Termination proof of /tmp/tmpRycqq6/c.05.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:

Cond_eval(TRUE, x, y) → eval(-@z(x, y), y)
Cond_eval2(TRUE, x, y) → eval(x, -@z(y, x))
Cond_eval3(TRUE, x, y) → eval(x, -@z(y, x))
Cond_eval1(TRUE, x, y) → eval(-@z(x, y), y)
eval(x, y) → Cond_eval1(&&(&&(>@z(x, y), >@z(x, 0@z)), >@z(y, 0@z)), x, y)
eval(x, y) → Cond_eval(&&(&&(&&(>@z(y, x), >@z(x, 0@z)), >@z(y, 0@z)), >@z(x, y)), x, y)
eval(x, y) → Cond_eval3(&&(&&(&&(>@z(y, x), >@z(x, 0@z)), >@z(y, 0@z)), >=@z(y, x)), x, y)
eval(x, y) → Cond_eval2(&&(&&(&&(>@z(x, y), >@z(x, 0@z)), >@z(y, 0@z)), >=@z(y, x)), x, y)

The set Q consists of the following terms:

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

Cond_eval(TRUE, x, y) → eval(-@z(x, y), y)
Cond_eval2(TRUE, x, y) → eval(x, -@z(y, x))
Cond_eval3(TRUE, x, y) → eval(x, -@z(y, x))
Cond_eval1(TRUE, x, y) → eval(-@z(x, y), y)
eval(x, y) → Cond_eval1(&&(&&(>@z(x, y), >@z(x, 0@z)), >@z(y, 0@z)), x, y)
eval(x, y) → Cond_eval(&&(&&(&&(>@z(y, x), >@z(x, 0@z)), >@z(y, 0@z)), >@z(x, y)), x, y)
eval(x, y) → Cond_eval3(&&(&&(&&(>@z(y, x), >@z(x, 0@z)), >@z(y, 0@z)), >=@z(y, x)), x, y)
eval(x, y) → Cond_eval2(&&(&&(&&(>@z(x, y), >@z(x, 0@z)), >@z(y, 0@z)), >=@z(y, x)), x, y)

The integer pair graph contains the following rules and edges:

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

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

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

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

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

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

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

(2) -> (3), if ((y[2] →^* y[3])∧(-@z(x[2], y[2]) →^* x[3]))

(2) -> (4), if ((y[2] →^* y[4])∧(-@z(x[2], y[2]) →^* x[4]))

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

(3) -> (6), if ((x[3] →^* x[6])∧(y[3] →^* y[6])∧(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])) →^* TRUE))

(4) -> (7), if ((x[4] →^* x[7])∧(y[4] →^* y[7])∧(&&(&&(&&(>@z(x[4], y[4]), >@z(x[4], 0@z)), >@z(y[4], 0@z)), >=@z(y[4], x[4])) →^* TRUE))

(5) -> (2), if ((x[5] →^* x[2])∧(y[5] →^* y[2])∧(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])) →^* TRUE))

(6) -> (0), if ((-@z(y[6], x[6]) →^* y[0])∧(x[6] →^* x[0]))

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

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

(6) -> (5), if ((-@z(y[6], x[6]) →^* y[5])∧(x[6] →^* x[5]))

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

(7) -> (3), if ((-@z(y[7], x[7]) →^* y[3])∧(x[7] →^* x[3]))

(7) -> (4), if ((-@z(y[7], x[7]) →^* y[4])∧(x[7] →^* x[4]))

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

The set Q consists of the following terms:

Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval1(TRUE, x0, x1)
eval(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): EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0])
(1): COND_EVAL1(TRUE, x[1], y[1]) → EVAL(-@z(x[1], y[1]), y[1])
(2): COND_EVAL(TRUE, x[2], y[2]) → EVAL(-@z(x[2], y[2]), y[2])
(3): EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])
(4): EVAL(x[4], y[4]) → COND_EVAL2(&&(&&(&&(>@z(x[4], y[4]), >@z(x[4], 0@z)), >@z(y[4], 0@z)), >=@z(y[4], x[4])), x[4], y[4])
(5): EVAL(x[5], y[5]) → COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])
(6): COND_EVAL3(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], x[6]))
(7): COND_EVAL2(TRUE, x[7], y[7]) → EVAL(x[7], -@z(y[7], x[7]))

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

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

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

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

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

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

(2) -> (3), if ((y[2] →^* y[3])∧(-@z(x[2], y[2]) →^* x[3]))

(2) -> (4), if ((y[2] →^* y[4])∧(-@z(x[2], y[2]) →^* x[4]))

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

(3) -> (6), if ((x[3] →^* x[6])∧(y[3] →^* y[6])∧(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])) →^* TRUE))

(4) -> (7), if ((x[4] →^* x[7])∧(y[4] →^* y[7])∧(&&(&&(&&(>@z(x[4], y[4]), >@z(x[4], 0@z)), >@z(y[4], 0@z)), >=@z(y[4], x[4])) →^* TRUE))

(5) -> (2), if ((x[5] →^* x[2])∧(y[5] →^* y[2])∧(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])) →^* TRUE))

(6) -> (0), if ((-@z(y[6], x[6]) →^* y[0])∧(x[6] →^* x[0]))

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

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

(6) -> (5), if ((-@z(y[6], x[6]) →^* y[5])∧(x[6] →^* x[5]))

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

(7) -> (3), if ((-@z(y[7], x[7]) →^* y[3])∧(x[7] →^* x[3]))

(7) -> (4), if ((-@z(y[7], x[7]) →^* y[4])∧(x[7] →^* x[4]))

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

The set Q consists of the following terms:

Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval1(TRUE, x0, x1)
eval(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, y) → COND_EVAL1(&&(&&(>@z(x, y), >@z(x, 0@z)), >@z(y, 0@z)), x, y) the following chains were created:

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

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

For Pair COND_EVAL1(TRUE, x, y) → EVAL(-@z(x, y), y) the following chains were created:

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

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

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

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

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

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

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

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

We consider the chain EVAL(x[5], y[5]) → COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]), COND_EVAL(TRUE, x[2], y[2]) → EVAL(-@z(x[2], y[2]), y[2]) which results in the following constraint:
(13)    (x[5]=x[2]∧&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5]))=TRUE∧y[5]=y[2] ⇒ COND_EVAL(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL(TRUE, x[2], y[2])≥EVAL(-@z(x[2], y[2]), y[2])∧(U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥))

We simplified constraint (13) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(14)    (>@z(x[5], y[5])=TRUE∧>@z(y[5], 0@z)=TRUE∧>@z(y[5], x[5])=TRUE∧>@z(x[5], 0@z)=TRUE ⇒ COND_EVAL(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL(TRUE, x[5], y[5])≥EVAL(-@z(x[5], y[5]), y[5])∧(U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥))

We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(15)    (-1 + x[5] + (-1)y[5] ≥ 0∧-1 + y[5] ≥ 0∧-1 + y[5] + (-1)x[5] ≥ 0∧-1 + x[5] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(16)    (-1 + x[5] + (-1)y[5] ≥ 0∧-1 + y[5] ≥ 0∧-1 + y[5] + (-1)x[5] ≥ 0∧-1 + x[5] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[2], y[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)    (-1 + x[5] + (-1)y[5] ≥ 0∧-1 + x[5] ≥ 0∧-1 + y[5] ≥ 0∧-1 + y[5] + (-1)x[5] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We solved constraint (17) using rule (IDP_SMT_SPLIT).

For Pair EVAL(x, y) → COND_EVAL3(&&(&&(&&(>@z(y, x), >@z(x, 0@z)), >@z(y, 0@z)), >=@z(y, x)), x, y) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(18)    (EVAL(x[3], y[3])≥NonInfC∧EVAL(x[3], y[3])≥COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])∧(U^Increasing(COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])), ≥))

We simplified constraint (18) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(19)    ((U^Increasing(COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (19) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(20)    ((U^Increasing(COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (20) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(21)    ((U^Increasing(COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (21) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(22)    (0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 = 0∧0 = 0)

For Pair EVAL(x, y) → COND_EVAL2(&&(&&(&&(>@z(x, y), >@z(x, 0@z)), >@z(y, 0@z)), >=@z(y, x)), x, y) the following chains were created:

We consider the chain EVAL(x[4], y[4]) → COND_EVAL2(&&(&&(&&(>@z(x[4], y[4]), >@z(x[4], 0@z)), >@z(y[4], 0@z)), >=@z(y[4], x[4])), x[4], y[4]) which results in the following constraint:
(23)    (EVAL(x[4], y[4])≥NonInfC∧EVAL(x[4], y[4])≥COND_EVAL2(&&(&&(&&(>@z(x[4], y[4]), >@z(x[4], 0@z)), >@z(y[4], 0@z)), >=@z(y[4], x[4])), x[4], y[4])∧(U^Increasing(COND_EVAL2(&&(&&(&&(>@z(x[4], y[4]), >@z(x[4], 0@z)), >@z(y[4], 0@z)), >=@z(y[4], x[4])), x[4], y[4])), ≥))

We simplified constraint (23) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(24)    ((U^Increasing(COND_EVAL2(&&(&&(&&(>@z(x[4], y[4]), >@z(x[4], 0@z)), >@z(y[4], 0@z)), >=@z(y[4], x[4])), x[4], y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (24) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(25)    ((U^Increasing(COND_EVAL2(&&(&&(&&(>@z(x[4], y[4]), >@z(x[4], 0@z)), >@z(y[4], 0@z)), >=@z(y[4], x[4])), x[4], y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(26)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_EVAL2(&&(&&(&&(>@z(x[4], y[4]), >@z(x[4], 0@z)), >@z(y[4], 0@z)), >=@z(y[4], x[4])), x[4], y[4])), ≥))

We simplified constraint (26) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(27)    (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL2(&&(&&(&&(>@z(x[4], y[4]), >@z(x[4], 0@z)), >@z(y[4], 0@z)), >=@z(y[4], x[4])), x[4], y[4])), ≥)∧0 ≥ 0)

For Pair EVAL(x, y) → COND_EVAL(&&(&&(&&(>@z(y, x), >@z(x, 0@z)), >@z(y, 0@z)), >@z(x, y)), x, y) the following chains were created:

We consider the chain EVAL(x[5], y[5]) → COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]) which results in the following constraint:
(28)    (EVAL(x[5], y[5])≥NonInfC∧EVAL(x[5], y[5])≥COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])∧(U^Increasing(COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥))

We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(29)    ((U^Increasing(COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(30)    ((U^Increasing(COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (30) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(31)    ((U^Increasing(COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (31) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(32)    (0 = 0∧0 = 0∧0 ≥ 0∧(U^Increasing(COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)

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

We consider the chain EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL3(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], x[6])) which results in the following constraint:
(33)    (y[3]=y[6]∧x[3]=x[6]∧&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3]))=TRUE ⇒ COND_EVAL3(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL3(TRUE, x[6], y[6])≥EVAL(x[6], -@z(y[6], x[6]))∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (33) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(34)    (>=@z(y[3], x[3])=TRUE∧>@z(y[3], 0@z)=TRUE∧>@z(y[3], x[3])=TRUE∧>@z(x[3], 0@z)=TRUE ⇒ COND_EVAL3(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL3(TRUE, x[3], y[3])≥EVAL(x[3], -@z(y[3], x[3]))∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (34) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(35)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (35) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(36)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (36) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(37)    (-1 + y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(38)    (-2 + y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧x[3] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (38) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(39)    (y[3] ≥ 0∧1 + x[3] + y[3] ≥ 0∧1 + y[3] ≥ 0∧x[3] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

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

We consider the chain EVAL(x[4], y[4]) → COND_EVAL2(&&(&&(&&(>@z(x[4], y[4]), >@z(x[4], 0@z)), >@z(y[4], 0@z)), >=@z(y[4], x[4])), x[4], y[4]), COND_EVAL2(TRUE, x[7], y[7]) → EVAL(x[7], -@z(y[7], x[7])) which results in the following constraint:
(40)    (&&(&&(&&(>@z(x[4], y[4]), >@z(x[4], 0@z)), >@z(y[4], 0@z)), >=@z(y[4], x[4]))=TRUE∧x[4]=x[7]∧y[4]=y[7] ⇒ COND_EVAL2(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL2(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], x[7]))∧(U^Increasing(EVAL(x[7], -@z(y[7], x[7]))), ≥))

We simplified constraint (40) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(41)    (>=@z(y[4], x[4])=TRUE∧>@z(y[4], 0@z)=TRUE∧>@z(x[4], y[4])=TRUE∧>@z(x[4], 0@z)=TRUE ⇒ COND_EVAL2(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL2(TRUE, x[4], y[4])≥EVAL(x[4], -@z(y[4], x[4]))∧(U^Increasing(EVAL(x[7], -@z(y[7], x[7]))), ≥))

We simplified constraint (41) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(42)    (y[4] + (-1)x[4] ≥ 0∧-1 + y[4] ≥ 0∧-1 + x[4] + (-1)y[4] ≥ 0∧-1 + x[4] ≥ 0 ⇒ (U^Increasing(EVAL(x[7], -@z(y[7], x[7]))), ≥)∧-1 + (-1)Bound ≥ 0∧-1 ≥ 0)

We simplified constraint (42) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(43)    (y[4] + (-1)x[4] ≥ 0∧-1 + y[4] ≥ 0∧-1 + x[4] + (-1)y[4] ≥ 0∧-1 + x[4] ≥ 0 ⇒ (U^Increasing(EVAL(x[7], -@z(y[7], x[7]))), ≥)∧-1 + (-1)Bound ≥ 0∧-1 ≥ 0)

We simplified constraint (43) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(44)    (-1 + x[4] + (-1)y[4] ≥ 0∧-1 + y[4] ≥ 0∧y[4] + (-1)x[4] ≥ 0∧-1 + x[4] ≥ 0 ⇒ -1 + (-1)Bound ≥ 0∧-1 ≥ 0∧(U^Increasing(EVAL(x[7], -@z(y[7], x[7]))), ≥))

We solved constraint (44) using rule (IDP_SMT_SPLIT).

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

EVAL(x, y) → COND_EVAL1(&&(&&(>@z(x, y), >@z(x, 0@z)), >@z(y, 0@z)), x, y)
- (0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0])), ≥)∧0 = 0∧0 = 0)
COND_EVAL1(TRUE, x, y) → EVAL(-@z(x, y), y)
- (y[0] ≥ 0∧1 + y[0] + x[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], y[1]), y[1])), ≥)∧0 ≥ 0)
COND_EVAL(TRUE, x, y) → EVAL(-@z(x, y), y)
EVAL(x, y) → COND_EVAL3(&&(&&(&&(>@z(y, x), >@z(x, 0@z)), >@z(y, 0@z)), >=@z(y, x)), x, y)
- (0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 = 0∧0 = 0)
EVAL(x, y) → COND_EVAL2(&&(&&(&&(>@z(x, y), >@z(x, 0@z)), >@z(y, 0@z)), >=@z(y, x)), x, y)
- (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL2(&&(&&(&&(>@z(x[4], y[4]), >@z(x[4], 0@z)), >@z(y[4], 0@z)), >=@z(y[4], x[4])), x[4], y[4])), ≥)∧0 ≥ 0)
EVAL(x, y) → COND_EVAL(&&(&&(&&(>@z(y, x), >@z(x, 0@z)), >@z(y, 0@z)), >@z(x, y)), x, y)
- (0 = 0∧0 = 0∧0 ≥ 0∧(U^Increasing(COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)
COND_EVAL3(TRUE, x, y) → EVAL(x, -@z(y, x))
- (y[3] ≥ 0∧1 + x[3] + y[3] ≥ 0∧1 + y[3] ≥ 0∧x[3] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))
COND_EVAL2(TRUE, x, y) → EVAL(x, -@z(y, x))

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(-@z(x₁, x₂)) = x₁ + (-1)x₂
POL(0@z) = 0
POL(COND_EVAL1(x₁, x₂, x₃)) = -1 + (-1)x₁
POL(TRUE) = 0
POL(&&(x₁, x₂)) = 0
POL(COND_EVAL(x₁, x₂, x₃)) = -1 + (-1)x₁
POL(FALSE) = 0
POL(>@z(x₁, x₂)) = -1
POL(COND_EVAL3(x₁, x₂, x₃)) = -1 + (-1)x₁
POL(>=@z(x₁, x₂)) = -1
POL(COND_EVAL2(x₁, x₂, x₃)) = -1 + (2)x₁
POL(EVAL(x₁, x₂)) = -1
POL(undefined) = -1

The following pairs are in P_>:

COND_EVAL2(TRUE, x[7], y[7]) → EVAL(x[7], -@z(y[7], x[7]))

The following pairs are in P_bound:

COND_EVAL2(TRUE, x[7], y[7]) → EVAL(x[7], -@z(y[7], x[7]))

The following pairs are in P_≥:

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

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

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

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

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

(6) -> (5), if ((-@z(y[6], x[6]) →^* y[5])∧(x[6] →^* x[5]))

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

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

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

(6) -> (0), if ((-@z(y[6], x[6]) →^* y[0])∧(x[6] →^* x[0]))

(3) -> (6), if ((x[3] →^* x[6])∧(y[3] →^* y[6])∧(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])) →^* TRUE))

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

(2) -> (4), if ((y[2] →^* y[4])∧(-@z(x[2], y[2]) →^* x[4]))

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

(5) -> (2), if ((x[5] →^* x[2])∧(y[5] →^* y[2])∧(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])) →^* TRUE))

(2) -> (3), if ((y[2] →^* y[3])∧(-@z(x[2], y[2]) →^* x[3]))

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ 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:

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

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

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

(6) -> (5), if ((-@z(y[6], x[6]) →^* y[5])∧(x[6] →^* x[5]))

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

(6) -> (0), if ((-@z(y[6], x[6]) →^* y[0])∧(x[6] →^* x[0]))

(3) -> (6), if ((x[3] →^* x[6])∧(y[3] →^* y[6])∧(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])) →^* TRUE))

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

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

(5) -> (2), if ((x[5] →^* x[2])∧(y[5] →^* y[2])∧(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])) →^* TRUE))

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

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

(2) -> (3), if ((y[2] →^* y[3])∧(-@z(x[2], y[2]) →^* x[3]))

The set Q consists of the following terms:

Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval1(TRUE, x0, x1)
eval(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_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@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_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])∧(U^Increasing(COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@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_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@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_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@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)    ((U^Increasing(COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧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_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)

For Pair COND_EVAL(TRUE, x[2], y[2]) → EVAL(-@z(x[2], y[2]), y[2]) the following chains were created:

We consider the chain EVAL(x[5], y[5]) → COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]), COND_EVAL(TRUE, x[2], y[2]) → EVAL(-@z(x[2], y[2]), y[2]), EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(6)    (x[5]=x[2]∧&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5]))=TRUE∧y[2]=y[3]∧y[5]=y[2]∧-@z(x[2], y[2])=x[3] ⇒ COND_EVAL(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL(TRUE, x[2], y[2])≥EVAL(-@z(x[2], y[2]), y[2])∧(U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥))

We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7)    (>@z(x[5], y[5])=TRUE∧>@z(y[5], 0@z)=TRUE∧>@z(y[5], x[5])=TRUE∧>@z(x[5], 0@z)=TRUE ⇒ COND_EVAL(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL(TRUE, x[5], y[5])≥EVAL(-@z(x[5], y[5]), y[5])∧(U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    (-1 + x[5] + (-1)y[5] ≥ 0∧-1 + y[5] ≥ 0∧-1 + y[5] + (-1)x[5] ≥ 0∧-1 + x[5] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥)∧-2 + (-1)Bound ≥ 0∧-2 ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    (-1 + x[5] + (-1)y[5] ≥ 0∧-1 + y[5] ≥ 0∧-1 + y[5] + (-1)x[5] ≥ 0∧-1 + x[5] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥)∧-2 + (-1)Bound ≥ 0∧-2 ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    (-1 + y[5] + (-1)x[5] ≥ 0∧-1 + x[5] + (-1)y[5] ≥ 0∧-1 + x[5] ≥ 0∧-1 + y[5] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥)∧-2 ≥ 0∧-2 + (-1)Bound ≥ 0)

We solved constraint (10) using rule (IDP_SMT_SPLIT).
We consider the chain EVAL(x[5], y[5]) → COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]), COND_EVAL(TRUE, x[2], y[2]) → EVAL(-@z(x[2], y[2]), y[2]), EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0]) which results in the following constraint:
(11)    (x[5]=x[2]∧&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5]))=TRUE∧-@z(x[2], y[2])=x[0]∧y[5]=y[2]∧y[2]=y[0] ⇒ COND_EVAL(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL(TRUE, x[2], y[2])≥EVAL(-@z(x[2], y[2]), y[2])∧(U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥))

We simplified constraint (11) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(12)    (>@z(x[5], y[5])=TRUE∧>@z(y[5], 0@z)=TRUE∧>@z(y[5], x[5])=TRUE∧>@z(x[5], 0@z)=TRUE ⇒ COND_EVAL(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL(TRUE, x[5], y[5])≥EVAL(-@z(x[5], y[5]), y[5])∧(U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥))

We simplified constraint (12) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(13)    (-1 + x[5] + (-1)y[5] ≥ 0∧-1 + y[5] ≥ 0∧-1 + y[5] + (-1)x[5] ≥ 0∧-1 + x[5] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥)∧-2 + (-1)Bound ≥ 0∧-2 ≥ 0)

We simplified constraint (13) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(14)    (-1 + x[5] + (-1)y[5] ≥ 0∧-1 + y[5] ≥ 0∧-1 + y[5] + (-1)x[5] ≥ 0∧-1 + x[5] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥)∧-2 + (-1)Bound ≥ 0∧-2 ≥ 0)

We simplified constraint (14) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(15)    (-1 + x[5] ≥ 0∧-1 + y[5] + (-1)x[5] ≥ 0∧-1 + x[5] + (-1)y[5] ≥ 0∧-1 + y[5] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥)∧-2 + (-1)Bound ≥ 0∧-2 ≥ 0)

We solved constraint (15) using rule (IDP_SMT_SPLIT).
We consider the chain EVAL(x[5], y[5]) → COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]), COND_EVAL(TRUE, x[2], y[2]) → EVAL(-@z(x[2], y[2]), y[2]), EVAL(x[5], y[5]) → COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]) which results in the following constraint:
(16)    (y[2]=y[5]1∧x[5]=x[2]∧&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5]))=TRUE∧y[5]=y[2]∧-@z(x[2], y[2])=x[5]1 ⇒ COND_EVAL(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL(TRUE, x[2], y[2])≥EVAL(-@z(x[2], y[2]), y[2])∧(U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥))

We simplified constraint (16) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(17)    (>@z(x[5], y[5])=TRUE∧>@z(y[5], 0@z)=TRUE∧>@z(y[5], x[5])=TRUE∧>@z(x[5], 0@z)=TRUE ⇒ COND_EVAL(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL(TRUE, x[5], y[5])≥EVAL(-@z(x[5], y[5]), y[5])∧(U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥))

We simplified constraint (17) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(18)    (-1 + x[5] + (-1)y[5] ≥ 0∧-1 + y[5] ≥ 0∧-1 + y[5] + (-1)x[5] ≥ 0∧-1 + x[5] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥)∧-2 + (-1)Bound ≥ 0∧-2 ≥ 0)

We simplified constraint (18) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(19)    (-1 + x[5] + (-1)y[5] ≥ 0∧-1 + y[5] ≥ 0∧-1 + y[5] + (-1)x[5] ≥ 0∧-1 + x[5] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥)∧-2 + (-1)Bound ≥ 0∧-2 ≥ 0)

We simplified constraint (19) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(20)    (-1 + x[5] ≥ 0∧-1 + y[5] + (-1)x[5] ≥ 0∧-1 + x[5] + (-1)y[5] ≥ 0∧-1 + y[5] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[2], y[2]), y[2])), ≥)∧-2 + (-1)Bound ≥ 0∧-2 ≥ 0)

We solved constraint (20) using rule (IDP_SMT_SPLIT).

For Pair EVAL(x[5], y[5]) → COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]) the following chains were created:

We consider the chain EVAL(x[5], y[5]) → COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]) which results in the following constraint:
(21)    (EVAL(x[5], y[5])≥NonInfC∧EVAL(x[5], y[5])≥COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])∧(U^Increasing(COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥))

We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(22)    ((U^Increasing(COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(23)    ((U^Increasing(COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(24)    (1 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥))

We simplified constraint (24) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(25)    (0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥)∧1 ≥ 0∧0 = 0∧0 = 0)

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

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

We simplified constraint (26) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(27)    (>@z(y[0], 0@z)=TRUE∧>@z(x[0], y[0])=TRUE∧>@z(x[0], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(-@z(x[0], y[0]), y[0])∧(U^Increasing(EVAL(-@z(x[1], y[1]), y[1])), ≥))

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

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

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

We simplified constraint (30) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(31)    (x[0] + -1 ≥ 0∧x[0] + -2 + (-1)y[0] ≥ 0∧y[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], y[1]), y[1])), ≥)∧0 ≥ 0)

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

We simplified constraint (32) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(33)    (1 + y[0] + x[0] ≥ 0∧x[0] ≥ 0∧y[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], y[1]), y[1])), ≥)∧0 ≥ 0)
We consider the chain EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0]), COND_EVAL1(TRUE, x[1], y[1]) → EVAL(-@z(x[1], y[1]), y[1]), EVAL(x[5], y[5]) → COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]) which results in the following constraint:
(34)    (&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z))=TRUE∧y[0]=y[1]∧y[1]=y[5]∧-@z(x[1], y[1])=x[5]∧x[0]=x[1] ⇒ COND_EVAL1(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL1(TRUE, x[1], y[1])≥EVAL(-@z(x[1], y[1]), y[1])∧(U^Increasing(EVAL(-@z(x[1], y[1]), y[1])), ≥))

We simplified constraint (34) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(35)    (>@z(y[0], 0@z)=TRUE∧>@z(x[0], y[0])=TRUE∧>@z(x[0], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(-@z(x[0], y[0]), y[0])∧(U^Increasing(EVAL(-@z(x[1], y[1]), y[1])), ≥))

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

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

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

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

We simplified constraint (39) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(40)    (1 + y[0] + x[0] ≥ 0∧y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], y[1]), y[1])), ≥)∧0 ≥ 0)
We consider the chain EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0]), COND_EVAL1(TRUE, x[1], y[1]) → EVAL(-@z(x[1], y[1]), y[1]), EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(41)    (y[1]=y[3]∧-@z(x[1], y[1])=x[3]∧&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z))=TRUE∧y[0]=y[1]∧x[0]=x[1] ⇒ COND_EVAL1(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL1(TRUE, x[1], y[1])≥EVAL(-@z(x[1], y[1]), y[1])∧(U^Increasing(EVAL(-@z(x[1], y[1]), y[1])), ≥))

We simplified constraint (41) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(42)    (>@z(y[0], 0@z)=TRUE∧>@z(x[0], y[0])=TRUE∧>@z(x[0], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(-@z(x[0], y[0]), y[0])∧(U^Increasing(EVAL(-@z(x[1], y[1]), y[1])), ≥))

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

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

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

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

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

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

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

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

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

We simplified constraint (50) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(51)    ((U^Increasing(COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (51) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(52)    (0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0])), ≥)∧0 = 0∧0 = 0)

For Pair COND_EVAL3(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], x[6])) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL3(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], x[6])), EVAL(x[5], y[5]) → COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5]) which results in the following constraint:
(53)    (y[3]=y[6]∧x[3]=x[6]∧&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3]))=TRUE∧-@z(y[6], x[6])=y[5]∧x[6]=x[5] ⇒ COND_EVAL3(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL3(TRUE, x[6], y[6])≥EVAL(x[6], -@z(y[6], x[6]))∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (53) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(54)    (>=@z(y[3], x[3])=TRUE∧>@z(y[3], 0@z)=TRUE∧>@z(y[3], x[3])=TRUE∧>@z(x[3], 0@z)=TRUE ⇒ COND_EVAL3(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL3(TRUE, x[3], y[3])≥EVAL(x[3], -@z(y[3], x[3]))∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (54) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(55)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (55) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(56)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (56) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(57)    (-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0∧-1 + y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧0 ≥ 0)

We simplified constraint (57) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(58)    (-2 + y[3] + (-1)x[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧x[3] ≥ 0∧-1 + y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧0 ≥ 0)

We simplified constraint (58) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(59)    (y[3] ≥ 0∧1 + y[3] ≥ 0∧x[3] ≥ 0∧1 + x[3] + y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧0 ≥ 0)
We consider the chain EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL3(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], x[6])), EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(60)    (y[3]=y[6]∧-@z(y[6], x[6])=y[3]1∧x[6]=x[3]1∧x[3]=x[6]∧&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3]))=TRUE ⇒ COND_EVAL3(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL3(TRUE, x[6], y[6])≥EVAL(x[6], -@z(y[6], x[6]))∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (60) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(61)    (>=@z(y[3], x[3])=TRUE∧>@z(y[3], 0@z)=TRUE∧>@z(y[3], x[3])=TRUE∧>@z(x[3], 0@z)=TRUE ⇒ COND_EVAL3(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL3(TRUE, x[3], y[3])≥EVAL(x[3], -@z(y[3], x[3]))∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (61) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(62)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (62) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(63)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (63) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(64)    (-1 + y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (64) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(65)    (-2 + y[3] + (-1)x[3] ≥ 0∧x[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (65) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(66)    (y[3] ≥ 0∧x[3] ≥ 0∧1 + y[3] ≥ 0∧1 + x[3] + y[3] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))
We consider the chain EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL3(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], x[6])), EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0]) which results in the following constraint:
(67)    (y[3]=y[6]∧x[3]=x[6]∧&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3]))=TRUE∧-@z(y[6], x[6])=y[0]∧x[6]=x[0] ⇒ COND_EVAL3(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL3(TRUE, x[6], y[6])≥EVAL(x[6], -@z(y[6], x[6]))∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (67) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(68)    (>=@z(y[3], x[3])=TRUE∧>@z(y[3], 0@z)=TRUE∧>@z(y[3], x[3])=TRUE∧>@z(x[3], 0@z)=TRUE ⇒ COND_EVAL3(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL3(TRUE, x[3], y[3])≥EVAL(x[3], -@z(y[3], x[3]))∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (68) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(69)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (69) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(70)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (70) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(71)    (-1 + x[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧0 ≥ 0)

We simplified constraint (71) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(72)    (x[3] ≥ 0∧-2 + y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧0 ≥ 0)

We simplified constraint (72) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(73)    (x[3] ≥ 0∧y[3] ≥ 0∧1 + x[3] + y[3] ≥ 0∧1 + y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧0 ≥ 0)

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

EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])
- (0 ≥ 0∧0 = 0∧(U^Increasing(COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
COND_EVAL(TRUE, x[2], y[2]) → EVAL(-@z(x[2], y[2]), y[2])
EVAL(x[5], y[5]) → COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])
- (0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL(&&(&&(&&(>@z(y[5], x[5]), >@z(x[5], 0@z)), >@z(y[5], 0@z)), >@z(x[5], y[5])), x[5], y[5])), ≥)∧1 ≥ 0∧0 = 0∧0 = 0)
COND_EVAL1(TRUE, x[1], y[1]) → EVAL(-@z(x[1], y[1]), y[1])
- (1 + y[0] + x[0] ≥ 0∧x[0] ≥ 0∧y[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], y[1]), y[1])), ≥)∧0 ≥ 0)
- (1 + y[0] + x[0] ≥ 0∧y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], y[1]), y[1])), ≥)∧0 ≥ 0)
- (y[0] ≥ 0∧1 + y[0] + x[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], y[1]), y[1])), ≥)∧0 ≥ 0)
EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0])
- (0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0])), ≥)∧0 = 0∧0 = 0)
COND_EVAL3(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], x[6]))
- (y[3] ≥ 0∧1 + y[3] ≥ 0∧x[3] ≥ 0∧1 + x[3] + y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧0 ≥ 0)
- (y[3] ≥ 0∧x[3] ≥ 0∧1 + y[3] ≥ 0∧1 + x[3] + y[3] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))
- (x[3] ≥ 0∧y[3] ≥ 0∧1 + x[3] + y[3] ≥ 0∧1 + y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧0 ≥ 0)

POL(-@z(x₁, x₂)) = x₁ + (-1)x₂
POL(>=@z(x₁, x₂)) = -1
POL(COND_EVAL3(x₁, x₂, x₃)) = 1 + (2)x₁
POL(0@z) = 0
POL(COND_EVAL1(x₁, x₂, x₃)) = 1 + (2)x₁
POL(TRUE) = -1
POL(&&(x₁, x₂)) = -1
POL(COND_EVAL(x₁, x₂, x₃)) = -1 + x₁
POL(EVAL(x₁, x₂)) = -1
POL(FALSE) = -1
POL(undefined) = -1
POL(>@z(x₁, x₂)) = -1

The following pairs are in P_>:

COND_EVAL(TRUE, x[2], y[2]) → EVAL(-@z(x[2], y[2]), y[2])

The following pairs are in P_bound:

COND_EVAL(TRUE, x[2], y[2]) → EVAL(-@z(x[2], y[2]), y[2])

The following pairs are in P_≥:

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

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ 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:

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

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

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

(6) -> (5), if ((-@z(y[6], x[6]) →^* y[5])∧(x[6] →^* x[5]))

(3) -> (6), if ((x[3] →^* x[6])∧(y[3] →^* y[6])∧(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])) →^* TRUE))

(6) -> (0), if ((-@z(y[6], x[6]) →^* y[0])∧(x[6] →^* x[0]))

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

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

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPNonInfProof

                    ↳ IDP

                      ↳ IDependencyGraphProof

                        ↳ IDP

                          ↳ IDPNonInfProof

I DP problem:
The following domains are used:

z

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

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

(6) -> (0), if ((-@z(y[6], x[6]) →^* y[0])∧(x[6] →^* x[0]))

(3) -> (6), if ((x[3] →^* x[6])∧(y[3] →^* y[6])∧(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])) →^* TRUE))

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

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

The set Q consists of the following terms:

Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval1(TRUE, x0, x1)
eval(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_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@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_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])∧(U^Increasing(COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@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_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@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_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@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_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@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∧(U^Increasing(COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)

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

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

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

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

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

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

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

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

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (y[0] ≥ 0∧x[0] ≥ 0∧1 + y[0] + x[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], y[1]), y[1])), ≥)∧y[0] ≥ 0)
We consider the chain EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0]), COND_EVAL1(TRUE, x[1], y[1]) → EVAL(-@z(x[1], y[1]), y[1]), EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(14)    (y[1]=y[3]∧-@z(x[1], y[1])=x[3]∧&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z))=TRUE∧y[0]=y[1]∧x[0]=x[1] ⇒ COND_EVAL1(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL1(TRUE, x[1], y[1])≥EVAL(-@z(x[1], y[1]), y[1])∧(U^Increasing(EVAL(-@z(x[1], y[1]), y[1])), ≥))

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

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

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

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

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

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

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

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

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

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

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

We simplified constraint (24) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(25)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0])), ≥))

We simplified constraint (25) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(26)    (0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0])), ≥)∧0 = 0)

For Pair COND_EVAL3(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], x[6])) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL3(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], x[6])), EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(27)    (y[3]=y[6]∧-@z(y[6], x[6])=y[3]1∧x[6]=x[3]1∧x[3]=x[6]∧&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3]))=TRUE ⇒ COND_EVAL3(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL3(TRUE, x[6], y[6])≥EVAL(x[6], -@z(y[6], x[6]))∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (27) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(28)    (>=@z(y[3], x[3])=TRUE∧>@z(y[3], 0@z)=TRUE∧>@z(y[3], x[3])=TRUE∧>@z(x[3], 0@z)=TRUE ⇒ COND_EVAL3(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL3(TRUE, x[3], y[3])≥EVAL(x[3], -@z(y[3], x[3]))∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(29)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧-1 + (-1)Bound + y[3] + x[3] ≥ 0∧x[3] ≥ 0)

We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(30)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧-1 + (-1)Bound + y[3] + x[3] ≥ 0∧x[3] ≥ 0)

We simplified constraint (30) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(31)    (-1 + y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ x[3] ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧-1 + (-1)Bound + y[3] + x[3] ≥ 0)

We simplified constraint (31) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(32)    (-2 + y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧x[3] ≥ 0 ⇒ 1 + x[3] ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧(-1)Bound + y[3] + x[3] ≥ 0)

We simplified constraint (32) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(33)    (-1 + y[3] ≥ 0∧x[3] + y[3] ≥ 0∧y[3] ≥ 0∧x[3] ≥ 0 ⇒ 1 + x[3] ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧1 + (-1)Bound + (2)x[3] + y[3] ≥ 0)

We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(34)    (y[3] ≥ 0∧1 + x[3] + y[3] ≥ 0∧1 + y[3] ≥ 0∧x[3] ≥ 0 ⇒ 1 + x[3] ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧2 + (-1)Bound + (2)x[3] + y[3] ≥ 0)
We consider the chain EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL3(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], x[6])), EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0]) which results in the following constraint:
(35)    (y[3]=y[6]∧x[3]=x[6]∧&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3]))=TRUE∧-@z(y[6], x[6])=y[0]∧x[6]=x[0] ⇒ COND_EVAL3(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL3(TRUE, x[6], y[6])≥EVAL(x[6], -@z(y[6], x[6]))∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (35) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(36)    (>=@z(y[3], x[3])=TRUE∧>@z(y[3], 0@z)=TRUE∧>@z(y[3], x[3])=TRUE∧>@z(x[3], 0@z)=TRUE ⇒ COND_EVAL3(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL3(TRUE, x[3], y[3])≥EVAL(x[3], -@z(y[3], x[3]))∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (36) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(37)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧-1 + (-1)Bound + y[3] + x[3] ≥ 0∧x[3] ≥ 0)

We simplified constraint (37) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(38)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧-1 + (-1)Bound + y[3] + x[3] ≥ 0∧x[3] ≥ 0)

We simplified constraint (38) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(39)    (-1 + y[3] ≥ 0∧-1 + x[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧y[3] + (-1)x[3] ≥ 0 ⇒ -1 + (-1)Bound + y[3] + x[3] ≥ 0∧x[3] ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (39) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(40)    (-1 + y[3] ≥ 0∧x[3] ≥ 0∧-2 + y[3] + (-1)x[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0 ⇒ (-1)Bound + y[3] + x[3] ≥ 0∧1 + x[3] ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (40) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(41)    (x[3] + y[3] ≥ 0∧x[3] ≥ 0∧-1 + y[3] ≥ 0∧y[3] ≥ 0 ⇒ 1 + (-1)Bound + (2)x[3] + y[3] ≥ 0∧1 + x[3] ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (41) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(42)    (1 + x[3] + y[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0∧1 + y[3] ≥ 0 ⇒ 2 + (-1)Bound + (2)x[3] + y[3] ≥ 0∧1 + x[3] ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

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

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

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

The following pairs are in P_>:

EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0])

The following pairs are in P_bound:

COND_EVAL3(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], x[6]))

The following pairs are in P_≥:

EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])
COND_EVAL1(TRUE, x[1], y[1]) → EVAL(-@z(x[1], y[1]), y[1])
COND_EVAL3(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], x[6]))

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPNonInfProof

                    ↳ IDP

                      ↳ IDependencyGraphProof

                        ↳ IDP

                          ↳ IDPNonInfProof

                            ↳ AND

                              ↳ IDP

                                ↳ IDependencyGraphProof

                              ↳ IDP

I DP problem:
The following domains are used:

z

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

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

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPNonInfProof

                    ↳ IDP

                      ↳ IDependencyGraphProof

                        ↳ 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:

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

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

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

The set Q consists of the following terms:

Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval1(TRUE, x0, x1)
eval(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[1], y[1]) → EVAL(-@z(x[1], y[1]), y[1]) the following chains were created:

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

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

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

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

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

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

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

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

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

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

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    ((U^Increasing(COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    ((U^Increasing(COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    ((U^Increasing(COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13)    (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0])), ≥)∧0 = 0)

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

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

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

The following pairs are in P_>:

EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], y[0]), >@z(x[0], 0@z)), >@z(y[0], 0@z)), x[0], y[0])

The following pairs are in P_bound:

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

The following pairs are in P_≥:

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

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPNonInfProof

                    ↳ IDP

                      ↳ IDependencyGraphProof

                        ↳ IDP

                          ↳ IDPNonInfProof

                            ↳ AND

                              ↳ IDP

                                ↳ IDependencyGraphProof

                                  ↳ IDP

                                    ↳ IDPNonInfProof

                                      ↳ AND

                                        ↳ IDP

                                          ↳ IDependencyGraphProof

                                        ↳ IDP

                              ↳ IDP

I DP problem:
The following domains are used:

z

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPNonInfProof

                    ↳ IDP

                      ↳ IDependencyGraphProof

                        ↳ 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:

(1): COND_EVAL1(TRUE, x[1], y[1]) → EVAL(-@z(x[1], y[1]), y[1])

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPNonInfProof

                    ↳ IDP

                      ↳ IDependencyGraphProof

                        ↳ IDP

                          ↳ IDPNonInfProof

                            ↳ AND

                              ↳ IDP

                              ↳ IDP

                                ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

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

(3) -> (6), if ((x[3] →^* x[6])∧(y[3] →^* y[6])∧(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])) →^* TRUE))

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPNonInfProof

                    ↳ IDP

                      ↳ IDependencyGraphProof

                        ↳ IDP

                          ↳ IDPNonInfProof

                            ↳ AND

                              ↳ IDP

                              ↳ IDP

                                ↳ IDependencyGraphProof

                                  ↳ IDP

                                    ↳ IDPNonInfProof

I DP problem:
The following domains are used:

z

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

(3) -> (6), if ((x[3] →^* x[6])∧(y[3] →^* y[6])∧(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])) →^* TRUE))

The set Q consists of the following terms:

Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
Cond_eval1(TRUE, x0, x1)
eval(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_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@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_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])∧(U^Increasing(COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@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_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@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_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@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_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@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∧(U^Increasing(COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0)

For Pair COND_EVAL3(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], x[6])) the following chains were created:

We consider the chain EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]), COND_EVAL3(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], x[6])), EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3]) which results in the following constraint:
(6)    (y[3]=y[6]∧-@z(y[6], x[6])=y[3]1∧x[6]=x[3]1∧x[3]=x[6]∧&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3]))=TRUE ⇒ COND_EVAL3(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL3(TRUE, x[6], y[6])≥EVAL(x[6], -@z(y[6], x[6]))∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

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], 0@z)=TRUE∧>@z(y[3], x[3])=TRUE∧>@z(x[3], 0@z)=TRUE ⇒ COND_EVAL3(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL3(TRUE, x[3], y[3])≥EVAL(x[3], -@z(y[3], x[3]))∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧(-1)Bound + (2)y[3] + (-1)x[3] ≥ 0∧-2 + (2)x[3] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    (y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧(-1)Bound + (2)y[3] + (-1)x[3] ≥ 0∧-2 + (2)x[3] ≥ 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∧-1 + y[3] ≥ 0∧-1 + x[3] ≥ 0 ⇒ -2 + (2)x[3] ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧(-1)Bound + (2)y[3] + (-1)x[3] ≥ 0)

We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11)    (-2 + y[3] + (-1)x[3] ≥ 0∧-1 + y[3] + (-1)x[3] ≥ 0∧-1 + y[3] ≥ 0∧x[3] ≥ 0 ⇒ (2)x[3] ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧-1 + (-1)Bound + (2)y[3] + (-1)x[3] ≥ 0)

We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12)    (y[3] ≥ 0∧1 + y[3] ≥ 0∧1 + x[3] + y[3] ≥ 0∧x[3] ≥ 0 ⇒ (2)x[3] ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧3 + (-1)Bound + x[3] + (2)y[3] ≥ 0)

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

EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])
- (0 ≥ 0∧(U^Increasing(COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0)
COND_EVAL3(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], x[6]))
- (y[3] ≥ 0∧1 + y[3] ≥ 0∧1 + x[3] + y[3] ≥ 0∧x[3] ≥ 0 ⇒ (2)x[3] ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], x[6]))), ≥)∧3 + (-1)Bound + x[3] + (2)y[3] ≥ 0)

POL(-@z(x₁, x₂)) = x₁ + (-1)x₂
POL(>=@z(x₁, x₂)) = -1
POL(COND_EVAL3(x₁, x₂, x₃)) = (2)x₃ + (-1)x₂
POL(0@z) = 0
POL(TRUE) = -1
POL(&&(x₁, x₂)) = 2
POL(EVAL(x₁, x₂)) = 1 + (2)x₂ + (-1)x₁
POL(FALSE) = 2
POL(undefined) = -1
POL(>@z(x₁, x₂)) = -1

The following pairs are in P_>:

EVAL(x[3], y[3]) → COND_EVAL3(&&(&&(&&(>@z(y[3], x[3]), >@z(x[3], 0@z)), >@z(y[3], 0@z)), >=@z(y[3], x[3])), x[3], y[3])
COND_EVAL3(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], x[6]))

The following pairs are in P_bound:

COND_EVAL3(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], x[6]))

The following pairs are in P_≥:
none

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPNonInfProof

                    ↳ IDP

                      ↳ IDependencyGraphProof

                        ↳ IDP

                          ↳ IDPNonInfProof

                            ↳ AND

                              ↳ IDP

                              ↳ IDP

                                ↳ IDependencyGraphProof

                                  ↳ IDP

                                    ↳ IDPNonInfProof

                                      ↳ AND

                                        ↳ IDP

                                          ↳ IDependencyGraphProof

                                        ↳ IDP

I DP problem:
The following domains are used:none

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

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ IDP

              ↳ IDependencyGraphProof

                ↳ IDP

                  ↳ IDPNonInfProof

                    ↳ IDP

                      ↳ IDependencyGraphProof

                        ↳ IDP

                          ↳ IDPNonInfProof

                            ↳ AND

                              ↳ IDP

                              ↳ IDP

                                ↳ IDependencyGraphProof

                                  ↳ IDP

                                    ↳ IDPNonInfProof

                                      ↳ AND

                                        ↳ IDP

                                        ↳ IDP

                                          ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

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

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