Termination proof of /tmp/tmpBPYab8/17.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_eval2(TRUE, x, y) → eval(-@z(x, 1@z), y)
Cond_eval(TRUE, x, y) → eval(-@z(x, 1@z), y)
Cond_eval1(TRUE, x, y) → eval(x, -@z(y, 1@z))
eval(x, y) → Cond_eval1(&&(&&(>@z(+@z(x, y), 0@z), >=@z(y, x)), >@z(y, x)), x, y)
eval(x, y) → Cond_eval3(&&(&&(>@z(+@z(x, y), 0@z), >=@z(y, x)), >@z(x, y)), x, y)
Cond_eval3(TRUE, x, y) → eval(x, -@z(y, 1@z))
eval(x, y) → Cond_eval(&&(>@z(+@z(x, y), 0@z), >@z(x, y)), x, y)
eval(x, y) → Cond_eval2(&&(&&(>@z(+@z(x, y), 0@z), >=@z(y, x)), =@z(x, y)), x, y)

The set Q consists of the following terms:

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

Added dependency pairs

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

I DP problem:
The following domains are used:

z

The ITRS R consists of the following rules:

Cond_eval2(TRUE, x, y) → eval(-@z(x, 1@z), y)
Cond_eval(TRUE, x, y) → eval(-@z(x, 1@z), y)
Cond_eval1(TRUE, x, y) → eval(x, -@z(y, 1@z))
eval(x, y) → Cond_eval1(&&(&&(>@z(+@z(x, y), 0@z), >=@z(y, x)), >@z(y, x)), x, y)
eval(x, y) → Cond_eval3(&&(&&(>@z(+@z(x, y), 0@z), >=@z(y, x)), >@z(x, y)), x, y)
Cond_eval3(TRUE, x, y) → eval(x, -@z(y, 1@z))
eval(x, y) → Cond_eval(&&(>@z(+@z(x, y), 0@z), >@z(x, y)), x, y)
eval(x, y) → Cond_eval2(&&(&&(>@z(+@z(x, y), 0@z), >=@z(y, x)), =@z(x, y)), x, y)

The integer pair graph contains the following rules and edges:

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

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

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

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

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

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

(2) -> (7), if ((x[2] →^* x[7])∧(y[2] →^* y[7])∧(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])) →^* TRUE))

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

(4) -> (5), if ((x[4] →^* x[5])∧(y[4] →^* y[5])∧(&&(&&(>@z(+@z(x[4], y[4]), 0@z), >=@z(y[4], x[4])), >@z(x[4], y[4])) →^* TRUE))

(5) -> (0), if ((-@z(y[5], 1@z) →^* y[0])∧(x[5] →^* x[0]))

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

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

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

(6) -> (0), if ((-@z(y[6], 1@z) →^* y[0])∧(x[6] →^* x[0]))

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

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

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

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

(7) -> (2), if ((y[7] →^* y[2])∧(-@z(x[7], 1@z) →^* x[2]))

(7) -> (3), if ((y[7] →^* y[3])∧(-@z(x[7], 1@z) →^* x[3]))

(7) -> (4), if ((y[7] →^* y[4])∧(-@z(x[7], 1@z) →^* x[4]))

The set Q consists of the following terms:

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

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

I DP problem:
The following domains are used:

z

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

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

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

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

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

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

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

(2) -> (7), if ((x[2] →^* x[7])∧(y[2] →^* y[7])∧(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])) →^* TRUE))

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

(4) -> (5), if ((x[4] →^* x[5])∧(y[4] →^* y[5])∧(&&(&&(>@z(+@z(x[4], y[4]), 0@z), >=@z(y[4], x[4])), >@z(x[4], y[4])) →^* TRUE))

(5) -> (0), if ((-@z(y[5], 1@z) →^* y[0])∧(x[5] →^* x[0]))

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

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

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

(6) -> (0), if ((-@z(y[6], 1@z) →^* y[0])∧(x[6] →^* x[0]))

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

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

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

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

(7) -> (2), if ((y[7] →^* y[2])∧(-@z(x[7], 1@z) →^* x[2]))

(7) -> (3), if ((y[7] →^* y[3])∧(-@z(x[7], 1@z) →^* x[3]))

(7) -> (4), if ((y[7] →^* y[4])∧(-@z(x[7], 1@z) →^* x[4]))

The set Q consists of the following terms:

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

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

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

We consider the chain EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), 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(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0])∧(U^Increasing(COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), 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(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), 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(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), 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(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), 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∧(U^Increasing(COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

We simplified constraint (18) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(19)    (0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧(U^Increasing(COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])), x[2], y[2])), ≥)∧0 = 0)

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

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

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

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

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

We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(24)    (0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0)

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

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

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

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

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

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

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

We consider the chain EVAL(x[4], y[4]) → COND_EVAL3(&&(&&(>@z(+@z(x[4], y[4]), 0@z), >=@z(y[4], x[4])), >@z(x[4], y[4])), x[4], y[4]), COND_EVAL3(TRUE, x[5], y[5]) → EVAL(x[5], -@z(y[5], 1@z)) which results in the following constraint:
(30)    (x[4]=x[5]∧y[4]=y[5]∧&&(&&(>@z(+@z(x[4], y[4]), 0@z), >=@z(y[4], x[4])), >@z(x[4], y[4]))=TRUE ⇒ COND_EVAL3(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL3(TRUE, x[5], y[5])≥EVAL(x[5], -@z(y[5], 1@z))∧(U^Increasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))

We simplified constraint (30) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(31)    (>@z(x[4], y[4])=TRUE∧>@z(+@z(x[4], y[4]), 0@z)=TRUE∧>=@z(y[4], x[4])=TRUE ⇒ COND_EVAL3(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL3(TRUE, x[4], y[4])≥EVAL(x[4], -@z(y[4], 1@z))∧(U^Increasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))

We simplified constraint (31) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(32)    (-1 + x[4] + (-1)y[4] ≥ 0∧-1 + x[4] + y[4] ≥ 0∧y[4] + (-1)x[4] ≥ 0 ⇒ (U^Increasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧-1 + (-1)Bound + y[4] + (2)x[4] ≥ 0∧0 ≥ 0)

We simplified constraint (32) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(33)    (-1 + x[4] + (-1)y[4] ≥ 0∧-1 + x[4] + y[4] ≥ 0∧y[4] + (-1)x[4] ≥ 0 ⇒ (U^Increasing(EVAL(x[5], -@z(y[5], 1@z))), ≥)∧-1 + (-1)Bound + y[4] + (2)x[4] ≥ 0∧0 ≥ 0)

We simplified constraint (33) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(34)    (y[4] + (-1)x[4] ≥ 0∧-1 + x[4] + (-1)y[4] ≥ 0∧-1 + x[4] + y[4] ≥ 0 ⇒ 0 ≥ 0∧-1 + (-1)Bound + y[4] + (2)x[4] ≥ 0∧(U^Increasing(EVAL(x[5], -@z(y[5], 1@z))), ≥))

We solved constraint (34) using rule (IDP_SMT_SPLIT).

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

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

We simplified constraint (35) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(36)    (>@z(y[0], x[0])=TRUE∧>@z(+@z(x[0], y[0]), 0@z)=TRUE∧>=@z(y[0], x[0])=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(x[0], -@z(y[0], 1@z))∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥))

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

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

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

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

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

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

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

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

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

We simplified constraint (44) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(45)    (>@z(+@z(x[2], y[2]), 0@z)=TRUE∧>=@z(y[2], x[2])=TRUE∧>=@z(x[2], y[2])=TRUE∧<=@z(x[2], y[2])=TRUE ⇒ COND_EVAL2(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL2(TRUE, x[2], y[2])≥EVAL(-@z(x[2], 1@z), y[2])∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥))

We simplified constraint (45) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(46)    (-1 + x[2] + y[2] ≥ 0∧y[2] + (-1)x[2] ≥ 0∧x[2] + (-1)y[2] ≥ 0∧y[2] + (-1)x[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (46) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(47)    (-1 + x[2] + y[2] ≥ 0∧y[2] + (-1)x[2] ≥ 0∧x[2] + (-1)y[2] ≥ 0∧y[2] + (-1)x[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (47) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(48)    (y[2] + (-1)x[2] ≥ 0∧x[2] + (-1)y[2] ≥ 0∧y[2] + (-1)x[2] ≥ 0∧-1 + x[2] + y[2] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥)∧0 ≥ 0)

We simplified constraint (48) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(49)    (x[2] ≥ 0∧(-1)x[2] ≥ 0∧x[2] ≥ 0∧-1 + (2)y[2] + (-1)x[2] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥)∧0 ≥ 0)

We simplified constraint (49) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(50)    (0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧-1 + (2)y[2] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥)∧0 ≥ 0)

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

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

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

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

The following pairs are in P_>:

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

The following pairs are in P_bound:

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

The following pairs are in P_≥:

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

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

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

              ↳ IDP

I DP problem:
The following domains are used:

z

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

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

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

(7) -> (4), if ((y[7] →^* y[4])∧(-@z(x[7], 1@z) →^* x[4]))

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

(7) -> (3), if ((y[7] →^* y[3])∧(-@z(x[7], 1@z) →^* x[3]))

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

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

(7) -> (2), if ((y[7] →^* y[2])∧(-@z(x[7], 1@z) →^* x[2]))

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

(2) -> (7), if ((x[2] →^* x[7])∧(y[2] →^* y[7])∧(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])) →^* TRUE))

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

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

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

(6) -> (0), if ((-@z(y[6], 1@z) →^* y[0])∧(x[6] →^* x[0]))

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

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

              ↳ IDP

I DP problem:
The following domains are used:

z

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

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

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

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

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

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

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

(7) -> (3), if ((y[7] →^* y[3])∧(-@z(x[7], 1@z) →^* x[3]))

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

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

(6) -> (0), if ((-@z(y[6], 1@z) →^* y[0])∧(x[6] →^* x[0]))

(7) -> (2), if ((y[7] →^* y[2])∧(-@z(x[7], 1@z) →^* x[2]))

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

(2) -> (7), if ((x[2] →^* x[7])∧(y[2] →^* y[7])∧(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])) →^* TRUE))

The set Q consists of the following terms:

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

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

For Pair COND_EVAL2(TRUE, x[7], y[7]) → EVAL(-@z(x[7], 1@z), y[7]) the following chains were created:

We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])), x[2], y[2]), COND_EVAL2(TRUE, x[7], y[7]) → EVAL(-@z(x[7], 1@z), y[7]), EVAL(x[3], y[3]) → COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3]) which results in the following constraint:
(1)    (&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2]))=TRUE∧y[2]=y[7]∧x[2]=x[7]∧y[7]=y[3]∧-@z(x[7], 1@z)=x[3] ⇒ COND_EVAL2(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL2(TRUE, x[7], y[7])≥EVAL(-@z(x[7], 1@z), y[7])∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥))

We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>@z(+@z(x[2], y[2]), 0@z)=TRUE∧>=@z(y[2], x[2])=TRUE∧>=@z(x[2], y[2])=TRUE∧<=@z(x[2], y[2])=TRUE ⇒ COND_EVAL2(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL2(TRUE, x[2], y[2])≥EVAL(-@z(x[2], 1@z), y[2])∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥))

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

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (-1 + x[2] + y[2] ≥ 0∧y[2] + (-1)x[2] ≥ 0∧x[2] + (-1)y[2] ≥ 0∧y[2] + (-1)x[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥)∧(-1)Bound + x[2] ≥ 0∧0 ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (y[2] + (-1)x[2] ≥ 0∧y[2] + (-1)x[2] ≥ 0∧x[2] + (-1)y[2] ≥ 0∧-1 + x[2] + y[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥)∧(-1)Bound + x[2] ≥ 0)

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

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧-1 + (2)y[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥)∧(-1)Bound + y[2] ≥ 0)
We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])), x[2], y[2]), COND_EVAL2(TRUE, x[7], y[7]) → EVAL(-@z(x[7], 1@z), y[7]), EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0]) which results in the following constraint:
(8)    (&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2]))=TRUE∧y[2]=y[7]∧x[2]=x[7]∧y[7]=y[0]∧-@z(x[7], 1@z)=x[0] ⇒ COND_EVAL2(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL2(TRUE, x[7], y[7])≥EVAL(-@z(x[7], 1@z), y[7])∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥))

We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(9)    (>@z(+@z(x[2], y[2]), 0@z)=TRUE∧>=@z(y[2], x[2])=TRUE∧>=@z(x[2], y[2])=TRUE∧<=@z(x[2], y[2])=TRUE ⇒ COND_EVAL2(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL2(TRUE, x[2], y[2])≥EVAL(-@z(x[2], 1@z), y[2])∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥))

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

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

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

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (1 + (-2)y[2] + x[2] ≥ 0∧(2)y[2] + -1 + (-1)x[2] ≥ 0∧(2)y[2] + -1 + (-1)x[2] ≥ 0∧x[2] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥)∧1 + (-1)Bound + (-1)y[2] + x[2] ≥ 0)
We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])), x[2], y[2]), COND_EVAL2(TRUE, x[7], y[7]) → EVAL(-@z(x[7], 1@z), y[7]), EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])), x[2], y[2]) which results in the following constraint:
(14)    (&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2]))=TRUE∧y[2]=y[7]∧x[2]=x[7]∧-@z(x[7], 1@z)=x[2]1∧y[7]=y[2]1 ⇒ COND_EVAL2(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL2(TRUE, x[7], y[7])≥EVAL(-@z(x[7], 1@z), y[7])∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥))

We simplified constraint (14) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(15)    (>@z(+@z(x[2], y[2]), 0@z)=TRUE∧>=@z(y[2], x[2])=TRUE∧>=@z(x[2], y[2])=TRUE∧<=@z(x[2], y[2])=TRUE ⇒ COND_EVAL2(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL2(TRUE, x[2], y[2])≥EVAL(-@z(x[2], 1@z), y[2])∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥))

We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16)    (-1 + x[2] + y[2] ≥ 0∧y[2] + (-1)x[2] ≥ 0∧x[2] + (-1)y[2] ≥ 0∧y[2] + (-1)x[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥)∧(-1)Bound + x[2] ≥ 0∧0 ≥ 0)

We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17)    (-1 + x[2] + y[2] ≥ 0∧y[2] + (-1)x[2] ≥ 0∧x[2] + (-1)y[2] ≥ 0∧y[2] + (-1)x[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥)∧(-1)Bound + x[2] ≥ 0∧0 ≥ 0)

We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18)    (x[2] + (-1)y[2] ≥ 0∧-1 + x[2] + y[2] ≥ 0∧y[2] + (-1)x[2] ≥ 0∧y[2] + (-1)x[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥)∧0 ≥ 0∧(-1)Bound + x[2] ≥ 0)

We simplified constraint (18) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(19)    (x[2] ≥ 0∧-1 + (2)y[2] + x[2] ≥ 0∧(-1)x[2] ≥ 0∧(-1)x[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥)∧0 ≥ 0∧(-1)Bound + y[2] + x[2] ≥ 0)

We simplified constraint (19) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(20)    (0 ≥ 0∧-1 + (2)y[2] ≥ 0∧0 ≥ 0∧0 ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥)∧0 ≥ 0∧(-1)Bound + y[2] ≥ 0)

For Pair EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])), x[2], y[2]) the following chains were created:

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

We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(22)    ((U^Increasing(COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(23)    ((U^Increasing(COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(24)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])), x[2], y[2])), ≥))

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_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])), x[2], y[2])), ≥)∧0 = 0∧0 = 0∧0 = 0)

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

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

We simplified constraint (26) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(27)    (>@z(+@z(x[3], y[3]), 0@z)=TRUE∧>@z(x[3], y[3])=TRUE ⇒ COND_EVAL(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL(TRUE, x[3], y[3])≥EVAL(-@z(x[3], 1@z), y[3])∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥))

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

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

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

We simplified constraint (30) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(31)    (x[3] ≥ 0∧(2)y[3] + x[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥)∧1 ≥ 0)

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

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

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

We simplified constraint (35) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(36)    (>@z(+@z(x[3], y[3]), 0@z)=TRUE∧>@z(x[3], y[3])=TRUE ⇒ COND_EVAL(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL(TRUE, x[3], y[3])≥EVAL(-@z(x[3], 1@z), y[3])∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥))

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

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

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

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

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

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

We simplified constraint (42) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(43)    ((2)y[3] + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥)∧1 ≥ 0)
We consider the chain EVAL(x[3], y[3]) → COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3]), COND_EVAL(TRUE, x[1], y[1]) → EVAL(-@z(x[1], 1@z), y[1]), EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0]) which results in the following constraint:
(44)    (&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3]))=TRUE∧x[3]=x[1]∧y[3]=y[1]∧-@z(x[1], 1@z)=x[0]∧y[1]=y[0] ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(-@z(x[1], 1@z), y[1])∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (44) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(45)    (>@z(+@z(x[3], y[3]), 0@z)=TRUE∧>@z(x[3], y[3])=TRUE ⇒ COND_EVAL(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL(TRUE, x[3], y[3])≥EVAL(-@z(x[3], 1@z), y[3])∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (45) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(46)    (-1 + x[3] + y[3] ≥ 0∧-1 + x[3] + (-1)y[3] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (46) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(47)    (-1 + x[3] + y[3] ≥ 0∧-1 + x[3] + (-1)y[3] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥)∧0 ≥ 0∧1 ≥ 0)

We simplified constraint (47) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(48)    (-1 + x[3] + (-1)y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥)∧0 ≥ 0)

We simplified constraint (48) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(49)    (x[3] ≥ 0∧(2)y[3] + x[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥)∧0 ≥ 0)

We simplified constraint (49) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(50)    (x[3] ≥ 0∧(2)y[3] + x[3] ≥ 0∧y[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥)∧0 ≥ 0)

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

We simplified constraint (51) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(52)    ((2)y[3] + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥)∧0 ≥ 0)

For Pair EVAL(x[3], y[3]) → COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3]) the following chains were created:

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

We simplified constraint (53) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(54)    ((U^Increasing(COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (54) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(55)    ((U^Increasing(COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (55) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(56)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3])), ≥))

We simplified constraint (56) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(57)    (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 = 0)

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

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

We simplified constraint (58) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(59)    ((U^Increasing(COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (59) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(60)    ((U^Increasing(COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (60) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(61)    (0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0])), ≥)∧0 ≥ 0)

We simplified constraint (61) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(62)    (0 = 0∧0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0)

For Pair COND_EVAL1(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z)) the following chains were created:

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

We simplified constraint (63) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(64)    (>@z(y[0], x[0])=TRUE∧>@z(+@z(x[0], y[0]), 0@z)=TRUE∧>=@z(y[0], x[0])=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(x[0], -@z(y[0], 1@z))∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥))

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

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

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

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

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

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

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

We simplified constraint (71) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(72)    (y[0] ≥ 0∧1 + (2)x[0] + y[0] ≥ 0∧(2)x[0] + y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0)
We consider the chain EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0]), COND_EVAL1(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z)), EVAL(x[3], y[3]) → COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3]) which results in the following constraint:
(73)    (-@z(y[6], 1@z)=y[3]∧x[6]=x[3]∧x[0]=x[6]∧y[0]=y[6]∧&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0]))=TRUE ⇒ COND_EVAL1(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL1(TRUE, x[6], y[6])≥EVAL(x[6], -@z(y[6], 1@z))∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥))

We simplified constraint (73) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(74)    (>@z(y[0], x[0])=TRUE∧>@z(+@z(x[0], y[0]), 0@z)=TRUE∧>=@z(y[0], x[0])=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(x[0], -@z(y[0], 1@z))∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥))

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

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

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

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

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

We simplified constraint (79) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(80)    ((2)x[0] + y[0] ≥ 0∧y[0] ≥ 0∧1 + y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥))

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

We simplified constraint (81) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(82)    (y[0] ≥ 0∧(2)x[0] + y[0] ≥ 0∧1 + (2)x[0] + y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥))
We consider the chain EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0]), COND_EVAL1(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z)), EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0]) which results in the following constraint:
(83)    (x[0]=x[6]∧-@z(y[6], 1@z)=y[0]1∧y[0]=y[6]∧&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0]))=TRUE∧x[6]=x[0]1 ⇒ COND_EVAL1(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL1(TRUE, x[6], y[6])≥EVAL(x[6], -@z(y[6], 1@z))∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥))

We simplified constraint (83) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(84)    (>@z(y[0], x[0])=TRUE∧>@z(+@z(x[0], y[0]), 0@z)=TRUE∧>=@z(y[0], x[0])=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(x[0], -@z(y[0], 1@z))∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥))

We simplified constraint (84) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(85)    (-1 + y[0] + (-1)x[0] ≥ 0∧x[0] + -1 + y[0] ≥ 0∧y[0] + (-1)x[0] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (85) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(86)    (-1 + y[0] + (-1)x[0] ≥ 0∧x[0] + -1 + y[0] ≥ 0∧y[0] + (-1)x[0] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (86) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(87)    (y[0] + (-1)x[0] ≥ 0∧x[0] + -1 + y[0] ≥ 0∧-1 + y[0] + (-1)x[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0)

We simplified constraint (87) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(88)    (1 + y[0] ≥ 0∧(2)x[0] + y[0] ≥ 0∧y[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0)

We simplified constraint (88) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(89)    (1 + y[0] ≥ 0∧(2)x[0] + y[0] ≥ 0∧y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0)

(90)    (1 + y[0] ≥ 0∧(-2)x[0] + y[0] ≥ 0∧y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0)

We simplified constraint (90) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(91)    (1 + (2)x[0] + y[0] ≥ 0∧y[0] ≥ 0∧(2)x[0] + y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0)

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

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

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

The following pairs are in P_>:

COND_EVAL2(TRUE, x[7], y[7]) → EVAL(-@z(x[7], 1@z), y[7])

The following pairs are in P_bound:

COND_EVAL2(TRUE, x[7], y[7]) → EVAL(-@z(x[7], 1@z), y[7])

The following pairs are in P_≥:

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

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ 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:

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

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

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

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

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

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

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

(6) -> (0), if ((-@z(y[6], 1@z) →^* y[0])∧(x[6] →^* x[0]))

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

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

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

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

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

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

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

(6) -> (0), if ((-@z(y[6], 1@z) →^* y[0])∧(x[6] →^* x[0]))

The set Q consists of the following terms:

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

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

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

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

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

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

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

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

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

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

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

We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(9)    (x[3] ≥ 0∧(2)y[3] + x[3] ≥ 0∧y[3] ≥ 0 ⇒ 1 ≥ 0∧1 + (-1)Bound + x[3] ≥ 0∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥))
We consider the chain EVAL(x[3], y[3]) → COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3]), COND_EVAL(TRUE, x[1], y[1]) → EVAL(-@z(x[1], 1@z), y[1]), EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0]) which results in the following constraint:
(10)    (&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3]))=TRUE∧x[3]=x[1]∧y[3]=y[1]∧-@z(x[1], 1@z)=x[0]∧y[1]=y[0] ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(-@z(x[1], 1@z), y[1])∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥))

We simplified constraint (10) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(11)    (>@z(+@z(x[3], y[3]), 0@z)=TRUE∧>@z(x[3], y[3])=TRUE ⇒ COND_EVAL(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL(TRUE, x[3], y[3])≥EVAL(-@z(x[3], 1@z), y[3])∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥))

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

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

We simplified constraint (13) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(14)    (-1 + x[3] + (-1)y[3] ≥ 0∧-1 + x[3] + y[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥)∧(-1)Bound + y[3] + x[3] ≥ 0)

We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(15)    ((-2)y[3] + x[3] ≥ 0∧x[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥)∧1 + (-1)Bound + x[3] ≥ 0)

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

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

We simplified constraint (16) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(18)    (x[3] ≥ 0∧(2)y[3] + x[3] ≥ 0∧y[3] ≥ 0 ⇒ 1 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥)∧1 + (-1)Bound + (2)y[3] + x[3] ≥ 0)

For Pair EVAL(x[3], y[3]) → COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3]) the following chains were created:

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

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

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

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

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

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

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

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

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

We simplified constraint (26) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(27)    (0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0])), ≥)∧0 ≥ 0)

We simplified constraint (27) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(28)    (0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0)

For Pair COND_EVAL1(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z)) the following chains were created:

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

We simplified constraint (29) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(30)    (>@z(y[0], x[0])=TRUE∧>@z(+@z(x[0], y[0]), 0@z)=TRUE∧>=@z(y[0], x[0])=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(x[0], -@z(y[0], 1@z))∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥))

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

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

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

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

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

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

We simplified constraint (36) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(37)    (y[0] ≥ 0∧(2)x[0] + y[0] ≥ 0∧1 + (2)x[0] + y[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We consider the chain EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0]), COND_EVAL1(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z)), EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0]) which results in the following constraint:
(38)    (x[0]=x[6]∧-@z(y[6], 1@z)=y[0]1∧y[0]=y[6]∧&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0]))=TRUE∧x[6]=x[0]1 ⇒ COND_EVAL1(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL1(TRUE, x[6], y[6])≥EVAL(x[6], -@z(y[6], 1@z))∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥))

We simplified constraint (38) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(39)    (>@z(y[0], x[0])=TRUE∧>@z(+@z(x[0], y[0]), 0@z)=TRUE∧>=@z(y[0], x[0])=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(x[0], -@z(y[0], 1@z))∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥))

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

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

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

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

We simplified constraint (43) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(44)    ((-2)x[0] + y[0] ≥ 0∧y[0] ≥ 0∧1 + (-2)x[0] + y[0] ≥ 0∧x[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0)

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

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

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

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

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

The following pairs are in P_>:

COND_EVAL1(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z))

The following pairs are in P_bound:

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

The following pairs are in P_≥:

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

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

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

                          ↳ IDP

                            ↳ IDPNonInfProof

                              ↳ AND

                                ↳ IDP

                                  ↳ IDependencyGraphProof

                                ↳ IDP

              ↳ IDP

I DP problem:
The following domains are used:

z

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

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

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

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

(6) -> (0), if ((-@z(y[6], 1@z) →^* y[0])∧(x[6] →^* x[0]))

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

                          ↳ IDP

                            ↳ IDPNonInfProof

                              ↳ AND

                                ↳ IDP

                                  ↳ IDependencyGraphProof

                                    ↳ IDP

                                      ↳ IDPNonInfProof

                                ↳ IDP

              ↳ IDP

I DP problem:
The following domains are used:

z

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

(6) -> (0), if ((-@z(y[6], 1@z) →^* y[0])∧(x[6] →^* x[0]))

The set Q consists of the following terms:

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

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

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

We consider the chain EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), 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(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0])∧(U^Increasing(COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), 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(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), 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(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), 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(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), 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∧0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0])), ≥)∧0 ≥ 0)

For Pair COND_EVAL1(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z)) the following chains were created:

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

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

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

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

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

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

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

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(13)    (1 + y[0] ≥ 0∧y[0] ≥ 0∧(2)x[0] + y[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0∧(-1)Bound + (2)x[0] + (2)y[0] ≥ 0)

(14)    (1 + y[0] ≥ 0∧y[0] ≥ 0∧(-2)x[0] + y[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0∧(-1)Bound + (-2)x[0] + (2)y[0] ≥ 0)

We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(15)    (1 + (2)x[0] + y[0] ≥ 0∧(2)x[0] + y[0] ≥ 0∧y[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0∧(-1)Bound + (2)x[0] + (2)y[0] ≥ 0)

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

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

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

The following pairs are in P_>:

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

The following pairs are in P_bound:

COND_EVAL1(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z))

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
&&(FALSE, TRUE)¹ → FALSE¹
&&(TRUE, FALSE)¹ → FALSE¹

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

                          ↳ IDP

                            ↳ IDPNonInfProof

                              ↳ AND

                                ↳ IDP

                                  ↳ IDependencyGraphProof

                                    ↳ IDP

                                      ↳ IDPNonInfProof

                                        ↳ AND

                                          ↳ IDP

                                            ↳ IDependencyGraphProof

                                          ↳ IDP

                                ↳ IDP

              ↳ IDP

I DP problem:
The following domains are used:none

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

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

                          ↳ IDP

                            ↳ IDPNonInfProof

                              ↳ AND

                                ↳ IDP

                                  ↳ IDependencyGraphProof

                                    ↳ IDP

                                      ↳ IDPNonInfProof

                                        ↳ AND

                                          ↳ IDP

                                          ↳ IDP

                                            ↳ IDependencyGraphProof

                                ↳ IDP

              ↳ IDP

I DP problem:
The following domains are used:

z

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

                          ↳ IDP

                            ↳ IDPNonInfProof

                              ↳ AND

                                ↳ IDP

                                ↳ IDP

                                  ↳ IDependencyGraphProof

              ↳ IDP

I DP problem:
The following domains are used:

z

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

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

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

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

                          ↳ IDP

                            ↳ IDPNonInfProof

                              ↳ AND

                                ↳ IDP

                                ↳ IDP

                                  ↳ IDependencyGraphProof

                                    ↳ IDP

                                      ↳ IDPNonInfProof

              ↳ IDP

I DP problem:
The following domains are used:

z

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

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

The set Q consists of the following terms:

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

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

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

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

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

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

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

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

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

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

(8)    (x[3] ≥ 0∧(-2)y[3] + x[3] ≥ 0∧y[3] ≥ 0 ⇒ 2 + (-1)Bound + (-4)y[3] + (2)x[3] ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥))

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

For Pair EVAL(x[3], y[3]) → COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3]) the following chains were created:

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

We simplified constraint (10) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(11)    ((U^Increasing(COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(12)    ((U^Increasing(COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(13)    (0 ≥ 0∧(U^Increasing(COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3])), ≥)∧0 ≥ 0)

We simplified constraint (13) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(14)    (0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3])), ≥))

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

COND_EVAL(TRUE, x[1], y[1]) → EVAL(-@z(x[1], 1@z), y[1])
- (x[3] ≥ 0∧(2)y[3] + x[3] ≥ 0∧y[3] ≥ 0 ⇒ 2 + (-1)Bound + (4)y[3] + (2)x[3] ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥))
- ((2)y[3] + x[3] ≥ 0∧x[3] ≥ 0∧y[3] ≥ 0 ⇒ 2 + (-1)Bound + (2)x[3] ≥ 0∧0 ≥ 0∧(U^Increasing(EVAL(-@z(x[1], 1@z), y[1])), ≥))
EVAL(x[3], y[3]) → COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3])
- (0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3])), ≥))

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

The following pairs are in P_>:

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

The following pairs are in P_bound:

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

The following pairs are in P_≥:
none

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

                          ↳ IDP

                            ↳ IDPNonInfProof

                              ↳ AND

                                ↳ IDP

                                ↳ IDP

                                  ↳ IDependencyGraphProof

                                    ↳ IDP

                                      ↳ IDPNonInfProof

                                        ↳ AND

                                          ↳ IDP

                                            ↳ IDependencyGraphProof

                                          ↳ IDP

              ↳ IDP

I DP problem:
The following domains are used:

z

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

(3): EVAL(x[3], y[3]) → COND_EVAL(&&(>@z(+@z(x[3], y[3]), 0@z), >@z(x[3], y[3])), x[3], y[3])

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ IDP

                        ↳ IDependencyGraphProof

                          ↳ IDP

                            ↳ IDPNonInfProof

                              ↳ AND

                                ↳ IDP

                                ↳ IDP

                                  ↳ IDependencyGraphProof

                                    ↳ IDP

                                      ↳ IDPNonInfProof

                                        ↳ AND

                                          ↳ IDP

                                          ↳ 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_eval2(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval3(TRUE, x0, x1)

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

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

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

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

(7) -> (4), if ((y[7] →^* y[4])∧(-@z(x[7], 1@z) →^* x[4]))

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

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

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

(7) -> (3), if ((y[7] →^* y[3])∧(-@z(x[7], 1@z) →^* x[3]))

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

(6) -> (0), if ((-@z(y[6], 1@z) →^* y[0])∧(x[6] →^* x[0]))

(7) -> (2), if ((y[7] →^* y[2])∧(-@z(x[7], 1@z) →^* x[2]))

(2) -> (7), if ((x[2] →^* x[7])∧(y[2] →^* y[7])∧(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])) →^* TRUE))

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ 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:

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

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

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

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

(6) -> (0), if ((-@z(y[6], 1@z) →^* y[0])∧(x[6] →^* x[0]))

(7) -> (2), if ((y[7] →^* y[2])∧(-@z(x[7], 1@z) →^* x[2]))

(2) -> (7), if ((x[2] →^* x[7])∧(y[2] →^* y[7])∧(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])) →^* TRUE))

The set Q consists of the following terms:

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

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

For Pair COND_EVAL2(TRUE, x[7], y[7]) → EVAL(-@z(x[7], 1@z), y[7]) the following chains were created:

We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])), x[2], y[2]), COND_EVAL2(TRUE, x[7], y[7]) → EVAL(-@z(x[7], 1@z), y[7]), EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0]) which results in the following constraint:
(1)    (&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2]))=TRUE∧y[2]=y[7]∧x[2]=x[7]∧y[7]=y[0]∧-@z(x[7], 1@z)=x[0] ⇒ COND_EVAL2(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL2(TRUE, x[7], y[7])≥EVAL(-@z(x[7], 1@z), y[7])∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥))

We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>@z(+@z(x[2], y[2]), 0@z)=TRUE∧>=@z(y[2], x[2])=TRUE∧>=@z(x[2], y[2])=TRUE∧<=@z(x[2], y[2])=TRUE ⇒ COND_EVAL2(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL2(TRUE, x[2], y[2])≥EVAL(-@z(x[2], 1@z), y[2])∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥))

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

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (-1 + x[2] + y[2] ≥ 0∧y[2] + (-1)x[2] ≥ 0∧x[2] + (-1)y[2] ≥ 0∧y[2] + (-1)x[2] ≥ 0 ⇒ (U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥)∧-1 + (-1)Bound + (2)y[2] ≥ 0∧0 ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x[2] + (-1)y[2] ≥ 0∧y[2] + (-1)x[2] ≥ 0∧-1 + x[2] + y[2] ≥ 0∧y[2] + (-1)x[2] ≥ 0 ⇒ -1 + (-1)Bound + (2)y[2] ≥ 0∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥)∧0 ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (1 + (-2)y[2] + x[2] ≥ 0∧(2)y[2] + -1 + (-1)x[2] ≥ 0∧x[2] ≥ 0∧(2)y[2] + -1 + (-1)x[2] ≥ 0 ⇒ -1 + (-1)Bound + (2)y[2] ≥ 0∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥)∧0 ≥ 0)
We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])), x[2], y[2]), COND_EVAL2(TRUE, x[7], y[7]) → EVAL(-@z(x[7], 1@z), y[7]), EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])), x[2], y[2]) which results in the following constraint:
(7)    (&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2]))=TRUE∧y[2]=y[7]∧x[2]=x[7]∧-@z(x[7], 1@z)=x[2]1∧y[7]=y[2]1 ⇒ COND_EVAL2(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL2(TRUE, x[7], y[7])≥EVAL(-@z(x[7], 1@z), y[7])∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥))

We simplified constraint (7) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(8)    (>@z(+@z(x[2], y[2]), 0@z)=TRUE∧>=@z(y[2], x[2])=TRUE∧>=@z(x[2], y[2])=TRUE∧<=@z(x[2], y[2])=TRUE ⇒ COND_EVAL2(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL2(TRUE, x[2], y[2])≥EVAL(-@z(x[2], 1@z), y[2])∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥))

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

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

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

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

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

For Pair EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])), x[2], y[2]) the following chains were created:

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

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

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

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

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

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

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

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

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

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

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

For Pair COND_EVAL1(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z)) the following chains were created:

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

We simplified constraint (24) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(25)    (>@z(y[0], x[0])=TRUE∧>@z(+@z(x[0], y[0]), 0@z)=TRUE∧>=@z(y[0], x[0])=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(x[0], -@z(y[0], 1@z))∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥))

We simplified constraint (25) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(26)    (-1 + y[0] + (-1)x[0] ≥ 0∧x[0] + -1 + y[0] ≥ 0∧y[0] + (-1)x[0] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (26) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(27)    (-1 + y[0] + (-1)x[0] ≥ 0∧x[0] + -1 + y[0] ≥ 0∧y[0] + (-1)x[0] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (27) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(28)    (-1 + y[0] + (-1)x[0] ≥ 0∧y[0] + (-1)x[0] ≥ 0∧x[0] + -1 + y[0] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (28) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(29)    (-1 + y[0] ≥ 0∧y[0] ≥ 0∧(2)x[0] + -1 + y[0] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (29) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(30)    ((-2)x[0] + y[0] ≥ 0∧1 + (-2)x[0] + y[0] ≥ 0∧y[0] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (30) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(31)    ((-2)x[0] + y[0] ≥ 0∧1 + (-2)x[0] + y[0] ≥ 0∧y[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)

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

We simplified constraint (31) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(33)    (y[0] ≥ 0∧1 + y[0] ≥ 0∧(2)x[0] + y[0] ≥ 0∧x[0] ≥ 0 ⇒ (U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We consider the chain EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0]), COND_EVAL1(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z)), EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0]) which results in the following constraint:
(34)    (x[0]=x[6]∧-@z(y[6], 1@z)=y[0]1∧y[0]=y[6]∧&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0]))=TRUE∧x[6]=x[0]1 ⇒ COND_EVAL1(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL1(TRUE, x[6], y[6])≥EVAL(x[6], -@z(y[6], 1@z))∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥))

We simplified constraint (34) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(35)    (>@z(y[0], x[0])=TRUE∧>@z(+@z(x[0], y[0]), 0@z)=TRUE∧>=@z(y[0], x[0])=TRUE ⇒ COND_EVAL1(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL1(TRUE, x[0], y[0])≥EVAL(x[0], -@z(y[0], 1@z))∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥))

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

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

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

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

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

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

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

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

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

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

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

The following pairs are in P_>:

COND_EVAL1(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z))

The following pairs are in P_bound:

COND_EVAL2(TRUE, x[7], y[7]) → EVAL(-@z(x[7], 1@z), y[7])

The following pairs are in P_≥:

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

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

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                          ↳ IDependencyGraphProof

                        ↳ IDP

I DP problem:
The following domains are used:

z

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

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

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

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

(6) -> (0), if ((-@z(y[6], 1@z) →^* y[0])∧(x[6] →^* x[0]))

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                          ↳ IDependencyGraphProof

                            ↳ IDP

                              ↳ IDPNonInfProof

                        ↳ IDP

I DP problem:
The following domains are used:

z

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

(6) -> (0), if ((-@z(y[6], 1@z) →^* y[0])∧(x[6] →^* x[0]))

The set Q consists of the following terms:

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

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

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

We consider the chain EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), 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(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0])∧(U^Increasing(COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), 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(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), 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(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), 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∧0 ≥ 0∧(U^Increasing(COND_EVAL1(&&(&&(>@z(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[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(+@z(x[0], y[0]), 0@z), >=@z(y[0], x[0])), >@z(y[0], x[0])), x[0], y[0])), ≥)∧0 = 0∧0 = 0)

For Pair COND_EVAL1(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z)) the following chains were created:

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

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

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

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

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

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

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

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

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    ((2)x[0] + y[0] ≥ 0∧1 + (2)x[0] + y[0] ≥ 0∧y[0] ≥ 0∧x[0] ≥ 0 ⇒ 1 ≥ 0∧2 + (-1)Bound + (2)x[0] + (2)y[0] ≥ 0∧(U^Increasing(EVAL(x[6], -@z(y[6], 1@z))), ≥))

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

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

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

The following pairs are in P_>:

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

The following pairs are in P_bound:

COND_EVAL1(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z))

The following pairs are in P_≥:

COND_EVAL1(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z))

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                          ↳ IDependencyGraphProof

                            ↳ IDP

                              ↳ IDPNonInfProof

                                ↳ AND

                                  ↳ IDP

                                    ↳ IDependencyGraphProof

                                  ↳ IDP

                        ↳ IDP

I DP problem:
The following domains are used:

z

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

(6): COND_EVAL1(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z))

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                          ↳ IDependencyGraphProof

                            ↳ IDP

                              ↳ IDPNonInfProof

                                ↳ AND

                                  ↳ IDP

                                  ↳ IDP

                                    ↳ IDependencyGraphProof

                        ↳ IDP

I DP problem:
The following domains are used:

z

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                        ↳ IDP

                          ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

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

(7) -> (2), if ((y[7] →^* y[2])∧(-@z(x[7], 1@z) →^* x[2]))

(2) -> (7), if ((x[2] →^* x[7])∧(y[2] →^* y[7])∧(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])) →^* TRUE))

The set Q consists of the following terms:

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                        ↳ IDP

                          ↳ IDependencyGraphProof

                            ↳ IDP

                              ↳ IDPNonInfProof

I DP problem:
The following domains are used:

z

(7) -> (2), if ((y[7] →^* y[2])∧(-@z(x[7], 1@z) →^* x[2]))

(2) -> (7), if ((x[2] →^* x[7])∧(y[2] →^* y[7])∧(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])) →^* TRUE))

The set Q consists of the following terms:

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

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

For Pair EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])), x[2], y[2]) the following chains were created:

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

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

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

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

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

For Pair COND_EVAL2(TRUE, x[7], y[7]) → EVAL(-@z(x[7], 1@z), y[7]) the following chains were created:

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

We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7)    (>@z(+@z(x[2], y[2]), 0@z)=TRUE∧>=@z(y[2], x[2])=TRUE∧>=@z(x[2], y[2])=TRUE∧<=@z(x[2], y[2])=TRUE ⇒ COND_EVAL2(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL2(TRUE, x[2], y[2])≥EVAL(-@z(x[2], 1@z), y[2])∧(U^Increasing(EVAL(-@z(x[7], 1@z), y[7])), ≥))

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

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

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

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

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

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

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

The following pairs are in P_>:

COND_EVAL2(TRUE, x[7], y[7]) → EVAL(-@z(x[7], 1@z), y[7])

The following pairs are in P_bound:

COND_EVAL2(TRUE, x[7], y[7]) → EVAL(-@z(x[7], 1@z), y[7])

The following pairs are in P_≥:

EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(+@z(x[2], y[2]), 0@z), >=@z(y[2], x[2])), =@z(x[2], y[2])), x[2], y[2])

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

            ↳ AND

              ↳ IDP

              ↳ IDP

                ↳ IDependencyGraphProof

                  ↳ IDP

                    ↳ IDPNonInfProof

                      ↳ AND

                        ↳ IDP

                        ↳ IDP

                          ↳ IDependencyGraphProof

                            ↳ IDP

                              ↳ IDPNonInfProof

                                ↳ IDP

                                  ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

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

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