 EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(1)    (EVAL(x[3], y[3])≥NonInfC∧EVAL(x[3], y[3])≥COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2)    ((U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3)    ((U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4)    (0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0)

We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5)    (0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0)

For Pair COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]), EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(6)    (x[3]=x[0]∧y[3]=y[0]∧&&(>@z(y[3], x[3]), >=@z(y[3], x[3]))=TRUE∧+@z(x[0], 1@z)=x[3]1∧y[0]=y[3]1 ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(+@z(x[0], 1@z), y[0])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7)    (>@z(y[3], x[3])=TRUE∧>=@z(y[3], x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(+@z(x[3], 1@z), y[3])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧-1 + (-1)Bound + y[3] + (-1)x[3] ≥ 0∧0 ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧-1 + (-1)Bound + y[3] + (-1)x[3] ≥ 0∧0 ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧-1 + (-1)Bound + y[3] + (-1)x[3] ≥ 0)

We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11)    (-1 + x[3] ≥ 0∧x[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧-1 + (-1)Bound + x[3] ≥ 0)

We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12)    (x[3] ≥ 0∧1 + x[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧(-1)Bound + x[3] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(13)    (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧(-1)Bound + x[3] ≥ 0)

(14)    (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧(-1)Bound + x[3] ≥ 0)

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

EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
- (0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0)
COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
- (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧(-1)Bound + x[3] ≥ 0)
- (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧(-1)Bound + x[3] ≥ 0)

POL(>=@z(x₁, x₂)) = -1
POL(COND_EVAL1(x₁, x₂, x₃)) = -1 + x₃ + (-1)x₂
POL(TRUE) = 2
POL(&&(x₁, x₂)) = -1
POL(EVAL(x₁, x₂)) = -1 + x₂ + (-1)x₁
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(FALSE) = 2
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x₁, x₂)) = -1

The following pairs are in P_>:

COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])

The following pairs are in P_bound:

COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])

The following pairs are in P_≥:

EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])

At least the following rules have been oriented under context sensitive arithmetic replacement:

FALSE¹ → &&(FALSE, FALSE)¹
TRUE¹ → &&(TRUE, TRUE)¹
+@z¹ ↔
FALSE¹ → &&(TRUE, FALSE)¹
FALSE¹ → &&(FALSE, TRUE)¹

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                          ↳ IDependencyGraphProof

                            ↳ IDP

                              ↳ IDPNonInfProof

                                ↳ AND

                                  ↳ IDP

                                  ↳ IDP

                                    ↳ IDPNonInfProof

                                      ↳ IDP

                                        ↳ IDependencyGraphProof

                        ↳ IDP

              ↳ IDP

I DP problem:
The following domains are used:

z

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                        ↳ IDP

                          ↳ IDependencyGraphProof

              ↳ IDP

I DP problem:
The following domains are used:

z

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

(4): EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])
(0): COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
(7): EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])

(0) -> (3), if ((y[0] →^* y[3])∧(+@z(x[0], 1@z) →^* x[3]))

(0) -> (7), if ((y[0] →^* y[7])∧(+@z(x[0], 1@z) →^* x[7]))

(0) -> (4), if ((y[0] →^* y[4])∧(+@z(x[0], 1@z) →^* x[4]))

(3) -> (0), if ((x[3] →^* x[0])∧(y[3] →^* y[0])∧(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])) →^* TRUE))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                        ↳ IDP

                          ↳ IDependencyGraphProof

                            ↳ IDP

                              ↳ IDPNonInfProof

              ↳ IDP

I DP problem:
The following domains are used:

z

(0) -> (3), if ((y[0] →^* y[3])∧(+@z(x[0], 1@z) →^* x[3]))

(3) -> (0), if ((x[3] →^* x[0])∧(y[3] →^* y[0])∧(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])) →^* TRUE))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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 EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(1)    (EVAL(x[3], y[3])≥NonInfC∧EVAL(x[3], y[3])≥COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2)    ((U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3)    ((U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4)    (0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0)

We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5)    (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0)

For Pair COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]), EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(6)    (x[3]=x[0]∧y[3]=y[0]∧&&(>@z(y[3], x[3]), >=@z(y[3], x[3]))=TRUE∧+@z(x[0], 1@z)=x[3]1∧y[0]=y[3]1 ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(+@z(x[0], 1@z), y[0])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7)    (>@z(y[3], x[3])=TRUE∧>=@z(y[3], x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(+@z(x[3], 1@z), y[3])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧(-1)Bound + y[3] + (-1)x[3] ≥ 0∧0 ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧(-1)Bound + y[3] + (-1)x[3] ≥ 0∧0 ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧(-1)Bound + y[3] + (-1)x[3] ≥ 0∧0 ≥ 0)

We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11)    (x[3] ≥ 0∧1 + x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 + (-1)Bound + x[3] ≥ 0∧0 ≥ 0)

We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(12)    (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 + (-1)Bound + x[3] ≥ 0∧0 ≥ 0)

(13)    (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 + (-1)Bound + x[3] ≥ 0∧0 ≥ 0)

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

EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
- (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0)
COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
- (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 + (-1)Bound + x[3] ≥ 0∧0 ≥ 0)
- (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 + (-1)Bound + x[3] ≥ 0∧0 ≥ 0)

POL(>=@z(x₁, x₂)) = -1
POL(COND_EVAL1(x₁, x₂, x₃)) = -1 + x₃ + (-1)x₂ + (-1)x₁
POL(TRUE) = -1
POL(&&(x₁, x₂)) = -1
POL(EVAL(x₁, x₂)) = x₂ + (-1)x₁
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x₁, x₂)) = -1

The following pairs are in P_>:

COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])

The following pairs are in P_bound:

COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])

The following pairs are in P_≥:

EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])

At least the following rules have been oriented under context sensitive arithmetic replacement:

&&(FALSE, FALSE)¹ ↔ FALSE¹
&&(TRUE, TRUE)¹ ↔ TRUE¹
+@z¹ ↔
FALSE¹ → &&(TRUE, FALSE)¹
&&(FALSE, TRUE)¹ ↔ FALSE¹

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                        ↳ IDP

                          ↳ IDependencyGraphProof

                            ↳ IDP

                              ↳ IDPNonInfProof

                                ↳ IDP

                                  ↳ IDependencyGraphProof

              ↳ IDP

I DP problem:
The following domains are used:

z

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

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

(0): COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
(1): COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1])
(2): EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
(4): EVAL(x[4], y[4]) → COND_EVAL3(&&(>@z(y[4], x[4]), >@z(x[4], y[4])), x[4], y[4])
(6): COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z))
(7): EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])

(1) -> (4), if ((y[1] →^* y[4])∧(+@z(x[1], 1@z) →^* x[4]))

(6) -> (7), if ((+@z(y[6], 1@z) →^* y[7])∧(x[6] →^* x[7]))

(0) -> (4), if ((y[0] →^* y[4])∧(+@z(x[0], 1@z) →^* x[4]))

(1) -> (7), if ((y[1] →^* y[7])∧(+@z(x[1], 1@z) →^* x[7]))

(6) -> (3), if ((+@z(y[6], 1@z) →^* y[3])∧(x[6] →^* x[3]))

(1) -> (3), if ((y[1] →^* y[3])∧(+@z(x[1], 1@z) →^* x[3]))

(7) -> (1), if ((x[7] →^* x[1])∧(y[7] →^* y[1])∧(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])) →^* TRUE))

(6) -> (4), if ((+@z(y[6], 1@z) →^* y[4])∧(x[6] →^* x[4]))

(2) -> (6), if ((x[2] →^* x[6])∧(y[2] →^* y[6])∧(>@z(x[2], y[2]) →^* TRUE))

(0) -> (2), if ((y[0] →^* y[2])∧(+@z(x[0], 1@z) →^* x[2]))

(0) -> (3), if ((y[0] →^* y[3])∧(+@z(x[0], 1@z) →^* x[3]))

(0) -> (7), if ((y[0] →^* y[7])∧(+@z(x[0], 1@z) →^* x[7]))

(3) -> (0), if ((x[3] →^* x[0])∧(y[3] →^* y[0])∧(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])) →^* TRUE))

(6) -> (2), if ((+@z(y[6], 1@z) →^* y[2])∧(x[6] →^* x[2]))

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

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

I DP problem:
The following domains are used:

z

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

(2): EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])
(0): COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
(1): COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1])
(7): EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])
(6): COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z))

(0) -> (3), if ((y[0] →^* y[3])∧(+@z(x[0], 1@z) →^* x[3]))

(0) -> (7), if ((y[0] →^* y[7])∧(+@z(x[0], 1@z) →^* x[7]))

(6) -> (7), if ((+@z(y[6], 1@z) →^* y[7])∧(x[6] →^* x[7]))

(1) -> (3), if ((y[1] →^* y[3])∧(+@z(x[1], 1@z) →^* x[3]))

(6) -> (3), if ((+@z(y[6], 1@z) →^* y[3])∧(x[6] →^* x[3]))

(1) -> (7), if ((y[1] →^* y[7])∧(+@z(x[1], 1@z) →^* x[7]))

(3) -> (0), if ((x[3] →^* x[0])∧(y[3] →^* y[0])∧(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])) →^* TRUE))

(6) -> (2), if ((+@z(y[6], 1@z) →^* y[2])∧(x[6] →^* x[2]))

(7) -> (1), if ((x[7] →^* x[1])∧(y[7] →^* y[1])∧(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])) →^* TRUE))

(2) -> (6), if ((x[2] →^* x[6])∧(y[2] →^* y[6])∧(>@z(x[2], y[2]) →^* TRUE))

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

(0) -> (2), if ((y[0] →^* y[2])∧(+@z(x[0], 1@z) →^* x[2]))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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 EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2]) the following chains were created:

We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2]) which results in the following constraint:
(1)    (EVAL(x[2], y[2])≥NonInfC∧EVAL(x[2], y[2])≥COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])∧(U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2)    ((U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3)    ((U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4)    (0 ≥ 0∧(U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0)

We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5)    (0 = 0∧(U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0)

For Pair COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]), EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]) which results in the following constraint:
(6)    (y[0]=y[7]∧x[3]=x[0]∧y[3]=y[0]∧&&(>@z(y[3], x[3]), >=@z(y[3], x[3]))=TRUE∧+@z(x[0], 1@z)=x[7] ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(+@z(x[0], 1@z), y[0])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7)    (>@z(y[3], x[3])=TRUE∧>=@z(y[3], x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(+@z(x[3], 1@z), y[3])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0∧0 ≥ 0)

We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11)    (x[3] ≥ 0∧1 + x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0∧0 ≥ 0)

We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(12)    (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0∧0 ≥ 0)

(13)    (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0∧0 ≥ 0)
We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]), EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2]) which results in the following constraint:
(14)    (y[0]=y[2]∧+@z(x[0], 1@z)=x[2]∧x[3]=x[0]∧y[3]=y[0]∧&&(>@z(y[3], x[3]), >=@z(y[3], x[3]))=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(+@z(x[0], 1@z), y[0])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (14) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(15)    (>@z(y[3], x[3])=TRUE∧>=@z(y[3], x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(+@z(x[3], 1@z), y[3])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0)

We simplified constraint (18) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(19)    (1 + x[3] ≥ 0∧x[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0)

We simplified constraint (19) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(20)    (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0)

(21)    (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0)
We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]), EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(22)    (x[3]=x[0]∧y[3]=y[0]∧&&(>@z(y[3], x[3]), >=@z(y[3], x[3]))=TRUE∧+@z(x[0], 1@z)=x[3]1∧y[0]=y[3]1 ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(+@z(x[0], 1@z), y[0])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (22) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(23)    (>@z(y[3], x[3])=TRUE∧>=@z(y[3], x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL1(TRUE, x[3], y[3])≥EVAL(+@z(x[3], 1@z), y[3])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (23) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(24)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (24) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(25)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(26)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(27)    (x[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (27) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(28)    (1 + x[3] ≥ 0∧x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (28) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(29)    (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

(30)    (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)

For Pair EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(31)    (EVAL(x[3], y[3])≥NonInfC∧EVAL(x[3], y[3])≥COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥))

We simplified constraint (31) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(32)    ((U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (32) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(33)    ((U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (33) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(34)    (0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0)

We simplified constraint (34) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(35)    (0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 = 0)

For Pair COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1]) the following chains were created:

We consider the chain EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]), COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1]), EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(36)    (y[1]=y[3]∧+@z(x[1], 1@z)=x[3]∧y[7]=y[1]∧x[7]=x[1]∧&&(>@z(x[7], y[7]), >=@z(y[7], x[7]))=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(+@z(x[1], 1@z), y[1])∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (36) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(37)    (>@z(x[7], y[7])=TRUE∧>=@z(y[7], x[7])=TRUE ⇒ COND_EVAL(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL(TRUE, x[7], y[7])≥EVAL(+@z(x[7], 1@z), y[7])∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (37) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(38)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧-1 + (-1)Bound + (-1)y[7] + (-1)x[7] ≥ 0∧0 ≥ 0)

We simplified constraint (38) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(39)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧-1 + (-1)Bound + (-1)y[7] + (-1)x[7] ≥ 0∧0 ≥ 0)

We simplified constraint (39) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(40)    (y[7] + (-1)x[7] ≥ 0∧-1 + x[7] + (-1)y[7] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧-1 + (-1)Bound + (-1)y[7] + (-1)x[7] ≥ 0)

We solved constraint (40) using rule (IDP_SMT_SPLIT).
We consider the chain EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]), COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1]), EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2]) which results in the following constraint:
(41)    (y[7]=y[1]∧x[7]=x[1]∧+@z(x[1], 1@z)=x[2]∧&&(>@z(x[7], y[7]), >=@z(y[7], x[7]))=TRUE∧y[1]=y[2] ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(+@z(x[1], 1@z), y[1])∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (41) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(42)    (>@z(x[7], y[7])=TRUE∧>=@z(y[7], x[7])=TRUE ⇒ COND_EVAL(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL(TRUE, x[7], y[7])≥EVAL(+@z(x[7], 1@z), y[7])∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (42) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(43)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧-1 + (-1)Bound + (-1)y[7] + (-1)x[7] ≥ 0∧0 ≥ 0)

We simplified constraint (43) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(44)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧-1 + (-1)Bound + (-1)y[7] + (-1)x[7] ≥ 0∧0 ≥ 0)

We simplified constraint (44) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(45)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ -1 + (-1)Bound + (-1)y[7] + (-1)x[7] ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We solved constraint (45) using rule (IDP_SMT_SPLIT).
We consider the chain EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]), COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1]), EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]) which results in the following constraint:
(46)    (+@z(x[1], 1@z)=x[7]1∧y[7]=y[1]∧x[7]=x[1]∧y[1]=y[7]1∧&&(>@z(x[7], y[7]), >=@z(y[7], x[7]))=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(+@z(x[1], 1@z), y[1])∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (46) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(47)    (>@z(x[7], y[7])=TRUE∧>=@z(y[7], x[7])=TRUE ⇒ COND_EVAL(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL(TRUE, x[7], y[7])≥EVAL(+@z(x[7], 1@z), y[7])∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (47) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(48)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧-1 + (-1)Bound + (-1)y[7] + (-1)x[7] ≥ 0∧0 ≥ 0)

We simplified constraint (48) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(49)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧-1 + (-1)Bound + (-1)y[7] + (-1)x[7] ≥ 0∧0 ≥ 0)

We simplified constraint (49) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(50)    (-1 + x[7] + (-1)y[7] ≥ 0∧y[7] + (-1)x[7] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[1], 1@z), y[1])), ≥)∧-1 + (-1)Bound + (-1)y[7] + (-1)x[7] ≥ 0)

We solved constraint (50) using rule (IDP_SMT_SPLIT).

For Pair EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]) the following chains were created:

We consider the chain EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]) which results in the following constraint:
(51)    (EVAL(x[7], y[7])≥NonInfC∧EVAL(x[7], y[7])≥COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])∧(U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥))

We simplified constraint (51) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(52)    ((U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (52) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(53)    ((U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (53) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(54)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥))

We simplified constraint (54) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(55)    (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥))

For Pair COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z)) the following chains were created:

We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2]), COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z)), EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(56)    (>@z(x[2], y[2])=TRUE∧x[6]=x[3]∧x[2]=x[6]∧+@z(y[6], 1@z)=y[3]∧y[2]=y[6] ⇒ COND_EVAL2(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL2(TRUE, x[6], y[6])≥EVAL(x[6], +@z(y[6], 1@z))∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (56) using rules (III), (IV) which results in the following new constraint:
(57)    (>@z(x[2], y[2])=TRUE ⇒ COND_EVAL2(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL2(TRUE, x[2], y[2])≥EVAL(x[2], +@z(y[2], 1@z))∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (57) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(58)    (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (58) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(59)    (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (59) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(60)    (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ 0 ≥ 0∧1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (60) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(61)    (x[2] ≥ 0 ⇒ 0 ≥ 0∧1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (61) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(62)    (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

(63)    (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))
We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2]), COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z)), EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2]) which results in the following constraint:
(64)    (>@z(x[2], y[2])=TRUE∧+@z(y[6], 1@z)=y[2]1∧x[2]=x[6]∧x[6]=x[2]1∧y[2]=y[6] ⇒ COND_EVAL2(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL2(TRUE, x[6], y[6])≥EVAL(x[6], +@z(y[6], 1@z))∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (64) using rules (III), (IV) which results in the following new constraint:
(65)    (>@z(x[2], y[2])=TRUE ⇒ COND_EVAL2(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL2(TRUE, x[2], y[2])≥EVAL(x[2], +@z(y[2], 1@z))∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (65) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(66)    (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (66) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(67)    (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (67) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(68)    (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (68) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(69)    (x[2] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (69) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(70)    (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

(71)    (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))
We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2]), COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z)), EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7]) which results in the following constraint:
(72)    (>@z(x[2], y[2])=TRUE∧x[2]=x[6]∧y[2]=y[6]∧+@z(y[6], 1@z)=y[7]∧x[6]=x[7] ⇒ COND_EVAL2(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL2(TRUE, x[6], y[6])≥EVAL(x[6], +@z(y[6], 1@z))∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (72) using rules (III), (IV) which results in the following new constraint:
(73)    (>@z(x[2], y[2])=TRUE ⇒ COND_EVAL2(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL2(TRUE, x[2], y[2])≥EVAL(x[2], +@z(y[2], 1@z))∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (73) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(74)    (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (74) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(75)    (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (75) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(76)    (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ 0 ≥ 0∧1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (76) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(77)    (x[2] ≥ 0 ⇒ 0 ≥ 0∧1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (77) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(78)    (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

(79)    (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

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

EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])
- (0 = 0∧(U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0)
COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
- (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0∧0 ≥ 0)
- (x[3] ≥ 0∧1 + x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0∧0 ≥ 0)
- (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0)
- (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧1 ≥ 0)
- (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)
- (1 + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0∧1 ≥ 0)
EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
- (0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 = 0)
COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1])
EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])
- (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(U^Increasing(COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])), ≥))
COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z))
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

POL(>=@z(x₁, x₂)) = -1
POL(COND_EVAL2(x₁, x₂, x₃)) = -1 + (-1)x₃ + (-1)x₂
POL(COND_EVAL1(x₁, x₂, x₃)) = -1 + (-1)x₃ + (-1)x₂
POL(TRUE) = 0
POL(&&(x₁, x₂)) = -1
POL(COND_EVAL(x₁, x₂, x₃)) = -1 + (-1)x₃ + (-1)x₂
POL(EVAL(x₁, x₂)) = -1 + (-1)x₂ + (-1)x₁
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(FALSE) = 2
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x₁, x₂)) = -1

The following pairs are in P_>:

COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1])

The following pairs are in P_bound:

COND_EVAL(TRUE, x[1], y[1]) → EVAL(+@z(x[1], 1@z), y[1])

The following pairs are in P_≥:

EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])
COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])
COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z))

At least the following rules have been oriented under context sensitive arithmetic replacement:

FALSE¹ → &&(FALSE, FALSE)¹
TRUE¹ → &&(TRUE, TRUE)¹
+@z¹ ↔
FALSE¹ → &&(FALSE, TRUE)¹
FALSE¹ → &&(TRUE, FALSE)¹

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

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

(2): EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])
(0): COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
(7): EVAL(x[7], y[7]) → COND_EVAL(&&(>@z(x[7], y[7]), >=@z(y[7], x[7])), x[7], y[7])
(6): COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z))

(0) -> (3), if ((y[0] →^* y[3])∧(+@z(x[0], 1@z) →^* x[3]))

(0) -> (7), if ((y[0] →^* y[7])∧(+@z(x[0], 1@z) →^* x[7]))

(6) -> (7), if ((+@z(y[6], 1@z) →^* y[7])∧(x[6] →^* x[7]))

(6) -> (3), if ((+@z(y[6], 1@z) →^* y[3])∧(x[6] →^* x[3]))

(3) -> (0), if ((x[3] →^* x[0])∧(y[3] →^* y[0])∧(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])) →^* TRUE))

(6) -> (2), if ((+@z(y[6], 1@z) →^* y[2])∧(x[6] →^* x[2]))

(2) -> (6), if ((x[2] →^* x[6])∧(y[2] →^* y[6])∧(>@z(x[2], y[2]) →^* TRUE))

(0) -> (2), if ((y[0] →^* y[2])∧(+@z(x[0], 1@z) →^* x[2]))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

                          ↳ IDP

                            ↳ IDPtoQDPProof

                            ↳ IDPNonInfProof

I DP problem:
The following domains are used:

z

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

(2): EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])
(0): COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
(6): COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z))

(0) -> (3), if ((y[0] →^* y[3])∧(+@z(x[0], 1@z) →^* x[3]))

(6) -> (3), if ((+@z(y[6], 1@z) →^* y[3])∧(x[6] →^* x[3]))

(3) -> (0), if ((x[3] →^* x[0])∧(y[3] →^* y[0])∧(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])) →^* TRUE))

(6) -> (2), if ((+@z(y[6], 1@z) →^* y[2])∧(x[6] →^* x[2]))

(2) -> (6), if ((x[2] →^* x[6])∧(y[2] →^* y[6])∧(>@z(x[2], y[2]) →^* TRUE))

(0) -> (2), if ((y[0] →^* y[2])∧(+@z(x[0], 1@z) →^* x[2]))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

Represented integers and predefined function symbols by Terms

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

                          ↳ IDP

                            ↳ IDPtoQDPProof

                              ↳ QDP

                                ↳ UsableRulesProof

                            ↳ IDPNonInfProof

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

EVAL(x[2], y[2]) → COND_EVAL2(greater_int(x[2], y[2]), x[2], y[2])
COND_EVAL1(true, x[0], y[0]) → EVAL(plus_int(pos(s(0)), x[0]), y[0])
EVAL(x[3], y[3]) → COND_EVAL1(and(greater_int(y[3], x[3]), greatereq_int(y[3], x[3])), x[3], y[3])
COND_EVAL2(true, x[6], y[6]) → EVAL(x[6], plus_int(pos(s(0)), y[6]))

The TRS R consists of the following rules:

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))
plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(neg(x), pos(y)) → minus_nat(y, x)
plus_int(neg(x), neg(y)) → neg(plus_nat(x, y))
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, y)
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
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:

eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, x1)
Cond_eval3(true, x0, x1)
Cond_eval2(true, x0, x1)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

                          ↳ IDP

                            ↳ IDPtoQDPProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QReductionProof

                            ↳ IDPNonInfProof

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

EVAL(x[2], y[2]) → COND_EVAL2(greater_int(x[2], y[2]), x[2], y[2])
COND_EVAL1(true, x[0], y[0]) → EVAL(plus_int(pos(s(0)), x[0]), y[0])
EVAL(x[3], y[3]) → COND_EVAL1(and(greater_int(y[3], x[3]), greatereq_int(y[3], x[3])), x[3], y[3])
COND_EVAL2(true, x[6], y[6]) → EVAL(x[6], plus_int(pos(s(0)), y[6]))

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, 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))
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))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, x1)
Cond_eval3(true, x0, x1)
Cond_eval2(true, x0, x1)
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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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].

eval(x0, x1)
Cond_eval1(true, x0, x1)
Cond_eval(true, x0, x1)
Cond_eval3(true, x0, x1)
Cond_eval2(true, x0, x1)

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

                          ↳ IDP

                            ↳ IDPtoQDPProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QReductionProof

                                      ↳ QDP

                            ↳ IDPNonInfProof

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

EVAL(x[2], y[2]) → COND_EVAL2(greater_int(x[2], y[2]), x[2], y[2])
COND_EVAL1(true, x[0], y[0]) → EVAL(plus_int(pos(s(0)), x[0]), y[0])
EVAL(x[3], y[3]) → COND_EVAL1(and(greater_int(y[3], x[3]), greatereq_int(y[3], x[3])), x[3], y[3])
COND_EVAL2(true, x[6], y[6]) → EVAL(x[6], plus_int(pos(s(0)), y[6]))

The TRS R consists of the following rules:

plus_int(pos(x), neg(y)) → minus_nat(x, y)
plus_int(pos(x), pos(y)) → pos(plus_nat(x, y))
plus_nat(0, x) → x
plus_nat(s(x), y) → s(plus_nat(x, y))
minus_nat(0, 0) → pos(0)
minus_nat(0, s(y)) → neg(s(y))
minus_nat(s(x), 0) → pos(s(x))
minus_nat(s(x), s(y)) → minus_nat(x, 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))
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))
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true

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)))
plus_int(pos(x0), neg(x1))
plus_int(neg(x0), pos(x1))
plus_int(neg(x0), neg(x1))
plus_int(pos(x0), pos(x1))
plus_nat(0, x0)
plus_nat(s(x0), x1)
minus_nat(0, 0)
minus_nat(0, s(x0))
minus_nat(s(x0), 0)
minus_nat(s(x0), s(x1))
and(false, false)
and(false, true)
and(true, false)
and(true, true)
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 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 EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2]) the following chains were created:

We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2]) which results in the following constraint:
(1)    (EVAL(x[2], y[2])≥NonInfC∧EVAL(x[2], y[2])≥COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])∧(U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2)    ((U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧max{(-1)x[2] + y[2], x[2] + (-1)y[2]} + (-1)max{(-1)x[2] + y[2], x[2] + (-1)y[2]} ≥ 0)

We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3)    ((U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧max{(-1)x[2] + y[2], x[2] + (-1)y[2]} + (-1)max{(-1)x[2] + y[2], x[2] + (-1)y[2]} ≥ 0)

We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraints:
(4)    ((-2)x[2] + (2)y[2] ≥ 0 ⇒ (U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

(5)    (-1 + (2)x[2] + (-2)y[2] ≥ 0 ⇒ (U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (4) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    ((2)x[2] ≥ 0 ⇒ (U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(7)    (-1 + (2)x[2] + (-2)y[2] ≥ 0∧x[2] ≥ 0 ⇒ (U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

(8)    (-1 + (-2)x[2] + (-2)y[2] ≥ 0∧x[2] ≥ 0 ⇒ (U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(9)    ((2)x[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

(10)    ((2)x[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(11)    (-1 + (2)x[2] + (-2)y[2] ≥ 0∧x[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

(12)    (-1 + (2)x[2] + (2)y[2] ≥ 0∧x[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (-1 + (-2)x[2] + (2)y[2] ≥ 0∧x[2] ≥ 0 ⇒ (U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

For Pair COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]) the following chains were created:

We consider the chain COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z)), EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]) which results in the following constraint:
(14)    (x[6]=x[3]∧+@z(y[6], 1@z)=y[3]∧x[3]=x[0]∧y[3]=y[0]∧&&(>@z(y[3], x[3]), >=@z(y[3], x[3]))=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(+@z(x[0], 1@z), y[0])∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (14) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(15)    (>@z(+@z(y[6], 1@z), x[3])=TRUE∧>=@z(+@z(y[6], 1@z), x[3])=TRUE ⇒ COND_EVAL1(TRUE, x[3], +@z(y[6], 1@z))≥NonInfC∧COND_EVAL1(TRUE, x[3], +@z(y[6], 1@z))≥EVAL(+@z(x[3], 1@z), +@z(y[6], 1@z))∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥))

We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16)    (y[6] + (-1)x[3] ≥ 0∧1 + y[6] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧-1 + (-1)Bound + max{1 + (-1)x[3] + y[6], -1 + x[3] + (-1)y[6]} ≥ 0∧-1 + max{1 + (-1)x[3] + y[6], -1 + x[3] + (-1)y[6]} + (-1)max{(-1)x[3] + y[6], x[3] + (-1)y[6]} ≥ 0)

We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17)    (y[6] + (-1)x[3] ≥ 0∧1 + y[6] + (-1)x[3] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧-1 + (-1)Bound + max{1 + (-1)x[3] + y[6], -1 + x[3] + (-1)y[6]} ≥ 0∧-1 + max{1 + (-1)x[3] + y[6], -1 + x[3] + (-1)y[6]} + (-1)max{(-1)x[3] + y[6], x[3] + (-1)y[6]} ≥ 0)

We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18)    (y[6] + (-1)x[3] ≥ 0∧(-2)x[3] + (2)y[6] ≥ 0∧1 + y[6] + (-1)x[3] ≥ 0∧2 + (-2)x[3] + (2)y[6] ≥ 0 ⇒ (-1)Bound + (-1)x[3] + y[6] ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

We simplified constraint (18) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(19)    (x[3] ≥ 0∧(2)x[3] ≥ 0∧1 + x[3] ≥ 0∧2 + (2)x[3] ≥ 0 ⇒ (-1)Bound + x[3] ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

We simplified constraint (19) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(20)    (x[3] ≥ 0∧(2)x[3] ≥ 0∧1 + x[3] ≥ 0∧2 + (2)x[3] ≥ 0∧y[6] ≥ 0 ⇒ (-1)Bound + x[3] ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)

(21)    (x[3] ≥ 0∧(2)x[3] ≥ 0∧1 + x[3] ≥ 0∧2 + (2)x[3] ≥ 0∧y[6] ≥ 0 ⇒ (-1)Bound + x[3] ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)
We consider the chain COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]), EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]) which results in the following constraint:
(22)    (y[0]=y[3]∧+@z(x[0], 1@z)=x[3]∧&&(>@z(y[3], x[3]), >=@z(y[3], x[3]))=TRUE∧x[3]=x[0]1∧y[3]=y[0]1 ⇒ COND_EVAL1(TRUE, x[0]1, y[0]1)≥NonInfC∧COND_EVAL1(TRUE, x[0]1, y[0]1)≥EVAL(+@z(x[0]1, 1@z), y[0]1)∧(U^Increasing(EVAL(+@z(x[0]1, 1@z), y[0]1)), ≥))

We simplified constraint (22) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(23)    (>@z(y[3], +@z(x[0], 1@z))=TRUE∧>=@z(y[3], +@z(x[0], 1@z))=TRUE ⇒ COND_EVAL1(TRUE, +@z(x[0], 1@z), y[3])≥NonInfC∧COND_EVAL1(TRUE, +@z(x[0], 1@z), y[3])≥EVAL(+@z(+@z(x[0], 1@z), 1@z), y[3])∧(U^Increasing(EVAL(+@z(x[0]1, 1@z), y[0]1)), ≥))

We simplified constraint (23) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(24)    (-2 + y[3] + (-1)x[0] ≥ 0∧-1 + y[3] + (-1)x[0] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0]1, 1@z), y[0]1)), ≥)∧-1 + (-1)Bound + max{-1 + (-1)x[0] + y[3], 1 + x[0] + (-1)y[3]} ≥ 0∧-1 + max{-1 + (-1)x[0] + y[3], 1 + x[0] + (-1)y[3]} + (-1)max{-2 + (-1)x[0] + y[3], 2 + x[0] + (-1)y[3]} ≥ 0)

We simplified constraint (24) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(25)    (-2 + y[3] + (-1)x[0] ≥ 0∧-1 + y[3] + (-1)x[0] ≥ 0 ⇒ (U^Increasing(EVAL(+@z(x[0]1, 1@z), y[0]1)), ≥)∧-1 + (-1)Bound + max{-1 + (-1)x[0] + y[3], 1 + x[0] + (-1)y[3]} ≥ 0∧-1 + max{-1 + (-1)x[0] + y[3], 1 + x[0] + (-1)y[3]} + (-1)max{-2 + (-1)x[0] + y[3], 2 + x[0] + (-1)y[3]} ≥ 0)

We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(26)    (-4 + (-2)x[0] + (2)y[3] ≥ 0∧-2 + (-2)x[0] + (2)y[3] ≥ 0∧-2 + y[3] + (-1)x[0] ≥ 0∧-1 + y[3] + (-1)x[0] ≥ 0 ⇒ -2 + (-1)Bound + (-1)x[0] + y[3] ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0]1, 1@z), y[0]1)), ≥))

We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(27)    ((2)y[3] ≥ 0∧2 + (2)y[3] ≥ 0∧y[3] ≥ 0∧1 + y[3] ≥ 0 ⇒ (-1)Bound + y[3] ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0]1, 1@z), y[0]1)), ≥))

We simplified constraint (27) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(28)    ((2)y[3] ≥ 0∧2 + (2)y[3] ≥ 0∧y[3] ≥ 0∧1 + y[3] ≥ 0∧x[0] ≥ 0 ⇒ (-1)Bound + y[3] ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0]1, 1@z), y[0]1)), ≥))

(29)    ((2)y[3] ≥ 0∧2 + (2)y[3] ≥ 0∧y[3] ≥ 0∧1 + y[3] ≥ 0∧x[0] ≥ 0 ⇒ (-1)Bound + y[3] ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0]1, 1@z), y[0]1)), ≥))

For Pair EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(30)    (EVAL(x[3], y[3])≥NonInfC∧EVAL(x[3], y[3])≥COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥))

We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(31)    ((U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧max{(-1)x[3] + y[3], x[3] + (-1)y[3]} + (-1)max{(-1)x[3] + y[3], x[3] + (-1)y[3]} ≥ 0)

We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(32)    ((U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧max{(-1)x[3] + y[3], x[3] + (-1)y[3]} + (-1)max{(-1)x[3] + y[3], x[3] + (-1)y[3]} ≥ 0)

We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraints:
(33)    ((-2)x[3] + (2)y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0)

(34)    (-1 + (2)x[3] + (-2)y[3] ≥ 0 ⇒ (U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(35)    ((2)x[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0)

We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(36)    (-1 + (2)x[3] + (-2)y[3] ≥ 0∧x[3] ≥ 0 ⇒ (U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

(37)    (-1 + (-2)x[3] + (-2)y[3] ≥ 0∧x[3] ≥ 0 ⇒ (U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (35) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(38)    ((2)x[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0)

(39)    ((2)x[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0)

We simplified constraint (36) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(40)    (-1 + (2)x[3] + (-2)y[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

(41)    (-1 + (2)x[3] + (2)y[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(42)    (-1 + (-2)x[3] + (2)y[3] ≥ 0∧x[3] ≥ 0 ⇒ (U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

For Pair COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z)) the following chains were created:

We consider the chain COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0]), EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2]), COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z)) which results in the following constraint:
(43)    (>@z(x[2], y[2])=TRUE∧y[0]=y[2]∧x[2]=x[6]∧+@z(x[0], 1@z)=x[2]∧y[2]=y[6] ⇒ COND_EVAL2(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL2(TRUE, x[6], y[6])≥EVAL(x[6], +@z(y[6], 1@z))∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (43) using rule (III) which results in the following new constraint:
(44)    (>@z(+@z(x[0], 1@z), y[2])=TRUE ⇒ COND_EVAL2(TRUE, +@z(x[0], 1@z), y[2])≥NonInfC∧COND_EVAL2(TRUE, +@z(x[0], 1@z), y[2])≥EVAL(+@z(x[0], 1@z), +@z(y[2], 1@z))∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (44) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(45)    (x[0] + (-1)y[2] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0∧max{-1 + (-1)x[0] + y[2], 1 + x[0] + (-1)y[2]} + (-1)max{(-1)x[0] + y[2], x[0] + (-1)y[2]} ≥ 0)

We simplified constraint (45) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(46)    (x[0] + (-1)y[2] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0∧max{-1 + (-1)x[0] + y[2], 1 + x[0] + (-1)y[2]} + (-1)max{(-1)x[0] + y[2], x[0] + (-1)y[2]} ≥ 0)

We simplified constraint (46) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraints:
(47)    ((-2)x[0] + (2)y[2] ≥ 0∧1 + (2)x[0] + (-2)y[2] ≥ 0∧x[0] + (-1)y[2] ≥ 0 ⇒ 1 + (2)x[0] + (-2)y[2] ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0)

(48)    (-1 + (2)x[0] + (-2)y[2] ≥ 0∧1 + (2)x[0] + (-2)y[2] ≥ 0∧x[0] + (-1)y[2] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0)

We simplified constraint (47) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(49)    ((2)y[2] ≥ 0∧1 + (-2)y[2] ≥ 0∧(-1)y[2] ≥ 0 ⇒ 1 + (-2)y[2] ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0)

We simplified constraint (48) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(50)    (-1 + (2)y[2] ≥ 0∧1 + (2)y[2] ≥ 0∧y[2] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0)

We simplified constraint (49) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(51)    (0 ≥ 0∧1 ≥ 0∧0 ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0)

We simplified constraint (51) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(52)    (0 ≥ 0∧1 ≥ 0∧0 ≥ 0∧x[0] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0)

(53)    (0 ≥ 0∧1 ≥ 0∧0 ≥ 0∧x[0] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0)

We simplified constraint (50) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(54)    (-1 + (2)y[2] ≥ 0∧1 + (2)y[2] ≥ 0∧y[2] ≥ 0∧x[0] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0)

(55)    (-1 + (2)y[2] ≥ 0∧1 + (2)y[2] ≥ 0∧y[2] ≥ 0∧x[0] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0)
We consider the chain COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z)), EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2]), COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z)) which results in the following constraint:
(56)    (>@z(x[2], y[2])=TRUE∧+@z(y[6], 1@z)=y[2]∧x[6]=x[2]∧x[2]=x[6]1∧y[2]=y[6]1 ⇒ COND_EVAL2(TRUE, x[6]1, y[6]1)≥NonInfC∧COND_EVAL2(TRUE, x[6]1, y[6]1)≥EVAL(x[6]1, +@z(y[6]1, 1@z))∧(U^Increasing(EVAL(x[6]1, +@z(y[6]1, 1@z))), ≥))

We simplified constraint (56) using rule (III) which results in the following new constraint:
(57)    (>@z(x[2], +@z(y[6], 1@z))=TRUE ⇒ COND_EVAL2(TRUE, x[2], +@z(y[6], 1@z))≥NonInfC∧COND_EVAL2(TRUE, x[2], +@z(y[6], 1@z))≥EVAL(x[2], +@z(+@z(y[6], 1@z), 1@z))∧(U^Increasing(EVAL(x[6]1, +@z(y[6]1, 1@z))), ≥))

We simplified constraint (57) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(58)    (-2 + x[2] + (-1)y[6] ≥ 0 ⇒ (U^Increasing(EVAL(x[6]1, +@z(y[6]1, 1@z))), ≥)∧0 ≥ 0∧max{1 + (-1)x[2] + y[6], -1 + x[2] + (-1)y[6]} + (-1)max{2 + (-1)x[2] + y[6], -2 + x[2] + (-1)y[6]} ≥ 0)

We simplified constraint (58) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(59)    (-2 + x[2] + (-1)y[6] ≥ 0 ⇒ (U^Increasing(EVAL(x[6]1, +@z(y[6]1, 1@z))), ≥)∧0 ≥ 0∧max{1 + (-1)x[2] + y[6], -1 + x[2] + (-1)y[6]} + (-1)max{2 + (-1)x[2] + y[6], -2 + x[2] + (-1)y[6]} ≥ 0)

We simplified constraint (59) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraints:
(60)    (-2 + x[2] + (-1)y[6] ≥ 0∧-3 + (2)x[2] + (-2)y[6] ≥ 0∧4 + (-2)x[2] + (2)y[6] ≥ 0 ⇒ (U^Increasing(EVAL(x[6]1, +@z(y[6]1, 1@z))), ≥)∧-3 + (2)x[2] + (-2)y[6] ≥ 0∧0 ≥ 0)

(61)    (-2 + x[2] + (-1)y[6] ≥ 0∧-5 + (2)x[2] + (-2)y[6] ≥ 0∧-3 + (2)x[2] + (-2)y[6] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(x[6]1, +@z(y[6]1, 1@z))), ≥)∧0 ≥ 0)

We simplified constraint (60) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(62)    (x[2] ≥ 0∧1 + (2)x[2] ≥ 0∧(-2)x[2] ≥ 0 ⇒ (U^Increasing(EVAL(x[6]1, +@z(y[6]1, 1@z))), ≥)∧1 + (2)x[2] ≥ 0∧0 ≥ 0)

We simplified constraint (61) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(63)    (x[2] ≥ 0∧-1 + (2)x[2] ≥ 0∧1 + (2)x[2] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(x[6]1, +@z(y[6]1, 1@z))), ≥)∧0 ≥ 0)

We simplified constraint (62) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(64)    (0 ≥ 0∧1 ≥ 0∧0 ≥ 0 ⇒ (U^Increasing(EVAL(x[6]1, +@z(y[6]1, 1@z))), ≥)∧1 ≥ 0∧0 ≥ 0)

We simplified constraint (64) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(65)    (0 ≥ 0∧1 ≥ 0∧0 ≥ 0∧y[6] ≥ 0 ⇒ (U^Increasing(EVAL(x[6]1, +@z(y[6]1, 1@z))), ≥)∧1 ≥ 0∧0 ≥ 0)

(66)    (0 ≥ 0∧1 ≥ 0∧0 ≥ 0∧y[6] ≥ 0 ⇒ (U^Increasing(EVAL(x[6]1, +@z(y[6]1, 1@z))), ≥)∧1 ≥ 0∧0 ≥ 0)

We simplified constraint (63) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(67)    (x[2] ≥ 0∧-1 + (2)x[2] ≥ 0∧1 + (2)x[2] ≥ 0∧y[6] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(x[6]1, +@z(y[6]1, 1@z))), ≥)∧0 ≥ 0)

(68)    (x[2] ≥ 0∧-1 + (2)x[2] ≥ 0∧1 + (2)x[2] ≥ 0∧y[6] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(x[6]1, +@z(y[6]1, 1@z))), ≥)∧0 ≥ 0)

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

EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])
- ((2)x[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
- ((2)x[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
- (-1 + (2)x[2] + (-2)y[2] ≥ 0∧x[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
- (-1 + (2)x[2] + (2)y[2] ≥ 0∧x[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
- (-1 + (-2)x[2] + (2)y[2] ≥ 0∧x[2] ≥ 0 ⇒ (U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])
- (x[3] ≥ 0∧(2)x[3] ≥ 0∧1 + x[3] ≥ 0∧2 + (2)x[3] ≥ 0∧y[6] ≥ 0 ⇒ (-1)Bound + x[3] ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)
- (x[3] ≥ 0∧(2)x[3] ≥ 0∧1 + x[3] ≥ 0∧2 + (2)x[3] ≥ 0∧y[6] ≥ 0 ⇒ (-1)Bound + x[3] ≥ 0∧(U^Increasing(EVAL(+@z(x[0], 1@z), y[0])), ≥)∧0 ≥ 0)
- ((2)y[3] ≥ 0∧2 + (2)y[3] ≥ 0∧y[3] ≥ 0∧1 + y[3] ≥ 0∧x[0] ≥ 0 ⇒ (-1)Bound + y[3] ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0]1, 1@z), y[0]1)), ≥))
- ((2)y[3] ≥ 0∧2 + (2)y[3] ≥ 0∧y[3] ≥ 0∧1 + y[3] ≥ 0∧x[0] ≥ 0 ⇒ (-1)Bound + y[3] ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(+@z(x[0]1, 1@z), y[0]1)), ≥))
EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
- ((2)x[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0)
- ((2)x[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0)
- (-1 + (2)x[3] + (-2)y[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
- (-1 + (2)x[3] + (2)y[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ (U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
- (-1 + (-2)x[3] + (2)y[3] ≥ 0∧x[3] ≥ 0 ⇒ (U^Increasing(COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z))
- (0 ≥ 0∧1 ≥ 0∧0 ≥ 0∧x[0] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0)
- (0 ≥ 0∧1 ≥ 0∧0 ≥ 0∧x[0] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0)
- (-1 + (2)y[2] ≥ 0∧1 + (2)y[2] ≥ 0∧y[2] ≥ 0∧x[0] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0)
- (-1 + (2)y[2] ≥ 0∧1 + (2)y[2] ≥ 0∧y[2] ≥ 0∧x[0] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧0 ≥ 0)
- (0 ≥ 0∧1 ≥ 0∧0 ≥ 0∧y[6] ≥ 0 ⇒ (U^Increasing(EVAL(x[6]1, +@z(y[6]1, 1@z))), ≥)∧1 ≥ 0∧0 ≥ 0)
- (0 ≥ 0∧1 ≥ 0∧0 ≥ 0∧y[6] ≥ 0 ⇒ (U^Increasing(EVAL(x[6]1, +@z(y[6]1, 1@z))), ≥)∧1 ≥ 0∧0 ≥ 0)
- (x[2] ≥ 0∧-1 + (2)x[2] ≥ 0∧1 + (2)x[2] ≥ 0∧y[6] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(x[6]1, +@z(y[6]1, 1@z))), ≥)∧0 ≥ 0)
- (x[2] ≥ 0∧-1 + (2)x[2] ≥ 0∧1 + (2)x[2] ≥ 0∧y[6] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(x[6]1, +@z(y[6]1, 1@z))), ≥)∧0 ≥ 0)

POL(>=@z(x₁, x₂)) = -1
POL(COND_EVAL2(x₁, x₂, x₃)) = -1 + max{(-1)x₂ + x₃, x₂ + (-1)x₃}
POL(COND_EVAL1(x₁, x₂, x₃)) = -1 + max{(-1)x₂ + x₃, x₂ + (-1)x₃}
POL(TRUE) = -1
POL(&&(x₁, x₂)) = -1
POL(EVAL(x₁, x₂)) = -1 + max{(-1)x₁ + x₂, x₁ + (-1)x₂}
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x₁, x₂)) = -1

The following pairs are in P_>:

COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])

The following pairs are in P_bound:

COND_EVAL1(TRUE, x[0], y[0]) → EVAL(+@z(x[0], 1@z), y[0])

The following pairs are in P_≥:

EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])
EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z))

At least the following rules have been oriented under context sensitive arithmetic replacement:

&&(FALSE, FALSE)¹ ↔ FALSE¹
&&(TRUE, TRUE)¹ ↔ TRUE¹
+@z¹ ↔
&&(FALSE, TRUE)¹ ↔ FALSE¹
&&(TRUE, FALSE)¹ ↔ FALSE¹

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

                          ↳ IDP

                            ↳ IDPtoQDPProof

                            ↳ IDPNonInfProof

                              ↳ IDP

                                ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

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

(2): EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])
(3): EVAL(x[3], y[3]) → COND_EVAL1(&&(>@z(y[3], x[3]), >=@z(y[3], x[3])), x[3], y[3])
(6): COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z))

(6) -> (3), if ((+@z(y[6], 1@z) →^* y[3])∧(x[6] →^* x[3]))

(6) -> (2), if ((+@z(y[6], 1@z) →^* y[2])∧(x[6] →^* x[2]))

(2) -> (6), if ((x[2] →^* x[6])∧(y[2] →^* y[6])∧(>@z(x[2], y[2]) →^* TRUE))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

                          ↳ IDP

                            ↳ IDPtoQDPProof

                            ↳ IDPNonInfProof

                              ↳ IDP

                                ↳ IDependencyGraphProof

                                  ↳ IDP

                                    ↳ IDPNonInfProof

I DP problem:
The following domains are used:

z

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

(2): EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])
(6): COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z))

(6) -> (2), if ((+@z(y[6], 1@z) →^* y[2])∧(x[6] →^* x[2]))

(2) -> (6), if ((x[2] →^* x[6])∧(y[2] →^* y[6])∧(>@z(x[2], y[2]) →^* TRUE))

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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 EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2]) the following chains were created:

We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2]) which results in the following constraint:
(1)    (EVAL(x[2], y[2])≥NonInfC∧EVAL(x[2], y[2])≥COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])∧(U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2)    ((U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3)    ((U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4)    ((U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5)    (0 = 0∧0 = 0∧(U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0)

For Pair COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z)) the following chains were created:

We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2]), COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z)), EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2]) which results in the following constraint:
(6)    (>@z(x[2], y[2])=TRUE∧+@z(y[6], 1@z)=y[2]1∧x[2]=x[6]∧x[6]=x[2]1∧y[2]=y[6] ⇒ COND_EVAL2(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL2(TRUE, x[6], y[6])≥EVAL(x[6], +@z(y[6], 1@z))∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (6) using rules (III), (IV) which results in the following new constraint:
(7)    (>@z(x[2], y[2])=TRUE ⇒ COND_EVAL2(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL2(TRUE, x[2], y[2])≥EVAL(x[2], +@z(y[2], 1@z))∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧-1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0∧0 ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧-1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0∧0 ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    (-1 + x[2] + (-1)y[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧-1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0)

We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11)    (x[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧(-1)Bound + x[2] ≥ 0)

We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(12)    (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧(-1)Bound + x[2] ≥ 0)

(13)    (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧(-1)Bound + x[2] ≥ 0)

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

EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])
- (0 = 0∧0 = 0∧(U^Increasing(COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0)
COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z))
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧(-1)Bound + x[2] ≥ 0)
- (x[2] ≥ 0∧y[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], +@z(y[6], 1@z))), ≥)∧(-1)Bound + x[2] ≥ 0)

POL(COND_EVAL2(x₁, x₂, x₃)) = -1 + (-1)x₃ + x₂
POL(TRUE) = 2
POL(EVAL(x₁, x₂)) = -1 + (-1)x₂ + x₁
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x₁, x₂)) = -1

The following pairs are in P_>:

COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z))

The following pairs are in P_bound:

COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], +@z(y[6], 1@z))

The following pairs are in P_≥:

EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])

At least the following rules have been oriented under context sensitive arithmetic replacement:

+@z¹ ↔

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

                          ↳ IDP

                            ↳ IDPtoQDPProof

                            ↳ IDPNonInfProof

                              ↳ IDP

                                ↳ IDependencyGraphProof

                                  ↳ IDP

                                    ↳ IDPNonInfProof

                                      ↳ IDP

                                        ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

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

(2): EVAL(x[2], y[2]) → COND_EVAL2(>@z(x[2], y[2]), x[2], y[2])

The set Q consists of the following terms:

eval(x0, x1)
Cond_eval1(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)

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