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_eval1(TRUE, x, y) → eval(-@z(x, 1@z), y)
eval(x, y) → Cond_eval5(&&(&&(>@z(x, 0@z), >=@z(0@z, x)), >=@z(0@z, y)), x, y)
Cond_eval5(TRUE, x, y) → eval(x, y)
eval(x, y) → Cond_eval4(&&(>@z(y, 0@z), >@z(x, 0@z)), x, y)
Cond_eval4(TRUE, x, y) → eval(-@z(x, 1@z), y)
eval(x, y) → Cond_eval3(&&(>@z(y, 0@z), >=@z(0@z, x)), x, y)
Cond_eval(TRUE, x, y) → eval(x, y)
Cond_eval2(TRUE, x, y) → eval(x, -@z(y, 1@z))
eval(x, y) → Cond_eval2(&&(&&(>@z(x, 0@z), >=@z(0@z, x)), >@z(y, 0@z)), x, y)
eval(x, y) → Cond_eval1(>@z(x, 0@z), x, y)
Cond_eval3(TRUE, x, y) → eval(x, -@z(y, 1@z))
eval(x, y) → Cond_eval(&&(&&(>@z(y, 0@z), >=@z(0@z, x)), >=@z(0@z, y)), x, y)
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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_eval1(TRUE, x, y) → eval(-@z(x, 1@z), y)
eval(x, y) → Cond_eval5(&&(&&(>@z(x, 0@z), >=@z(0@z, x)), >=@z(0@z, y)), x, y)
Cond_eval5(TRUE, x, y) → eval(x, y)
eval(x, y) → Cond_eval4(&&(>@z(y, 0@z), >@z(x, 0@z)), x, y)
Cond_eval4(TRUE, x, y) → eval(-@z(x, 1@z), y)
eval(x, y) → Cond_eval3(&&(>@z(y, 0@z), >=@z(0@z, x)), x, y)
Cond_eval(TRUE, x, y) → eval(x, y)
Cond_eval2(TRUE, x, y) → eval(x, -@z(y, 1@z))
eval(x, y) → Cond_eval2(&&(&&(>@z(x, 0@z), >=@z(0@z, x)), >@z(y, 0@z)), x, y)
eval(x, y) → Cond_eval1(>@z(x, 0@z), x, y)
Cond_eval3(TRUE, x, y) → eval(x, -@z(y, 1@z))
eval(x, y) → Cond_eval(&&(&&(>@z(y, 0@z), >=@z(0@z, x)), >=@z(0@z, y)), x, y)
The integer pair graph contains the following rules and edges:
(0): COND_EVAL(TRUE, x[0], y[0]) → EVAL(x[0], y[0])
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])
(2): EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(x[2], 0@z), >=@z(0@z, x[2])), >@z(y[2], 0@z)), x[2], y[2])
(3): EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(5): COND_EVAL5(TRUE, x[5], y[5]) → EVAL(x[5], y[5])
(6): COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z))
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(0) -> (1), if ((y[0] →* y[1])∧(x[0] →* x[1]))
(0) -> (2), if ((y[0] →* y[2])∧(x[0] →* x[2]))
(0) -> (3), if ((y[0] →* y[3])∧(x[0] →* x[3]))
(0) -> (7), if ((y[0] →* y[7])∧(x[0] →* x[7]))
(0) -> (9), if ((y[0] →* y[9])∧(x[0] →* x[9]))
(0) -> (10), if ((y[0] →* y[10])∧(x[0] →* x[10]))
(1) -> (0), if ((x[1] →* x[0])∧(y[1] →* y[0])∧(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])) →* TRUE))
(2) -> (6), if ((x[2] →* x[6])∧(y[2] →* y[6])∧(&&(&&(>@z(x[2], 0@z), >=@z(0@z, x[2])), >@z(y[2], 0@z)) →* TRUE))
(3) -> (5), if ((x[3] →* x[5])∧(y[3] →* y[5])∧(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])) →* TRUE))
(4) -> (1), if ((y[4] →* y[1])∧(-@z(x[4], 1@z) →* x[1]))
(4) -> (2), if ((y[4] →* y[2])∧(-@z(x[4], 1@z) →* x[2]))
(4) -> (3), if ((y[4] →* y[3])∧(-@z(x[4], 1@z) →* x[3]))
(4) -> (7), if ((y[4] →* y[7])∧(-@z(x[4], 1@z) →* x[7]))
(4) -> (9), if ((y[4] →* y[9])∧(-@z(x[4], 1@z) →* x[9]))
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(5) -> (1), if ((y[5] →* y[1])∧(x[5] →* x[1]))
(5) -> (2), if ((y[5] →* y[2])∧(x[5] →* x[2]))
(5) -> (3), if ((y[5] →* y[3])∧(x[5] →* x[3]))
(5) -> (7), if ((y[5] →* y[7])∧(x[5] →* x[7]))
(5) -> (9), if ((y[5] →* y[9])∧(x[5] →* x[9]))
(5) -> (10), if ((y[5] →* y[10])∧(x[5] →* x[10]))
(6) -> (1), if ((-@z(y[6], 1@z) →* y[1])∧(x[6] →* x[1]))
(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) -> (7), if ((-@z(y[6], 1@z) →* y[7])∧(x[6] →* x[7]))
(6) -> (9), if ((-@z(y[6], 1@z) →* y[9])∧(x[6] →* x[9]))
(6) -> (10), if ((-@z(y[6], 1@z) →* y[10])∧(x[6] →* x[10]))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
(8) -> (1), if ((y[8] →* y[1])∧(-@z(x[8], 1@z) →* x[1]))
(8) -> (2), if ((y[8] →* y[2])∧(-@z(x[8], 1@z) →* x[2]))
(8) -> (3), if ((y[8] →* y[3])∧(-@z(x[8], 1@z) →* x[3]))
(8) -> (7), if ((y[8] →* y[7])∧(-@z(x[8], 1@z) →* x[7]))
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(8) -> (10), if ((y[8] →* y[10])∧(-@z(x[8], 1@z) →* x[10]))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(11) -> (1), if ((-@z(y[11], 1@z) →* y[1])∧(x[11] →* x[1]))
(11) -> (2), if ((-@z(y[11], 1@z) →* y[2])∧(x[11] →* x[2]))
(11) -> (3), if ((-@z(y[11], 1@z) →* y[3])∧(x[11] →* x[3]))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(11) -> (9), if ((-@z(y[11], 1@z) →* y[9])∧(x[11] →* x[9]))
(11) -> (10), if ((-@z(y[11], 1@z) →* y[10])∧(x[11] →* x[10]))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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): COND_EVAL(TRUE, x[0], y[0]) → EVAL(x[0], y[0])
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])
(2): EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(x[2], 0@z), >=@z(0@z, x[2])), >@z(y[2], 0@z)), x[2], y[2])
(3): EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(5): COND_EVAL5(TRUE, x[5], y[5]) → EVAL(x[5], y[5])
(6): COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z))
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(0) -> (1), if ((y[0] →* y[1])∧(x[0] →* x[1]))
(0) -> (2), if ((y[0] →* y[2])∧(x[0] →* x[2]))
(0) -> (3), if ((y[0] →* y[3])∧(x[0] →* x[3]))
(0) -> (7), if ((y[0] →* y[7])∧(x[0] →* x[7]))
(0) -> (9), if ((y[0] →* y[9])∧(x[0] →* x[9]))
(0) -> (10), if ((y[0] →* y[10])∧(x[0] →* x[10]))
(1) -> (0), if ((x[1] →* x[0])∧(y[1] →* y[0])∧(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])) →* TRUE))
(2) -> (6), if ((x[2] →* x[6])∧(y[2] →* y[6])∧(&&(&&(>@z(x[2], 0@z), >=@z(0@z, x[2])), >@z(y[2], 0@z)) →* TRUE))
(3) -> (5), if ((x[3] →* x[5])∧(y[3] →* y[5])∧(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])) →* TRUE))
(4) -> (1), if ((y[4] →* y[1])∧(-@z(x[4], 1@z) →* x[1]))
(4) -> (2), if ((y[4] →* y[2])∧(-@z(x[4], 1@z) →* x[2]))
(4) -> (3), if ((y[4] →* y[3])∧(-@z(x[4], 1@z) →* x[3]))
(4) -> (7), if ((y[4] →* y[7])∧(-@z(x[4], 1@z) →* x[7]))
(4) -> (9), if ((y[4] →* y[9])∧(-@z(x[4], 1@z) →* x[9]))
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(5) -> (1), if ((y[5] →* y[1])∧(x[5] →* x[1]))
(5) -> (2), if ((y[5] →* y[2])∧(x[5] →* x[2]))
(5) -> (3), if ((y[5] →* y[3])∧(x[5] →* x[3]))
(5) -> (7), if ((y[5] →* y[7])∧(x[5] →* x[7]))
(5) -> (9), if ((y[5] →* y[9])∧(x[5] →* x[9]))
(5) -> (10), if ((y[5] →* y[10])∧(x[5] →* x[10]))
(6) -> (1), if ((-@z(y[6], 1@z) →* y[1])∧(x[6] →* x[1]))
(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) -> (7), if ((-@z(y[6], 1@z) →* y[7])∧(x[6] →* x[7]))
(6) -> (9), if ((-@z(y[6], 1@z) →* y[9])∧(x[6] →* x[9]))
(6) -> (10), if ((-@z(y[6], 1@z) →* y[10])∧(x[6] →* x[10]))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
(8) -> (1), if ((y[8] →* y[1])∧(-@z(x[8], 1@z) →* x[1]))
(8) -> (2), if ((y[8] →* y[2])∧(-@z(x[8], 1@z) →* x[2]))
(8) -> (3), if ((y[8] →* y[3])∧(-@z(x[8], 1@z) →* x[3]))
(8) -> (7), if ((y[8] →* y[7])∧(-@z(x[8], 1@z) →* x[7]))
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(8) -> (10), if ((y[8] →* y[10])∧(-@z(x[8], 1@z) →* x[10]))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(11) -> (1), if ((-@z(y[11], 1@z) →* y[1])∧(x[11] →* x[1]))
(11) -> (2), if ((-@z(y[11], 1@z) →* y[2])∧(x[11] →* x[2]))
(11) -> (3), if ((-@z(y[11], 1@z) →* y[3])∧(x[11] →* x[3]))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(11) -> (9), if ((-@z(y[11], 1@z) →* y[9])∧(x[11] →* x[9]))
(11) -> (10), if ((-@z(y[11], 1@z) →* y[10])∧(x[11] →* x[10]))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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, y) → EVAL(x, y) the following chains were created:
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1]), COND_EVAL(TRUE, x[0], y[0]) → EVAL(x[0], y[0]) which results in the following constraint:
(1) (y[1]=y[0]∧x[1]=x[0]∧&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1]))=TRUE ⇒ COND_EVAL(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL(TRUE, x[0], y[0])≥EVAL(x[0], y[0])∧(UIncreasing(EVAL(x[0], y[0])), ≥))
We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(2) (>=@z(0@z, y[1])=TRUE∧>@z(y[1], 0@z)=TRUE∧>=@z(0@z, x[1])=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(x[1], y[1])∧(UIncreasing(EVAL(x[0], y[0])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) ((-1)y[1] ≥ 0∧-1 + y[1] ≥ 0∧(-1)x[1] ≥ 0 ⇒ (UIncreasing(EVAL(x[0], y[0])), ≥)∧-2 + (-1)Bound ≥ 0∧-1 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) ((-1)y[1] ≥ 0∧-1 + y[1] ≥ 0∧(-1)x[1] ≥ 0 ⇒ (UIncreasing(EVAL(x[0], y[0])), ≥)∧-2 + (-1)Bound ≥ 0∧-1 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + y[1] ≥ 0∧(-1)y[1] ≥ 0∧(-1)x[1] ≥ 0 ⇒ (UIncreasing(EVAL(x[0], y[0])), ≥)∧-1 ≥ 0∧-2 + (-1)Bound ≥ 0)
We solved constraint (5) using rule (IDP_SMT_SPLIT).
For Pair EVAL(x, y) → COND_EVAL(&&(&&(>@z(y, 0@z), >=@z(0@z, x)), >=@z(0@z, y)), x, y) the following chains were created:
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1]) which results in the following constraint:
(6) (EVAL(x[1], y[1])≥NonInfC∧EVAL(x[1], y[1])≥COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])∧(UIncreasing(COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])), ≥))
We simplified constraint (6) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(7) ((UIncreasing(COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (7) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(8) ((UIncreasing(COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (8) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(9) ((UIncreasing(COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])), ≥)∧1 ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(10) ((UIncreasing(COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧1 ≥ 0∧0 = 0)
For Pair EVAL(x, y) → COND_EVAL2(&&(&&(>@z(x, 0@z), >=@z(0@z, x)), >@z(y, 0@z)), x, y) the following chains were created:
- We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(x[2], 0@z), >=@z(0@z, x[2])), >@z(y[2], 0@z)), x[2], y[2]) which results in the following constraint:
(11) (EVAL(x[2], y[2])≥NonInfC∧EVAL(x[2], y[2])≥COND_EVAL2(&&(&&(>@z(x[2], 0@z), >=@z(0@z, x[2])), >@z(y[2], 0@z)), x[2], y[2])∧(UIncreasing(COND_EVAL2(&&(&&(>@z(x[2], 0@z), >=@z(0@z, x[2])), >@z(y[2], 0@z)), x[2], y[2])), ≥))
We simplified constraint (11) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(12) ((UIncreasing(COND_EVAL2(&&(&&(>@z(x[2], 0@z), >=@z(0@z, x[2])), >@z(y[2], 0@z)), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (12) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(13) ((UIncreasing(COND_EVAL2(&&(&&(>@z(x[2], 0@z), >=@z(0@z, x[2])), >@z(y[2], 0@z)), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (13) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(14) (0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL2(&&(&&(>@z(x[2], 0@z), >=@z(0@z, x[2])), >@z(y[2], 0@z)), x[2], y[2])), ≥))
We simplified constraint (14) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(15) (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL2(&&(&&(>@z(x[2], 0@z), >=@z(0@z, x[2])), >@z(y[2], 0@z)), x[2], y[2])), ≥))
For Pair EVAL(x, y) → COND_EVAL5(&&(&&(>@z(x, 0@z), >=@z(0@z, x)), >=@z(0@z, y)), x, y) the following chains were created:
- We consider the chain EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3]) which results in the following constraint:
(16) (EVAL(x[3], y[3])≥NonInfC∧EVAL(x[3], y[3])≥COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])∧(UIncreasing(COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])), ≥))
We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17) ((UIncreasing(COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18) ((UIncreasing(COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19) ((UIncreasing(COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (19) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(20) (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])), ≥)∧0 = 0∧1 ≥ 0)
For Pair COND_EVAL1(TRUE, x, y) → EVAL(-@z(x, 1@z), y) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]), COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]) which results in the following constraint:
(21) (x[10]=x[4]∧y[10]=y[4]∧>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL1(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (21) using rule (III) which results in the following new constraint:
(22) (>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL1(TRUE, x[10], y[10])≥EVAL(-@z(x[10], 1@z), y[10])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (22) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(23) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (23) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(24) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (24) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(25) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (25) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(26) (-1 + x[10] ≥ 0 ⇒ 0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(27) (x[10] ≥ 0 ⇒ 0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
For Pair COND_EVAL5(TRUE, x, y) → EVAL(x, y) the following chains were created:
- We consider the chain EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3]), COND_EVAL5(TRUE, x[5], y[5]) → EVAL(x[5], y[5]) which results in the following constraint:
(28) (y[3]=y[5]∧x[3]=x[5]∧&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3]))=TRUE ⇒ COND_EVAL5(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL5(TRUE, x[5], y[5])≥EVAL(x[5], y[5])∧(UIncreasing(EVAL(x[5], y[5])), ≥))
We simplified constraint (28) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(29) (>=@z(0@z, y[3])=TRUE∧>@z(x[3], 0@z)=TRUE∧>=@z(0@z, x[3])=TRUE ⇒ COND_EVAL5(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL5(TRUE, x[3], y[3])≥EVAL(x[3], y[3])∧(UIncreasing(EVAL(x[5], y[5])), ≥))
We simplified constraint (29) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(30) ((-1)y[3] ≥ 0∧-1 + x[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], y[5])), ≥)∧0 ≥ 0∧-1 ≥ 0)
We simplified constraint (30) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(31) ((-1)y[3] ≥ 0∧-1 + x[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], y[5])), ≥)∧0 ≥ 0∧-1 ≥ 0)
We simplified constraint (31) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(32) ((-1)x[3] ≥ 0∧-1 + x[3] ≥ 0∧(-1)y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], y[5])), ≥)∧-1 ≥ 0∧0 ≥ 0)
We solved constraint (32) using rule (IDP_SMT_SPLIT).
For Pair COND_EVAL2(TRUE, x, y) → EVAL(x, -@z(y, 1@z)) the following chains were created:
- We consider the chain EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(x[2], 0@z), >=@z(0@z, x[2])), >@z(y[2], 0@z)), x[2], y[2]), COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z)) which results in the following constraint:
(33) (x[2]=x[6]∧y[2]=y[6]∧&&(&&(>@z(x[2], 0@z), >=@z(0@z, x[2])), >@z(y[2], 0@z))=TRUE ⇒ COND_EVAL2(TRUE, x[6], y[6])≥NonInfC∧COND_EVAL2(TRUE, x[6], y[6])≥EVAL(x[6], -@z(y[6], 1@z))∧(UIncreasing(EVAL(x[6], -@z(y[6], 1@z))), ≥))
We simplified constraint (33) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(34) (>@z(y[2], 0@z)=TRUE∧>@z(x[2], 0@z)=TRUE∧>=@z(0@z, x[2])=TRUE ⇒ COND_EVAL2(TRUE, x[2], y[2])≥NonInfC∧COND_EVAL2(TRUE, x[2], y[2])≥EVAL(x[2], -@z(y[2], 1@z))∧(UIncreasing(EVAL(x[6], -@z(y[6], 1@z))), ≥))
We simplified constraint (34) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(35) (y[2] + -1 ≥ 0∧-1 + x[2] ≥ 0∧(-1)x[2] ≥ 0 ⇒ (UIncreasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧-2 + (-1)Bound ≥ 0∧-1 ≥ 0)
We simplified constraint (35) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(36) (y[2] + -1 ≥ 0∧-1 + x[2] ≥ 0∧(-1)x[2] ≥ 0 ⇒ (UIncreasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧-2 + (-1)Bound ≥ 0∧-1 ≥ 0)
We simplified constraint (36) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(37) (-1 + x[2] ≥ 0∧(-1)x[2] ≥ 0∧y[2] + -1 ≥ 0 ⇒ -1 ≥ 0∧(UIncreasing(EVAL(x[6], -@z(y[6], 1@z))), ≥)∧-2 + (-1)Bound ≥ 0)
We solved constraint (37) using rule (IDP_SMT_SPLIT).
For Pair EVAL(x, y) → COND_EVAL3(&&(>@z(y, 0@z), >=@z(0@z, x)), x, y) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(38) (EVAL(x[7], y[7])≥NonInfC∧EVAL(x[7], y[7])≥COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥))
We simplified constraint (38) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(39) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (39) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(40) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (40) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(41) (0 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0)
We simplified constraint (41) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(42) (0 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
For Pair COND_EVAL4(TRUE, x, y) → EVAL(-@z(x, 1@z), y) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]), COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]) which results in the following constraint:
(43) (y[9]=y[8]∧&&(>@z(y[9], 0@z), >@z(x[9], 0@z))=TRUE∧x[9]=x[8] ⇒ COND_EVAL4(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL4(TRUE, x[8], y[8])≥EVAL(-@z(x[8], 1@z), y[8])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (43) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(44) (>@z(y[9], 0@z)=TRUE∧>@z(x[9], 0@z)=TRUE ⇒ COND_EVAL4(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL4(TRUE, x[9], y[9])≥EVAL(-@z(x[9], 1@z), y[9])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (44) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(45) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (45) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(46) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (46) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(47) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
We simplified constraint (47) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(48) (y[9] ≥ 0∧-1 + x[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
We simplified constraint (48) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(49) (y[9] ≥ 0∧x[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
For Pair EVAL(x, y) → COND_EVAL4(&&(>@z(y, 0@z), >@z(x, 0@z)), x, y) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(50) (EVAL(x[9], y[9])≥NonInfC∧EVAL(x[9], y[9])≥COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥))
We simplified constraint (50) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(51) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (51) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(52) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (52) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(53) (0 ≥ 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0)
We simplified constraint (53) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(54) (0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
For Pair EVAL(x, y) → COND_EVAL1(>@z(x, 0@z), x, y) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(55) (EVAL(x[10], y[10])≥NonInfC∧EVAL(x[10], y[10])≥COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥))
We simplified constraint (55) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(56) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (56) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(57) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (57) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(58) (0 ≥ 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0)
We simplified constraint (58) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(59) (0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧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[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]), COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)) which results in the following constraint:
(60) (&&(>@z(y[7], 0@z), >=@z(0@z, x[7]))=TRUE∧y[7]=y[11]∧x[7]=x[11] ⇒ COND_EVAL3(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL3(TRUE, x[11], y[11])≥EVAL(x[11], -@z(y[11], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (60) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(61) (>@z(y[7], 0@z)=TRUE∧>=@z(0@z, x[7])=TRUE ⇒ COND_EVAL3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL3(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (61) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(62) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (62) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(63) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (63) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(64) ((-1)x[7] ≥ 0∧-1 + y[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
We simplified constraint (64) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(65) (x[7] ≥ 0∧-1 + y[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
We simplified constraint (65) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(66) (x[7] ≥ 0∧y[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL(TRUE, x, y) → EVAL(x, y)
- EVAL(x, y) → COND_EVAL(&&(&&(>@z(y, 0@z), >=@z(0@z, x)), >=@z(0@z, y)), x, y)
- ((UIncreasing(COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧1 ≥ 0∧0 = 0)
- EVAL(x, y) → COND_EVAL2(&&(&&(>@z(x, 0@z), >=@z(0@z, x)), >@z(y, 0@z)), x, y)
- (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL2(&&(&&(>@z(x[2], 0@z), >=@z(0@z, x[2])), >@z(y[2], 0@z)), x[2], y[2])), ≥))
- EVAL(x, y) → COND_EVAL5(&&(&&(>@z(x, 0@z), >=@z(0@z, x)), >=@z(0@z, y)), x, y)
- (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])), ≥)∧0 = 0∧1 ≥ 0)
- COND_EVAL1(TRUE, x, y) → EVAL(-@z(x, 1@z), y)
- (x[10] ≥ 0 ⇒ 0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
- COND_EVAL5(TRUE, x, y) → EVAL(x, y)
- COND_EVAL2(TRUE, x, y) → EVAL(x, -@z(y, 1@z))
- EVAL(x, y) → COND_EVAL3(&&(>@z(y, 0@z), >=@z(0@z, x)), x, y)
- (0 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
- COND_EVAL4(TRUE, x, y) → EVAL(-@z(x, 1@z), y)
- (y[9] ≥ 0∧x[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
- EVAL(x, y) → COND_EVAL4(&&(>@z(y, 0@z), >@z(x, 0@z)), x, y)
- (0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
- EVAL(x, y) → COND_EVAL1(>@z(x, 0@z), x, y)
- (0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 = 0∧0 = 0)
- COND_EVAL3(TRUE, x, y) → EVAL(x, -@z(y, 1@z))
- (x[7] ≥ 0∧y[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
The constraints for P> respective Pbound 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 Pbound.
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(COND_EVAL5(x1, x2, x3)) = -1 + (-1)x1
POL(0@z) = 0
POL(COND_EVAL1(x1, x2, x3)) = -1
POL(TRUE) = 1
POL(&&(x1, x2)) = 1
POL(COND_EVAL(x1, x2, x3)) = -1 + (-1)x1
POL(COND_EVAL4(x1, x2, x3)) = (-1)x1
POL(FALSE) = 2
POL(>@z(x1, x2)) = -1
POL(COND_EVAL3(x1, x2, x3)) = (-1)x1
POL(>=@z(x1, x2)) = -1
POL(COND_EVAL2(x1, x2, x3)) = -1 + (-1)x1
POL(EVAL(x1, x2)) = -1
POL(1@z) = 1
POL(undefined) = -1
The following pairs are in P>:
EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(x[2], 0@z), >=@z(0@z, x[2])), >@z(y[2], 0@z)), x[2], y[2])
The following pairs are in Pbound:
COND_EVAL(TRUE, x[0], y[0]) → EVAL(x[0], y[0])
COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z))
The following pairs are in P≥:
COND_EVAL(TRUE, x[0], y[0]) → EVAL(x[0], y[0])
EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])
EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])
COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
COND_EVAL5(TRUE, x[5], y[5]) → EVAL(x[5], y[5])
COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z))
EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 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
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])
(2): EVAL(x[2], y[2]) → COND_EVAL2(&&(&&(>@z(x[2], 0@z), >=@z(0@z, x[2])), >@z(y[2], 0@z)), x[2], y[2])
(3): EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(5): COND_EVAL5(TRUE, x[5], y[5]) → EVAL(x[5], y[5])
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(4) -> (3), if ((y[4] →* y[3])∧(-@z(x[4], 1@z) →* x[3]))
(11) -> (2), if ((-@z(y[11], 1@z) →* y[2])∧(x[11] →* x[2]))
(11) -> (1), if ((-@z(y[11], 1@z) →* y[1])∧(x[11] →* x[1]))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(11) -> (9), if ((-@z(y[11], 1@z) →* y[9])∧(x[11] →* x[9]))
(5) -> (2), if ((y[5] →* y[2])∧(x[5] →* x[2]))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
(4) -> (2), if ((y[4] →* y[2])∧(-@z(x[4], 1@z) →* x[2]))
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(5) -> (1), if ((y[5] →* y[1])∧(x[5] →* x[1]))
(8) -> (7), if ((y[8] →* y[7])∧(-@z(x[8], 1@z) →* x[7]))
(11) -> (3), if ((-@z(y[11], 1@z) →* y[3])∧(x[11] →* x[3]))
(4) -> (1), if ((y[4] →* y[1])∧(-@z(x[4], 1@z) →* x[1]))
(8) -> (10), if ((y[8] →* y[10])∧(-@z(x[8], 1@z) →* x[10]))
(8) -> (2), if ((y[8] →* y[2])∧(-@z(x[8], 1@z) →* x[2]))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
(11) -> (10), if ((-@z(y[11], 1@z) →* y[10])∧(x[11] →* x[10]))
(5) -> (3), if ((y[5] →* y[3])∧(x[5] →* x[3]))
(5) -> (9), if ((y[5] →* y[9])∧(x[5] →* x[9]))
(4) -> (7), if ((y[4] →* y[7])∧(-@z(x[4], 1@z) →* x[7]))
(4) -> (9), if ((y[4] →* y[9])∧(-@z(x[4], 1@z) →* x[9]))
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(5) -> (7), if ((y[5] →* y[7])∧(x[5] →* x[7]))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(3) -> (5), if ((x[3] →* x[5])∧(y[3] →* y[5])∧(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])) →* TRUE))
(8) -> (3), if ((y[8] →* y[3])∧(-@z(x[8], 1@z) →* x[3]))
(8) -> (1), if ((y[8] →* y[1])∧(-@z(x[8], 1@z) →* x[1]))
(5) -> (10), if ((y[5] →* y[10])∧(x[5] →* x[10]))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(5): COND_EVAL5(TRUE, x[5], y[5]) → EVAL(x[5], y[5])
(3): EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(11) -> (10), if ((-@z(y[11], 1@z) →* y[10])∧(x[11] →* x[10]))
(5) -> (3), if ((y[5] →* y[3])∧(x[5] →* x[3]))
(4) -> (3), if ((y[4] →* y[3])∧(-@z(x[4], 1@z) →* x[3]))
(5) -> (9), if ((y[5] →* y[9])∧(x[5] →* x[9]))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(11) -> (9), if ((-@z(y[11], 1@z) →* y[9])∧(x[11] →* x[9]))
(4) -> (7), if ((y[4] →* y[7])∧(-@z(x[4], 1@z) →* x[7]))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
(4) -> (9), if ((y[4] →* y[9])∧(-@z(x[4], 1@z) →* x[9]))
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(8) -> (7), if ((y[8] →* y[7])∧(-@z(x[8], 1@z) →* x[7]))
(11) -> (3), if ((-@z(y[11], 1@z) →* y[3])∧(x[11] →* x[3]))
(8) -> (10), if ((y[8] →* y[10])∧(-@z(x[8], 1@z) →* x[10]))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(5) -> (7), if ((y[5] →* y[7])∧(x[5] →* x[7]))
(3) -> (5), if ((x[3] →* x[5])∧(y[3] →* y[5])∧(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])) →* TRUE))
(8) -> (3), if ((y[8] →* y[3])∧(-@z(x[8], 1@z) →* x[3]))
(5) -> (10), if ((y[5] →* y[10])∧(x[5] →* x[10]))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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_EVAL5(TRUE, x[5], y[5]) → EVAL(x[5], y[5]) the following chains were created:
- We consider the chain EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3]), COND_EVAL5(TRUE, x[5], y[5]) → EVAL(x[5], y[5]) which results in the following constraint:
(1) (y[3]=y[5]∧x[3]=x[5]∧&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3]))=TRUE ⇒ COND_EVAL5(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL5(TRUE, x[5], y[5])≥EVAL(x[5], y[5])∧(UIncreasing(EVAL(x[5], y[5])), ≥))
We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(2) (>=@z(0@z, y[3])=TRUE∧>@z(x[3], 0@z)=TRUE∧>=@z(0@z, x[3])=TRUE ⇒ COND_EVAL5(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL5(TRUE, x[3], y[3])≥EVAL(x[3], y[3])∧(UIncreasing(EVAL(x[5], y[5])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) ((-1)y[3] ≥ 0∧-1 + x[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], y[5])), ≥)∧-1 + (-1)Bound + y[3] ≥ 0∧-1 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) ((-1)y[3] ≥ 0∧-1 + x[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], y[5])), ≥)∧-1 + (-1)Bound + y[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] ≥ 0∧-1 + x[3] ≥ 0∧(-1)y[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], y[5])), ≥)∧-1 ≥ 0∧-1 + (-1)Bound + y[3] ≥ 0)
We solved constraint (5) using rule (IDP_SMT_SPLIT).
For Pair EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3]) the following chains were created:
- We consider the chain EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3]) which results in the following constraint:
(6) (EVAL(x[3], y[3])≥NonInfC∧EVAL(x[3], y[3])≥COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])∧(UIncreasing(COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])), ≥))
We simplified constraint (6) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(7) ((UIncreasing(COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (7) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(8) ((UIncreasing(COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (8) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(9) (0 ≥ 0∧(UIncreasing(COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])), ≥)∧0 ≥ 0)
We simplified constraint (9) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(10) ((UIncreasing(COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0)
For Pair COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]), COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]) which results in the following constraint:
(11) (x[10]=x[4]∧y[10]=y[4]∧>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL1(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (11) using rule (III) which results in the following new constraint:
(12) (>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL1(TRUE, x[10], y[10])≥EVAL(-@z(x[10], 1@z), y[10])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (12) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(13) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (13) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(14) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (14) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(15) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
We simplified constraint (15) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(16) (-1 + x[10] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 = 0)
We simplified constraint (16) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(17) (x[10] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 = 0)
For Pair EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(18) (EVAL(x[10], y[10])≥NonInfC∧EVAL(x[10], y[10])≥COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥))
We simplified constraint (18) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(19) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (19) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(20) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (20) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(21) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (21) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(22) (0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
For Pair EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(23) (EVAL(x[7], y[7])≥NonInfC∧EVAL(x[7], y[7])≥COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥))
We simplified constraint (23) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(24) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (24) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(25) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(26) (0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥))
We simplified constraint (26) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(27) (0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0)
For Pair COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]), COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]) which results in the following constraint:
(28) (y[9]=y[8]∧&&(>@z(y[9], 0@z), >@z(x[9], 0@z))=TRUE∧x[9]=x[8] ⇒ COND_EVAL4(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL4(TRUE, x[8], y[8])≥EVAL(-@z(x[8], 1@z), y[8])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (28) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(29) (>@z(y[9], 0@z)=TRUE∧>@z(x[9], 0@z)=TRUE ⇒ COND_EVAL4(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL4(TRUE, x[9], y[9])≥EVAL(-@z(x[9], 1@z), y[9])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (29) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(30) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧-1 + (-1)Bound + y[9] ≥ 0∧0 ≥ 0)
We simplified constraint (30) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(31) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧-1 + (-1)Bound + y[9] ≥ 0∧0 ≥ 0)
We simplified constraint (31) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(32) (-1 + x[9] ≥ 0∧y[9] + -1 ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧-1 + (-1)Bound + y[9] ≥ 0)
We simplified constraint (32) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(33) (-1 + x[9] ≥ 0∧y[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧(-1)Bound + y[9] ≥ 0)
We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(34) (x[9] ≥ 0∧y[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧(-1)Bound + y[9] ≥ 0)
For Pair EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(35) (EVAL(x[9], y[9])≥NonInfC∧EVAL(x[9], y[9])≥COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥))
We simplified constraint (35) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(36) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (36) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(37) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (37) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(38) (0 ≥ 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0)
We simplified constraint (38) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(39) (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 = 0)
For Pair COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]), COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)) which results in the following constraint:
(40) (&&(>@z(y[7], 0@z), >=@z(0@z, x[7]))=TRUE∧y[7]=y[11]∧x[7]=x[11] ⇒ COND_EVAL3(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL3(TRUE, x[11], y[11])≥EVAL(x[11], -@z(y[11], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (40) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(41) (>@z(y[7], 0@z)=TRUE∧>=@z(0@z, x[7])=TRUE ⇒ COND_EVAL3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL3(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (41) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(42) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (42) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(43) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (43) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(44) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
We simplified constraint (44) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(45) (-1 + y[7] ≥ 0∧x[7] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
We simplified constraint (45) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(46) (y[7] ≥ 0∧x[7] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL5(TRUE, x[5], y[5]) → EVAL(x[5], y[5])
- EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])
- ((UIncreasing(COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0)
- COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
- (x[10] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 = 0)
- EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
- (0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
- EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
- (0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0)
- COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
- (x[9] ≥ 0∧y[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧(-1)Bound + y[9] ≥ 0)
- EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
- (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 = 0)
- COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
- (y[7] ≥ 0∧x[7] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:
POL(COND_EVAL5(x1, x2, x3)) = -1 + x3
POL(-@z(x1, x2)) = x1 + (-1)x2
POL(0@z) = 0
POL(COND_EVAL1(x1, x2, x3)) = -1 + x3
POL(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(COND_EVAL4(x1, x2, x3)) = -1 + x3
POL(FALSE) = 1
POL(>@z(x1, x2)) = -1
POL(>=@z(x1, x2)) = -1
POL(COND_EVAL3(x1, x2, x3)) = -1 + x3
POL(EVAL(x1, x2)) = -1 + x2
POL(1@z) = 1
POL(undefined) = -1
The following pairs are in P>:
COND_EVAL5(TRUE, x[5], y[5]) → EVAL(x[5], y[5])
The following pairs are in Pbound:
COND_EVAL5(TRUE, x[5], y[5]) → EVAL(x[5], y[5])
COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
The following pairs are in P≥:
EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])
COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
At least the following rules have been oriented under context sensitive arithmetic replacement:
FALSE1 → &&(FALSE, FALSE)1
-@z1 ↔
FALSE1 → &&(FALSE, TRUE)1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(3): EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(11) -> (10), if ((-@z(y[11], 1@z) →* y[10])∧(x[11] →* x[10]))
(4) -> (3), if ((y[4] →* y[3])∧(-@z(x[4], 1@z) →* x[3]))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(11) -> (9), if ((-@z(y[11], 1@z) →* y[9])∧(x[11] →* x[9]))
(4) -> (7), if ((y[4] →* y[7])∧(-@z(x[4], 1@z) →* x[7]))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
(4) -> (9), if ((y[4] →* y[9])∧(-@z(x[4], 1@z) →* x[9]))
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(8) -> (7), if ((y[8] →* y[7])∧(-@z(x[8], 1@z) →* x[7]))
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(11) -> (3), if ((-@z(y[11], 1@z) →* y[3])∧(x[11] →* x[3]))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(8) -> (10), if ((y[8] →* y[10])∧(-@z(x[8], 1@z) →* x[10]))
(8) -> (3), if ((y[8] →* y[3])∧(-@z(x[8], 1@z) →* x[3]))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(11) -> (10), if ((-@z(y[11], 1@z) →* y[10])∧(x[11] →* x[10]))
(4) -> (9), if ((y[4] →* y[9])∧(-@z(x[4], 1@z) →* x[9]))
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(8) -> (7), if ((y[8] →* y[7])∧(-@z(x[8], 1@z) →* x[7]))
(8) -> (10), if ((y[8] →* y[10])∧(-@z(x[8], 1@z) →* x[10]))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(11) -> (9), if ((-@z(y[11], 1@z) →* y[9])∧(x[11] →* x[9]))
(4) -> (7), if ((y[4] →* y[7])∧(-@z(x[4], 1@z) →* x[7]))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]), COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(1) (-@z(x[4], 1@z)=x[9]∧x[10]=x[4]∧y[10]=y[4]∧y[4]=y[9]∧>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL1(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2) (>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL1(TRUE, x[10], y[10])≥EVAL(-@z(x[10], 1@z), y[10])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6) (-1 + x[10] ≥ 0 ⇒ 0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 = 0)
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (x[10] ≥ 0 ⇒ 0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 = 0)
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]), COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(8) (-@z(x[4], 1@z)=x[7]∧x[10]=x[4]∧y[10]=y[4]∧>@z(x[10], 0@z)=TRUE∧y[4]=y[7] ⇒ COND_EVAL1(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL1(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:
(9) (>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL1(TRUE, x[10], y[10])≥EVAL(-@z(x[10], 1@z), y[10])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 = 0∧0 = 0)
We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14) (x[10] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 = 0∧0 = 0)
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]), COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(15) (-@z(x[4], 1@z)=x[10]1∧x[10]=x[4]∧y[10]=y[4]∧y[4]=y[10]1∧>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL1(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (15) using rules (III), (IV) which results in the following new constraint:
(16) (>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL1(TRUE, x[10], y[10])≥EVAL(-@z(x[10], 1@z), y[10])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
We simplified constraint (19) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(20) (-1 + x[10] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
We simplified constraint (20) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(21) (x[10] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
For Pair EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(22) (EVAL(x[10], y[10])≥NonInfC∧EVAL(x[10], y[10])≥COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥))
We simplified constraint (22) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(23) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (23) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(24) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (24) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(25) (0 ≥ 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0)
We simplified constraint (25) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(26) (0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
For Pair EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(27) (EVAL(x[7], y[7])≥NonInfC∧EVAL(x[7], y[7])≥COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥))
We simplified constraint (27) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(28) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (28) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(29) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (29) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(30) (0 ≥ 0∧1 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥))
We simplified constraint (30) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(31) (0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 = 0∧0 = 0∧1 ≥ 0∧0 = 0)
For Pair COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]), COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]), EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(32) (y[9]=y[8]∧y[8]=y[9]1∧&&(>@z(y[9], 0@z), >@z(x[9], 0@z))=TRUE∧-@z(x[8], 1@z)=x[9]1∧x[9]=x[8] ⇒ COND_EVAL4(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL4(TRUE, x[8], y[8])≥EVAL(-@z(x[8], 1@z), y[8])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (32) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(33) (>@z(y[9], 0@z)=TRUE∧>@z(x[9], 0@z)=TRUE ⇒ COND_EVAL4(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL4(TRUE, x[9], y[9])≥EVAL(-@z(x[9], 1@z), y[9])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (33) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(34) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧1 + (-1)Bound + y[9] + x[9] ≥ 0∧1 ≥ 0)
We simplified constraint (34) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(35) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧1 + (-1)Bound + y[9] + x[9] ≥ 0∧1 ≥ 0)
We simplified constraint (35) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(36) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ 1 + (-1)Bound + y[9] + x[9] ≥ 0∧1 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (36) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(37) (y[9] ≥ 0∧-1 + x[9] ≥ 0 ⇒ 2 + (-1)Bound + y[9] + x[9] ≥ 0∧1 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(38) (y[9] ≥ 0∧x[9] ≥ 0 ⇒ 3 + (-1)Bound + y[9] + x[9] ≥ 0∧1 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]), COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]), EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(39) (y[9]=y[8]∧-@z(x[8], 1@z)=x[10]∧y[8]=y[10]∧&&(>@z(y[9], 0@z), >@z(x[9], 0@z))=TRUE∧x[9]=x[8] ⇒ COND_EVAL4(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL4(TRUE, x[8], y[8])≥EVAL(-@z(x[8], 1@z), y[8])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (39) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(40) (>@z(y[9], 0@z)=TRUE∧>@z(x[9], 0@z)=TRUE ⇒ COND_EVAL4(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL4(TRUE, x[9], y[9])≥EVAL(-@z(x[9], 1@z), y[9])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (40) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(41) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧1 + (-1)Bound + y[9] + x[9] ≥ 0∧1 ≥ 0)
We simplified constraint (41) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(42) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧1 + (-1)Bound + y[9] + x[9] ≥ 0∧1 ≥ 0)
We simplified constraint (42) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(43) (-1 + x[9] ≥ 0∧y[9] + -1 ≥ 0 ⇒ 1 + (-1)Bound + y[9] + x[9] ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧1 ≥ 0)
We simplified constraint (43) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(44) (-1 + x[9] ≥ 0∧y[9] ≥ 0 ⇒ 2 + (-1)Bound + y[9] + x[9] ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧1 ≥ 0)
We simplified constraint (44) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(45) (x[9] ≥ 0∧y[9] ≥ 0 ⇒ 3 + (-1)Bound + y[9] + x[9] ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧1 ≥ 0)
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]), COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]), EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(46) (-@z(x[8], 1@z)=x[7]∧y[8]=y[7]∧y[9]=y[8]∧&&(>@z(y[9], 0@z), >@z(x[9], 0@z))=TRUE∧x[9]=x[8] ⇒ COND_EVAL4(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL4(TRUE, x[8], y[8])≥EVAL(-@z(x[8], 1@z), y[8])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (46) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(47) (>@z(y[9], 0@z)=TRUE∧>@z(x[9], 0@z)=TRUE ⇒ COND_EVAL4(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL4(TRUE, x[9], y[9])≥EVAL(-@z(x[9], 1@z), y[9])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (47) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(48) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧1 + (-1)Bound + y[9] + x[9] ≥ 0∧1 ≥ 0)
We simplified constraint (48) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(49) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧1 + (-1)Bound + y[9] + x[9] ≥ 0∧1 ≥ 0)
We simplified constraint (49) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(50) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧1 + (-1)Bound + y[9] + x[9] ≥ 0)
We simplified constraint (50) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(51) (y[9] ≥ 0∧-1 + x[9] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧2 + (-1)Bound + y[9] + x[9] ≥ 0)
We simplified constraint (51) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(52) (y[9] ≥ 0∧x[9] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧3 + (-1)Bound + y[9] + x[9] ≥ 0)
For Pair EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(53) (EVAL(x[9], y[9])≥NonInfC∧EVAL(x[9], y[9])≥COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥))
We simplified constraint (53) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(54) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (54) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(55) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (55) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(56) (0 ≥ 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0)
We simplified constraint (56) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(57) (0 = 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
For Pair COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]), COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)), EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(58) (&&(>@z(y[7], 0@z), >=@z(0@z, x[7]))=TRUE∧-@z(y[11], 1@z)=y[7]1∧y[7]=y[11]∧x[11]=x[7]1∧x[7]=x[11] ⇒ COND_EVAL3(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL3(TRUE, x[11], y[11])≥EVAL(x[11], -@z(y[11], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (58) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(59) (>@z(y[7], 0@z)=TRUE∧>=@z(0@z, x[7])=TRUE ⇒ COND_EVAL3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL3(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (59) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(60) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (60) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(61) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (61) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(62) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (62) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(63) (-1 + y[7] ≥ 0∧x[7] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (63) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(64) (y[7] ≥ 0∧x[7] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]), COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)), EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(65) (&&(>@z(y[7], 0@z), >=@z(0@z, x[7]))=TRUE∧y[7]=y[11]∧x[11]=x[9]∧-@z(y[11], 1@z)=y[9]∧x[7]=x[11] ⇒ COND_EVAL3(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL3(TRUE, x[11], y[11])≥EVAL(x[11], -@z(y[11], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (65) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(66) (>@z(y[7], 0@z)=TRUE∧>=@z(0@z, x[7])=TRUE ⇒ COND_EVAL3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL3(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (66) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(67) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (67) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(68) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (68) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(69) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (69) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(70) (-1 + y[7] ≥ 0∧x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (70) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(71) (y[7] ≥ 0∧x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]), COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)), EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(72) (-@z(y[11], 1@z)=y[10]∧&&(>@z(y[7], 0@z), >=@z(0@z, x[7]))=TRUE∧y[7]=y[11]∧x[11]=x[10]∧x[7]=x[11] ⇒ COND_EVAL3(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL3(TRUE, x[11], y[11])≥EVAL(x[11], -@z(y[11], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (72) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(73) (>@z(y[7], 0@z)=TRUE∧>=@z(0@z, x[7])=TRUE ⇒ COND_EVAL3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL3(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (73) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(74) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (74) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(75) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (75) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(76) ((-1)x[7] ≥ 0∧-1 + y[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (76) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(77) (x[7] ≥ 0∧-1 + y[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (77) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(78) (x[7] ≥ 0∧y[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
- (x[10] ≥ 0 ⇒ 0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 = 0)
- (x[10] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 = 0∧0 = 0)
- (x[10] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
- EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
- (0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
- EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
- (0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 = 0∧0 = 0∧1 ≥ 0∧0 = 0)
- COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
- (y[9] ≥ 0∧x[9] ≥ 0 ⇒ 3 + (-1)Bound + y[9] + x[9] ≥ 0∧1 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
- (x[9] ≥ 0∧y[9] ≥ 0 ⇒ 3 + (-1)Bound + y[9] + x[9] ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧1 ≥ 0)
- (y[9] ≥ 0∧x[9] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧3 + (-1)Bound + y[9] + x[9] ≥ 0)
- EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
- (0 = 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
- COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
- (y[7] ≥ 0∧x[7] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
- (y[7] ≥ 0∧x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
- (x[7] ≥ 0∧y[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
The constraints for P> respective Pbound 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 Pbound.
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)) = x3 + x2
POL(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(COND_EVAL4(x1, x2, x3)) = x3 + x2 + (-1)x1
POL(FALSE) = -1
POL(>@z(x1, x2)) = 1
POL(COND_EVAL3(x1, x2, x3)) = -1 + x3 + x2 + (-1)x1
POL(>=@z(x1, x2)) = -1
POL(EVAL(x1, x2)) = 1 + x2 + x1
POL(1@z) = 1
POL(undefined) = -1
The following pairs are in P>:
EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
The following pairs are in Pbound:
COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
The following pairs are in P≥:
COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
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
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(11) -> (10), if ((-@z(y[11], 1@z) →* y[10])∧(x[11] →* x[10]))
(4) -> (9), if ((y[4] →* y[9])∧(-@z(x[4], 1@z) →* x[9]))
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(11) -> (9), if ((-@z(y[11], 1@z) →* y[9])∧(x[11] →* x[9]))
(4) -> (7), if ((y[4] →* y[7])∧(-@z(x[4], 1@z) →* x[7]))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(11) -> (10), if ((-@z(y[11], 1@z) →* y[10])∧(x[11] →* x[10]))
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(4) -> (7), if ((y[4] →* y[7])∧(-@z(x[4], 1@z) →* x[7]))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]), COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(1) (-@z(x[4], 1@z)=x[7]∧x[10]=x[4]∧y[10]=y[4]∧>@z(x[10], 0@z)=TRUE∧y[4]=y[7] ⇒ COND_EVAL1(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL1(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2) (>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL1(TRUE, x[10], y[10])≥EVAL(-@z(x[10], 1@z), y[10])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + x[10] ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + x[10] ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧-1 + (-1)Bound + x[10] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6) (-1 + x[10] ≥ 0 ⇒ 0 = 0∧0 = 0∧-1 + (-1)Bound + x[10] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (x[10] ≥ 0 ⇒ 0 = 0∧0 = 0∧(-1)Bound + x[10] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]), COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(8) (-@z(x[4], 1@z)=x[10]1∧x[10]=x[4]∧y[10]=y[4]∧y[4]=y[10]1∧>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL1(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:
(9) (>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL1(TRUE, x[10], y[10])≥EVAL(-@z(x[10], 1@z), y[10])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + x[10] ≥ 0∧0 ≥ 0)
We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + x[10] ≥ 0∧0 ≥ 0)
We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧-1 + (-1)Bound + x[10] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 = 0∧-1 + (-1)Bound + x[10] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14) (x[10] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 = 0∧(-1)Bound + x[10] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
For Pair EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(15) (EVAL(x[10], y[10])≥NonInfC∧EVAL(x[10], y[10])≥COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥))
We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (18) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(19) (0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
For Pair EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(20) (EVAL(x[7], y[7])≥NonInfC∧EVAL(x[7], y[7])≥COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥))
We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23) (0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥))
We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(24) (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 = 0)
For Pair COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]), COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)), EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(25) (&&(>@z(y[7], 0@z), >=@z(0@z, x[7]))=TRUE∧-@z(y[11], 1@z)=y[7]1∧y[7]=y[11]∧x[11]=x[7]1∧x[7]=x[11] ⇒ COND_EVAL3(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL3(TRUE, x[11], y[11])≥EVAL(x[11], -@z(y[11], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (25) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(26) (>@z(y[7], 0@z)=TRUE∧>=@z(0@z, x[7])=TRUE ⇒ COND_EVAL3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL3(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (26) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(27) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(28) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(29) ((-1)x[7] ≥ 0∧-1 + y[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (29) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(30) (x[7] ≥ 0∧-1 + y[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (30) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(31) (x[7] ≥ 0∧y[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]), COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)), EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(32) (-@z(y[11], 1@z)=y[10]∧&&(>@z(y[7], 0@z), >=@z(0@z, x[7]))=TRUE∧y[7]=y[11]∧x[11]=x[10]∧x[7]=x[11] ⇒ COND_EVAL3(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL3(TRUE, x[11], y[11])≥EVAL(x[11], -@z(y[11], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (32) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(33) (>@z(y[7], 0@z)=TRUE∧>=@z(0@z, x[7])=TRUE ⇒ COND_EVAL3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL3(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (33) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(34) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (34) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(35) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (35) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(36) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
We simplified constraint (36) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(37) (-1 + y[7] ≥ 0∧x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(38) (y[7] ≥ 0∧x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
- (x[10] ≥ 0 ⇒ 0 = 0∧0 = 0∧(-1)Bound + x[10] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
- (x[10] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 = 0∧(-1)Bound + x[10] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
- (0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
- EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
- (0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 = 0)
- COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
- (x[7] ≥ 0∧y[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
- (y[7] ≥ 0∧x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
The constraints for P> respective Pbound 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 Pbound.
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(>=@z(x1, x2)) = -1
POL(COND_EVAL3(x1, x2, x3)) = -1 + x2
POL(0@z) = 0
POL(COND_EVAL1(x1, x2, x3)) = -1 + x2
POL(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(EVAL(x1, x2)) = -1 + x1
POL(FALSE) = 0
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
The following pairs are in Pbound:
COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
The following pairs are in P≥:
EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
At least the following rules have been oriented under context sensitive arithmetic replacement:
FALSE1 → &&(FALSE, FALSE)1
-@z1 ↔
&&(TRUE, TRUE)1 ↔ TRUE1
FALSE1 → &&(FALSE, TRUE)1
FALSE1 → &&(TRUE, FALSE)1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(11) -> (10), if ((-@z(y[11], 1@z) →* y[10])∧(x[11] →* x[10]))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(1) (EVAL(x[7], y[7])≥NonInfC∧EVAL(x[7], y[7])≥COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥))
We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4) (0 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧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∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 = 0∧0 = 0)
For Pair COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]), COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)), EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(6) (&&(>@z(y[7], 0@z), >=@z(0@z, x[7]))=TRUE∧-@z(y[11], 1@z)=y[7]1∧y[7]=y[11]∧x[11]=x[7]1∧x[7]=x[11] ⇒ COND_EVAL3(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL3(TRUE, x[11], y[11])≥EVAL(x[11], -@z(y[11], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7) (>@z(y[7], 0@z)=TRUE∧>=@z(0@z, x[7])=TRUE ⇒ COND_EVAL3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL3(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧(-1)Bound + y[7] + (-1)x[7] ≥ 0∧0 ≥ 0)
We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧(-1)Bound + y[7] + (-1)x[7] ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10) ((-1)x[7] ≥ 0∧-1 + y[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧(-1)Bound + y[7] + (-1)x[7] ≥ 0∧0 ≥ 0)
We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11) (x[7] ≥ 0∧-1 + y[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧(-1)Bound + y[7] + x[7] ≥ 0∧0 ≥ 0)
We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12) (x[7] ≥ 0∧y[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧1 + (-1)Bound + y[7] + x[7] ≥ 0∧0 ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
- (0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 = 0∧0 = 0)
- COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
- (x[7] ≥ 0∧y[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧1 + (-1)Bound + y[7] + x[7] ≥ 0∧0 ≥ 0)
The constraints for P> respective Pbound 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 Pbound.
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(>=@z(x1, x2)) = -1
POL(COND_EVAL3(x1, x2, x3)) = 2 + x3 + (-1)x2 + (-1)x1
POL(0@z) = 0
POL(TRUE) = 2
POL(&&(x1, x2)) = 2
POL(EVAL(x1, x2)) = x2 + (-1)x1
POL(FALSE) = 2
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
The following pairs are in Pbound:
COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
The following pairs are in P≥:
EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], 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 → &&(TRUE, FALSE)1
&&(FALSE, TRUE)1 ↔ FALSE1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ 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:
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(4) -> (9), if ((y[4] →* y[9])∧(-@z(x[4], 1@z) →* x[9]))
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(8) -> (7), if ((y[8] →* y[7])∧(-@z(x[8], 1@z) →* x[7]))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(11) -> (9), if ((-@z(y[11], 1@z) →* y[9])∧(x[11] →* x[9]))
(4) -> (7), if ((y[4] →* y[7])∧(-@z(x[4], 1@z) →* x[7]))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(8) -> (7), if ((y[8] →* y[7])∧(-@z(x[8], 1@z) →* x[7]))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(11) -> (9), if ((-@z(y[11], 1@z) →* y[9])∧(x[11] →* x[9]))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(1) (EVAL(x[7], y[7])≥NonInfC∧EVAL(x[7], y[7])≥COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥))
We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4) (0 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧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∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 = 0∧0 = 0)
For Pair COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]), COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]), EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(6) (y[9]=y[8]∧y[8]=y[9]1∧&&(>@z(y[9], 0@z), >@z(x[9], 0@z))=TRUE∧-@z(x[8], 1@z)=x[9]1∧x[9]=x[8] ⇒ COND_EVAL4(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL4(TRUE, x[8], y[8])≥EVAL(-@z(x[8], 1@z), y[8])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7) (>@z(y[9], 0@z)=TRUE∧>@z(x[9], 0@z)=TRUE ⇒ COND_EVAL4(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL4(TRUE, x[9], y[9])≥EVAL(-@z(x[9], 1@z), y[9])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧-2 + (-1)Bound + x[9] ≥ 0∧0 ≥ 0)
We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧-2 + (-1)Bound + x[9] ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10) (-1 + x[9] ≥ 0∧y[9] + -1 ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧-2 + (-1)Bound + x[9] ≥ 0)
We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11) (-1 + x[9] ≥ 0∧y[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧-2 + (-1)Bound + x[9] ≥ 0)
We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12) (x[9] ≥ 0∧y[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧-1 + (-1)Bound + x[9] ≥ 0)
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]), COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]), EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(13) (-@z(x[8], 1@z)=x[7]∧y[8]=y[7]∧y[9]=y[8]∧&&(>@z(y[9], 0@z), >@z(x[9], 0@z))=TRUE∧x[9]=x[8] ⇒ COND_EVAL4(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL4(TRUE, x[8], y[8])≥EVAL(-@z(x[8], 1@z), y[8])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (13) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(14) (>@z(y[9], 0@z)=TRUE∧>@z(x[9], 0@z)=TRUE ⇒ COND_EVAL4(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL4(TRUE, x[9], y[9])≥EVAL(-@z(x[9], 1@z), y[9])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(15) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧-2 + (-1)Bound + x[9] ≥ 0∧0 ≥ 0)
We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(16) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧-2 + (-1)Bound + x[9] ≥ 0∧0 ≥ 0)
We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(17) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧-2 + (-1)Bound + x[9] ≥ 0)
We simplified constraint (17) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(18) (y[9] ≥ 0∧-1 + x[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧-2 + (-1)Bound + x[9] ≥ 0)
We simplified constraint (18) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(19) (y[9] ≥ 0∧x[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧-1 + (-1)Bound + x[9] ≥ 0)
For Pair EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(20) (EVAL(x[9], y[9])≥NonInfC∧EVAL(x[9], y[9])≥COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥))
We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(24) (0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
For Pair COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]), COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)), EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(25) (&&(>@z(y[7], 0@z), >=@z(0@z, x[7]))=TRUE∧-@z(y[11], 1@z)=y[7]1∧y[7]=y[11]∧x[11]=x[7]1∧x[7]=x[11] ⇒ COND_EVAL3(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL3(TRUE, x[11], y[11])≥EVAL(x[11], -@z(y[11], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (25) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(26) (>@z(y[7], 0@z)=TRUE∧>=@z(0@z, x[7])=TRUE ⇒ COND_EVAL3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL3(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (26) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(27) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(28) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(29) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
We simplified constraint (29) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(30) (-1 + y[7] ≥ 0∧x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
We simplified constraint (30) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(31) (y[7] ≥ 0∧x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]), COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)), EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(32) (&&(>@z(y[7], 0@z), >=@z(0@z, x[7]))=TRUE∧y[7]=y[11]∧x[11]=x[9]∧-@z(y[11], 1@z)=y[9]∧x[7]=x[11] ⇒ COND_EVAL3(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL3(TRUE, x[11], y[11])≥EVAL(x[11], -@z(y[11], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (32) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(33) (>@z(y[7], 0@z)=TRUE∧>=@z(0@z, x[7])=TRUE ⇒ COND_EVAL3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL3(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (33) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(34) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (34) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(35) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (35) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(36) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
We simplified constraint (36) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(37) (-1 + y[7] ≥ 0∧x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(38) (y[7] ≥ 0∧x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
- (0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 = 0∧0 = 0)
- COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
- (x[9] ≥ 0∧y[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧-1 + (-1)Bound + x[9] ≥ 0)
- (y[9] ≥ 0∧x[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧-1 + (-1)Bound + x[9] ≥ 0)
- EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
- (0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
- COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
- (y[7] ≥ 0∧x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
- (y[7] ≥ 0∧x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
The constraints for P> respective Pbound 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 Pbound.
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(>=@z(x1, x2)) = -1
POL(COND_EVAL3(x1, x2, x3)) = x2 + x1
POL(0@z) = 0
POL(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(COND_EVAL4(x1, x2, x3)) = -1 + x2 + x1
POL(EVAL(x1, x2)) = -1 + x1
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
The following pairs are in Pbound:
COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
The following pairs are in P≥:
EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
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
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(11) -> (9), if ((-@z(y[11], 1@z) →* y[9])∧(x[11] →* x[9]))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(1) (EVAL(x[7], y[7])≥NonInfC∧EVAL(x[7], y[7])≥COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥))
We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5) (0 ≥ 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 = 0∧0 = 0∧0 = 0)
For Pair COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]), COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)), EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(6) (&&(>@z(y[7], 0@z), >=@z(0@z, x[7]))=TRUE∧-@z(y[11], 1@z)=y[7]1∧y[7]=y[11]∧x[11]=x[7]1∧x[7]=x[11] ⇒ COND_EVAL3(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL3(TRUE, x[11], y[11])≥EVAL(x[11], -@z(y[11], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7) (>@z(y[7], 0@z)=TRUE∧>=@z(0@z, x[7])=TRUE ⇒ COND_EVAL3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL3(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-1 + (-1)Bound + (2)y[7] + (-1)x[7] ≥ 0∧0 ≥ 0)
We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-1 + (-1)Bound + (2)y[7] + (-1)x[7] ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-1 + (-1)Bound + (2)y[7] + (-1)x[7] ≥ 0)
We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11) (-1 + y[7] ≥ 0∧x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-1 + (-1)Bound + (2)y[7] + x[7] ≥ 0)
We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12) (y[7] ≥ 0∧x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧1 + (-1)Bound + (2)y[7] + x[7] ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
- (0 ≥ 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 = 0∧0 = 0∧0 = 0)
- COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
- (y[7] ≥ 0∧x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧1 + (-1)Bound + (2)y[7] + x[7] ≥ 0)
The constraints for P> respective Pbound 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 Pbound.
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(>=@z(x1, x2)) = -1
POL(COND_EVAL3(x1, x2, x3)) = -1 + (2)x3 + (-1)x2
POL(0@z) = 0
POL(TRUE) = -1
POL(&&(x1, x2)) = 2
POL(EVAL(x1, x2)) = (2)x2 + (-1)x1
POL(FALSE) = 2
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
The following pairs are in Pbound:
COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
The following pairs are in P≥:
none
At least the following rules have been oriented under context sensitive arithmetic replacement:
FALSE1 → &&(FALSE, FALSE)1
-@z1 ↔
&&(TRUE, TRUE)1 → TRUE1
FALSE1 → &&(FALSE, TRUE)1
&&(TRUE, FALSE)1 → FALSE1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ 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_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ 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:
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(8) -> (7), if ((y[8] →* y[7])∧(-@z(x[8], 1@z) →* x[7]))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(1) (EVAL(x[7], y[7])≥NonInfC∧EVAL(x[7], y[7])≥COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥))
We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4) (0 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0)
We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5) (0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
For Pair COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]), COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)), EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(6) (&&(>@z(y[7], 0@z), >=@z(0@z, x[7]))=TRUE∧-@z(y[11], 1@z)=y[7]1∧y[7]=y[11]∧x[11]=x[7]1∧x[7]=x[11] ⇒ COND_EVAL3(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL3(TRUE, x[11], y[11])≥EVAL(x[11], -@z(y[11], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7) (>@z(y[7], 0@z)=TRUE∧>=@z(0@z, x[7])=TRUE ⇒ COND_EVAL3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL3(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧(-1)Bound + y[7] + (-1)x[7] ≥ 0∧0 ≥ 0)
We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧(-1)Bound + y[7] + (-1)x[7] ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10) ((-1)x[7] ≥ 0∧-1 + y[7] ≥ 0 ⇒ (-1)Bound + y[7] + (-1)x[7] ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11) (x[7] ≥ 0∧-1 + y[7] ≥ 0 ⇒ (-1)Bound + y[7] + x[7] ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12) (x[7] ≥ 0∧y[7] ≥ 0 ⇒ 1 + (-1)Bound + y[7] + x[7] ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
- (0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
- COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
- (x[7] ≥ 0∧y[7] ≥ 0 ⇒ 1 + (-1)Bound + y[7] + x[7] ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0)
The constraints for P> respective Pbound 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 Pbound.
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(>=@z(x1, x2)) = -1
POL(COND_EVAL3(x1, x2, x3)) = -1 + x3 + (-1)x2 + (-1)x1
POL(0@z) = 0
POL(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(EVAL(x1, x2)) = 1 + x2 + (-1)x1
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
The following pairs are in Pbound:
COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
The following pairs are in P≥:
COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
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
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDP
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ 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:
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(3): EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(11) -> (10), if ((-@z(y[11], 1@z) →* y[10])∧(x[11] →* x[10]))
(4) -> (9), if ((y[4] →* y[9])∧(-@z(x[4], 1@z) →* x[9]))
(4) -> (3), if ((y[4] →* y[3])∧(-@z(x[4], 1@z) →* x[3]))
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(11) -> (3), if ((-@z(y[11], 1@z) →* y[3])∧(x[11] →* x[3]))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(11) -> (9), if ((-@z(y[11], 1@z) →* y[9])∧(x[11] →* x[9]))
(4) -> (7), if ((y[4] →* y[7])∧(-@z(x[4], 1@z) →* x[7]))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ 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:
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(11) -> (10), if ((-@z(y[11], 1@z) →* y[10])∧(x[11] →* x[10]))
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(4) -> (7), if ((y[4] →* y[7])∧(-@z(x[4], 1@z) →* x[7]))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]), COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(1) (-@z(x[4], 1@z)=x[7]∧x[10]=x[4]∧y[10]=y[4]∧>@z(x[10], 0@z)=TRUE∧y[4]=y[7] ⇒ COND_EVAL1(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL1(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2) (>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL1(TRUE, x[10], y[10])≥EVAL(-@z(x[10], 1@z), y[10])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (x[10] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]), COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(8) (-@z(x[4], 1@z)=x[10]1∧x[10]=x[4]∧y[10]=y[4]∧y[4]=y[10]1∧>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL1(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:
(9) (>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL1(TRUE, x[10], y[10])≥EVAL(-@z(x[10], 1@z), y[10])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0)
We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 = 0)
We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14) (x[10] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 = 0)
For Pair EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(15) (EVAL(x[10], y[10])≥NonInfC∧EVAL(x[10], y[10])≥COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥))
We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18) (0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥))
We simplified constraint (18) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(19) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
For Pair EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(20) (EVAL(x[7], y[7])≥NonInfC∧EVAL(x[7], y[7])≥COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥))
We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23) (0 ≥ 0∧0 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥))
We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(24) (0 = 0∧0 = 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0)
For Pair COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]), COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)), EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(25) (&&(>@z(y[7], 0@z), >=@z(0@z, x[7]))=TRUE∧-@z(y[11], 1@z)=y[7]1∧y[7]=y[11]∧x[11]=x[7]1∧x[7]=x[11] ⇒ COND_EVAL3(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL3(TRUE, x[11], y[11])≥EVAL(x[11], -@z(y[11], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (25) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(26) (>@z(y[7], 0@z)=TRUE∧>=@z(0@z, x[7])=TRUE ⇒ COND_EVAL3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL3(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (26) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(27) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-1 + (-1)Bound + y[7] ≥ 0∧0 ≥ 0)
We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(28) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-1 + (-1)Bound + y[7] ≥ 0∧0 ≥ 0)
We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(29) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-1 + (-1)Bound + y[7] ≥ 0∧0 ≥ 0)
We simplified constraint (29) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(30) (-1 + y[7] ≥ 0∧x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-1 + (-1)Bound + y[7] ≥ 0∧0 ≥ 0)
We simplified constraint (30) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(31) (y[7] ≥ 0∧x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧(-1)Bound + y[7] ≥ 0∧0 ≥ 0)
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]), COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)), EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(32) (-@z(y[11], 1@z)=y[10]∧&&(>@z(y[7], 0@z), >=@z(0@z, x[7]))=TRUE∧y[7]=y[11]∧x[11]=x[10]∧x[7]=x[11] ⇒ COND_EVAL3(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL3(TRUE, x[11], y[11])≥EVAL(x[11], -@z(y[11], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (32) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(33) (>@z(y[7], 0@z)=TRUE∧>=@z(0@z, x[7])=TRUE ⇒ COND_EVAL3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL3(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (33) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(34) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-1 + (-1)Bound + y[7] ≥ 0∧0 ≥ 0)
We simplified constraint (34) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(35) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-1 + (-1)Bound + y[7] ≥ 0∧0 ≥ 0)
We simplified constraint (35) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(36) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧-1 + (-1)Bound + y[7] ≥ 0)
We simplified constraint (36) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(37) (-1 + y[7] ≥ 0∧x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧-1 + (-1)Bound + y[7] ≥ 0)
We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(38) (y[7] ≥ 0∧x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧(-1)Bound + y[7] ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
- (x[10] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- (x[10] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 = 0)
- EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
- ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
- EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
- (0 = 0∧0 = 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0)
- COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
- (y[7] ≥ 0∧x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧(-1)Bound + y[7] ≥ 0∧0 ≥ 0)
- (y[7] ≥ 0∧x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧(-1)Bound + y[7] ≥ 0)
The constraints for P> respective Pbound 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 Pbound.
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(>=@z(x1, x2)) = -1
POL(COND_EVAL3(x1, x2, x3)) = -1 + x3
POL(0@z) = 0
POL(COND_EVAL1(x1, x2, x3)) = -1 + x3
POL(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(EVAL(x1, x2)) = -1 + x2
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = 0
The following pairs are in P>:
COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
The following pairs are in Pbound:
COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
The following pairs are in P≥:
COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], 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 → &&(TRUE, FALSE)1
FALSE1 → &&(FALSE, TRUE)1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(4) -> (7), if ((y[4] →* y[7])∧(-@z(x[4], 1@z) →* x[7]))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ AND
↳ IDP
↳ 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:
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]), COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(1) (-@z(x[4], 1@z)=x[10]1∧x[10]=x[4]∧y[10]=y[4]∧y[4]=y[10]1∧>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL1(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2) (>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL1(TRUE, x[10], y[10])≥EVAL(-@z(x[10], 1@z), y[10])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + (2)x[10] ≥ 0∧1 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + (2)x[10] ≥ 0∧1 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + x[10] ≥ 0 ⇒ -1 + (-1)Bound + (2)x[10] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧1 ≥ 0)
We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6) (-1 + x[10] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + (2)x[10] ≥ 0∧0 = 0∧0 = 0)
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (x[10] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧1 + (-1)Bound + (2)x[10] ≥ 0∧0 = 0∧0 = 0)
For Pair EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(8) (EVAL(x[10], y[10])≥NonInfC∧EVAL(x[10], y[10])≥COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥))
We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11) (0 ≥ 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0)
We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12) (0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
- (x[10] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧1 + (-1)Bound + (2)x[10] ≥ 0∧0 = 0∧0 = 0)
- EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
- (0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0)
The constraints for P> respective Pbound 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 Pbound.
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 + (2)x2
POL(TRUE) = -1
POL(EVAL(x1, x2)) = -1 + (2)x1
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
The following pairs are in Pbound:
COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
The following pairs are in P≥:
EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
At least the following rules have been oriented under context sensitive arithmetic replacement:
-@z1 ↔
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ 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:
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(0): COND_EVAL(TRUE, x[0], y[0]) → EVAL(x[0], y[0])
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])
(3): EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(5): COND_EVAL5(TRUE, x[5], y[5]) → EVAL(x[5], y[5])
(6): COND_EVAL2(TRUE, x[6], y[6]) → EVAL(x[6], -@z(y[6], 1@z))
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(4) -> (3), if ((y[4] →* y[3])∧(-@z(x[4], 1@z) →* x[3]))
(11) -> (1), if ((-@z(y[11], 1@z) →* y[1])∧(x[11] →* x[1]))
(0) -> (10), if ((y[0] →* y[10])∧(x[0] →* x[10]))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(11) -> (9), if ((-@z(y[11], 1@z) →* y[9])∧(x[11] →* x[9]))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
(1) -> (0), if ((x[1] →* x[0])∧(y[1] →* y[0])∧(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])) →* TRUE))
(0) -> (7), if ((y[0] →* y[7])∧(x[0] →* x[7]))
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(5) -> (1), if ((y[5] →* y[1])∧(x[5] →* x[1]))
(6) -> (3), if ((-@z(y[6], 1@z) →* y[3])∧(x[6] →* x[3]))
(6) -> (1), if ((-@z(y[6], 1@z) →* y[1])∧(x[6] →* x[1]))
(8) -> (7), if ((y[8] →* y[7])∧(-@z(x[8], 1@z) →* x[7]))
(11) -> (3), if ((-@z(y[11], 1@z) →* y[3])∧(x[11] →* x[3]))
(4) -> (1), if ((y[4] →* y[1])∧(-@z(x[4], 1@z) →* x[1]))
(0) -> (1), if ((y[0] →* y[1])∧(x[0] →* x[1]))
(8) -> (10), if ((y[8] →* y[10])∧(-@z(x[8], 1@z) →* x[10]))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
(11) -> (10), if ((-@z(y[11], 1@z) →* y[10])∧(x[11] →* x[10]))
(5) -> (3), if ((y[5] →* y[3])∧(x[5] →* x[3]))
(0) -> (3), if ((y[0] →* y[3])∧(x[0] →* x[3]))
(5) -> (9), if ((y[5] →* y[9])∧(x[5] →* x[9]))
(6) -> (10), if ((-@z(y[6], 1@z) →* y[10])∧(x[6] →* x[10]))
(6) -> (9), if ((-@z(y[6], 1@z) →* y[9])∧(x[6] →* x[9]))
(4) -> (7), if ((y[4] →* y[7])∧(-@z(x[4], 1@z) →* x[7]))
(0) -> (9), if ((y[0] →* y[9])∧(x[0] →* x[9]))
(4) -> (9), if ((y[4] →* y[9])∧(-@z(x[4], 1@z) →* x[9]))
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(5) -> (7), if ((y[5] →* y[7])∧(x[5] →* x[7]))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(3) -> (5), if ((x[3] →* x[5])∧(y[3] →* y[5])∧(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])) →* TRUE))
(6) -> (7), if ((-@z(y[6], 1@z) →* y[7])∧(x[6] →* x[7]))
(8) -> (3), if ((y[8] →* y[3])∧(-@z(x[8], 1@z) →* x[3]))
(8) -> (1), if ((y[8] →* y[1])∧(-@z(x[8], 1@z) →* x[1]))
(5) -> (10), if ((y[5] →* y[10])∧(x[5] →* x[10]))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(0): COND_EVAL(TRUE, x[0], y[0]) → EVAL(x[0], y[0])
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])
(5): COND_EVAL5(TRUE, x[5], y[5]) → EVAL(x[5], y[5])
(3): EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(4) -> (3), if ((y[4] →* y[3])∧(-@z(x[4], 1@z) →* x[3]))
(11) -> (1), if ((-@z(y[11], 1@z) →* y[1])∧(x[11] →* x[1]))
(0) -> (10), if ((y[0] →* y[10])∧(x[0] →* x[10]))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(11) -> (9), if ((-@z(y[11], 1@z) →* y[9])∧(x[11] →* x[9]))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
(1) -> (0), if ((x[1] →* x[0])∧(y[1] →* y[0])∧(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])) →* TRUE))
(0) -> (7), if ((y[0] →* y[7])∧(x[0] →* x[7]))
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(5) -> (1), if ((y[5] →* y[1])∧(x[5] →* x[1]))
(8) -> (7), if ((y[8] →* y[7])∧(-@z(x[8], 1@z) →* x[7]))
(11) -> (3), if ((-@z(y[11], 1@z) →* y[3])∧(x[11] →* x[3]))
(4) -> (1), if ((y[4] →* y[1])∧(-@z(x[4], 1@z) →* x[1]))
(0) -> (1), if ((y[0] →* y[1])∧(x[0] →* x[1]))
(8) -> (10), if ((y[8] →* y[10])∧(-@z(x[8], 1@z) →* x[10]))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
(11) -> (10), if ((-@z(y[11], 1@z) →* y[10])∧(x[11] →* x[10]))
(0) -> (3), if ((y[0] →* y[3])∧(x[0] →* x[3]))
(5) -> (3), if ((y[5] →* y[3])∧(x[5] →* x[3]))
(5) -> (9), if ((y[5] →* y[9])∧(x[5] →* x[9]))
(4) -> (7), if ((y[4] →* y[7])∧(-@z(x[4], 1@z) →* x[7]))
(0) -> (9), if ((y[0] →* y[9])∧(x[0] →* x[9]))
(4) -> (9), if ((y[4] →* y[9])∧(-@z(x[4], 1@z) →* x[9]))
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(5) -> (7), if ((y[5] →* y[7])∧(x[5] →* x[7]))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(3) -> (5), if ((x[3] →* x[5])∧(y[3] →* y[5])∧(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])) →* TRUE))
(8) -> (3), if ((y[8] →* y[3])∧(-@z(x[8], 1@z) →* x[3]))
(8) -> (1), if ((y[8] →* y[1])∧(-@z(x[8], 1@z) →* x[1]))
(5) -> (10), if ((y[5] →* y[10])∧(x[5] →* x[10]))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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[0], y[0]) → EVAL(x[0], y[0]) the following chains were created:
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1]), COND_EVAL(TRUE, x[0], y[0]) → EVAL(x[0], y[0]) which results in the following constraint:
(1) (y[1]=y[0]∧x[1]=x[0]∧&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1]))=TRUE ⇒ COND_EVAL(TRUE, x[0], y[0])≥NonInfC∧COND_EVAL(TRUE, x[0], y[0])≥EVAL(x[0], y[0])∧(UIncreasing(EVAL(x[0], y[0])), ≥))
We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(2) (>=@z(0@z, y[1])=TRUE∧>@z(y[1], 0@z)=TRUE∧>=@z(0@z, x[1])=TRUE ⇒ COND_EVAL(TRUE, x[1], y[1])≥NonInfC∧COND_EVAL(TRUE, x[1], y[1])≥EVAL(x[1], y[1])∧(UIncreasing(EVAL(x[0], y[0])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) ((-1)y[1] ≥ 0∧-1 + y[1] ≥ 0∧(-1)x[1] ≥ 0 ⇒ (UIncreasing(EVAL(x[0], y[0])), ≥)∧-2 + (-1)Bound ≥ 0∧-2 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) ((-1)y[1] ≥ 0∧-1 + y[1] ≥ 0∧(-1)x[1] ≥ 0 ⇒ (UIncreasing(EVAL(x[0], y[0])), ≥)∧-2 + (-1)Bound ≥ 0∧-2 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) ((-1)x[1] ≥ 0∧-1 + y[1] ≥ 0∧(-1)y[1] ≥ 0 ⇒ (UIncreasing(EVAL(x[0], y[0])), ≥)∧-2 ≥ 0∧-2 + (-1)Bound ≥ 0)
We solved constraint (5) using rule (IDP_SMT_SPLIT).
For Pair EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1]) the following chains were created:
- We consider the chain EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1]) which results in the following constraint:
(6) (EVAL(x[1], y[1])≥NonInfC∧EVAL(x[1], y[1])≥COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])∧(UIncreasing(COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])), ≥))
We simplified constraint (6) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(7) ((UIncreasing(COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (7) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(8) ((UIncreasing(COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (8) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(9) ((UIncreasing(COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])), ≥)∧1 ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(10) (0 ≥ 0∧(UIncreasing(COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])), ≥)∧0 = 0∧1 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
For Pair COND_EVAL5(TRUE, x[5], y[5]) → EVAL(x[5], y[5]) the following chains were created:
- We consider the chain EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3]), COND_EVAL5(TRUE, x[5], y[5]) → EVAL(x[5], y[5]) which results in the following constraint:
(11) (y[3]=y[5]∧x[3]=x[5]∧&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3]))=TRUE ⇒ COND_EVAL5(TRUE, x[5], y[5])≥NonInfC∧COND_EVAL5(TRUE, x[5], y[5])≥EVAL(x[5], y[5])∧(UIncreasing(EVAL(x[5], y[5])), ≥))
We simplified constraint (11) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(12) (>=@z(0@z, y[3])=TRUE∧>@z(x[3], 0@z)=TRUE∧>=@z(0@z, x[3])=TRUE ⇒ COND_EVAL5(TRUE, x[3], y[3])≥NonInfC∧COND_EVAL5(TRUE, x[3], y[3])≥EVAL(x[3], y[3])∧(UIncreasing(EVAL(x[5], y[5])), ≥))
We simplified constraint (12) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(13) ((-1)y[3] ≥ 0∧-1 + x[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], y[5])), ≥)∧-1 + (-1)Bound ≥ 0∧-1 ≥ 0)
We simplified constraint (13) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(14) ((-1)y[3] ≥ 0∧-1 + x[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ (UIncreasing(EVAL(x[5], y[5])), ≥)∧-1 + (-1)Bound ≥ 0∧-1 ≥ 0)
We simplified constraint (14) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(15) (-1 + x[3] ≥ 0∧(-1)y[3] ≥ 0∧(-1)x[3] ≥ 0 ⇒ -1 ≥ 0∧(UIncreasing(EVAL(x[5], y[5])), ≥)∧-1 + (-1)Bound ≥ 0)
We solved constraint (15) using rule (IDP_SMT_SPLIT).
For Pair EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3]) the following chains were created:
- We consider the chain EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3]) which results in the following constraint:
(16) (EVAL(x[3], y[3])≥NonInfC∧EVAL(x[3], y[3])≥COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])∧(UIncreasing(COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])), ≥))
We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17) ((UIncreasing(COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18) ((UIncreasing(COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19) ((UIncreasing(COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (19) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(20) (0 ≥ 0∧(UIncreasing(COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
For Pair COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]), COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]) which results in the following constraint:
(21) (x[10]=x[4]∧y[10]=y[4]∧>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL1(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (21) using rule (III) which results in the following new constraint:
(22) (>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL1(TRUE, x[10], y[10])≥EVAL(-@z(x[10], 1@z), y[10])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (22) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(23) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (23) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(24) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (24) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(25) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (25) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(26) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(27) (x[10] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
For Pair EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(28) (EVAL(x[10], y[10])≥NonInfC∧EVAL(x[10], y[10])≥COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥))
We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(29) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(30) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (30) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(31) (0 ≥ 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0)
We simplified constraint (31) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(32) (0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 = 0)
For Pair COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]), COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)) which results in the following constraint:
(33) (&&(>@z(y[7], 0@z), >=@z(0@z, x[7]))=TRUE∧y[7]=y[11]∧x[7]=x[11] ⇒ COND_EVAL3(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL3(TRUE, x[11], y[11])≥EVAL(x[11], -@z(y[11], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (33) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(34) (>@z(y[7], 0@z)=TRUE∧>=@z(0@z, x[7])=TRUE ⇒ COND_EVAL3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL3(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (34) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(35) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (35) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(36) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (36) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(37) ((-1)x[7] ≥ 0∧-1 + y[7] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(38) (x[7] ≥ 0∧-1 + y[7] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (38) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(39) (x[7] ≥ 0∧y[7] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
For Pair EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(40) (EVAL(x[7], y[7])≥NonInfC∧EVAL(x[7], y[7])≥COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥))
We simplified constraint (40) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(41) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (41) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(42) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (42) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(43) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (43) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(44) (0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0)
For Pair COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]), COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]) which results in the following constraint:
(45) (y[9]=y[8]∧&&(>@z(y[9], 0@z), >@z(x[9], 0@z))=TRUE∧x[9]=x[8] ⇒ COND_EVAL4(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL4(TRUE, x[8], y[8])≥EVAL(-@z(x[8], 1@z), y[8])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (45) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(46) (>@z(y[9], 0@z)=TRUE∧>@z(x[9], 0@z)=TRUE ⇒ COND_EVAL4(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL4(TRUE, x[9], y[9])≥EVAL(-@z(x[9], 1@z), y[9])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (46) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(47) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (47) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(48) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (48) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(49) (-1 + x[9] ≥ 0∧y[9] + -1 ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (49) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(50) (-1 + x[9] ≥ 0∧y[9] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (50) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(51) (x[9] ≥ 0∧y[9] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
For Pair EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(52) (EVAL(x[9], y[9])≥NonInfC∧EVAL(x[9], y[9])≥COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥))
We simplified constraint (52) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(53) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (53) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(54) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (54) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(55) (0 ≥ 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0)
We simplified constraint (55) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(56) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL(TRUE, x[0], y[0]) → EVAL(x[0], y[0])
- EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])
- (0 ≥ 0∧(UIncreasing(COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])), ≥)∧0 = 0∧1 ≥ 0∧0 = 0∧0 = 0∧0 = 0)
- COND_EVAL5(TRUE, x[5], y[5]) → EVAL(x[5], y[5])
- EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])
- (0 ≥ 0∧(UIncreasing(COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
- COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
- (x[10] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
- (0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 = 0)
- COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
- (x[7] ≥ 0∧y[7] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
- EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
- (0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0)
- COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
- (x[9] ≥ 0∧y[9] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
- EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
- ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0)
The constraints for P> respective Pbound 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 Pbound.
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(COND_EVAL5(x1, x2, x3)) = -1
POL(0@z) = 0
POL(COND_EVAL1(x1, x2, x3)) = -1
POL(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(COND_EVAL4(x1, x2, x3)) = -1
POL(COND_EVAL(x1, x2, x3)) = -1 + x1
POL(FALSE) = -1
POL(>@z(x1, x2)) = -1
POL(COND_EVAL3(x1, x2, x3)) = x1
POL(>=@z(x1, x2)) = -1
POL(EVAL(x1, x2)) = -1
POL(1@z) = 1
POL(undefined) = -1
The following pairs are in P>:
COND_EVAL(TRUE, x[0], y[0]) → EVAL(x[0], y[0])
COND_EVAL5(TRUE, x[5], y[5]) → EVAL(x[5], y[5])
The following pairs are in Pbound:
COND_EVAL(TRUE, x[0], y[0]) → EVAL(x[0], y[0])
COND_EVAL5(TRUE, x[5], y[5]) → EVAL(x[5], y[5])
The following pairs are in P≥:
EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])
EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])
COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
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
↳ IDP
↳ IDependencyGraphProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(y[1], 0@z), >=@z(0@z, x[1])), >=@z(0@z, y[1])), x[1], y[1])
(3): EVAL(x[3], y[3]) → COND_EVAL5(&&(&&(>@z(x[3], 0@z), >=@z(0@z, x[3])), >=@z(0@z, y[3])), x[3], y[3])
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(11) -> (10), if ((-@z(y[11], 1@z) →* y[10])∧(x[11] →* x[10]))
(4) -> (3), if ((y[4] →* y[3])∧(-@z(x[4], 1@z) →* x[3]))
(11) -> (1), if ((-@z(y[11], 1@z) →* y[1])∧(x[11] →* x[1]))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(11) -> (9), if ((-@z(y[11], 1@z) →* y[9])∧(x[11] →* x[9]))
(4) -> (7), if ((y[4] →* y[7])∧(-@z(x[4], 1@z) →* x[7]))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
(4) -> (9), if ((y[4] →* y[9])∧(-@z(x[4], 1@z) →* x[9]))
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(8) -> (7), if ((y[8] →* y[7])∧(-@z(x[8], 1@z) →* x[7]))
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(11) -> (3), if ((-@z(y[11], 1@z) →* y[3])∧(x[11] →* x[3]))
(4) -> (1), if ((y[4] →* y[1])∧(-@z(x[4], 1@z) →* x[1]))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(8) -> (10), if ((y[8] →* y[10])∧(-@z(x[8], 1@z) →* x[10]))
(8) -> (3), if ((y[8] →* y[3])∧(-@z(x[8], 1@z) →* x[3]))
(8) -> (1), if ((y[8] →* y[1])∧(-@z(x[8], 1@z) →* x[1]))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ 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:
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(11) -> (10), if ((-@z(y[11], 1@z) →* y[10])∧(x[11] →* x[10]))
(4) -> (9), if ((y[4] →* y[9])∧(-@z(x[4], 1@z) →* x[9]))
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(8) -> (7), if ((y[8] →* y[7])∧(-@z(x[8], 1@z) →* x[7]))
(8) -> (10), if ((y[8] →* y[10])∧(-@z(x[8], 1@z) →* x[10]))
(11) -> (7), if ((-@z(y[11], 1@z) →* y[7])∧(x[11] →* x[7]))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(11) -> (9), if ((-@z(y[11], 1@z) →* y[9])∧(x[11] →* x[9]))
(4) -> (7), if ((y[4] →* y[7])∧(-@z(x[4], 1@z) →* x[7]))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
(7) -> (11), if ((x[7] →* x[11])∧(y[7] →* y[11])∧(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]), COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(1) (-@z(x[4], 1@z)=x[9]∧x[10]=x[4]∧y[10]=y[4]∧y[4]=y[9]∧>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL1(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2) (>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL1(TRUE, x[10], y[10])≥EVAL(-@z(x[10], 1@z), y[10])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6) (-1 + x[10] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (x[10] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]), COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(8) (-@z(x[4], 1@z)=x[7]∧x[10]=x[4]∧y[10]=y[4]∧>@z(x[10], 0@z)=TRUE∧y[4]=y[7] ⇒ COND_EVAL1(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL1(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:
(9) (>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL1(TRUE, x[10], y[10])≥EVAL(-@z(x[10], 1@z), y[10])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 = 0)
We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14) (x[10] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 = 0)
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]), COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(15) (-@z(x[4], 1@z)=x[10]1∧x[10]=x[4]∧y[10]=y[4]∧y[4]=y[10]1∧>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL1(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (15) using rules (III), (IV) which results in the following new constraint:
(16) (>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL1(TRUE, x[10], y[10])≥EVAL(-@z(x[10], 1@z), y[10])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (19) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(20) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 = 0)
We simplified constraint (20) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(21) (x[10] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 = 0)
For Pair EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(22) (EVAL(x[10], y[10])≥NonInfC∧EVAL(x[10], y[10])≥COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥))
We simplified constraint (22) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(23) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (23) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(24) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (24) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(25) (0 ≥ 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0)
We simplified constraint (25) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(26) (0 = 0∧0 = 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0)
For Pair EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(27) (EVAL(x[7], y[7])≥NonInfC∧EVAL(x[7], y[7])≥COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥))
We simplified constraint (27) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(28) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (28) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(29) ((UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (29) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(30) (0 ≥ 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0)
We simplified constraint (30) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(31) (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0)
For Pair COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]), COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]), EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(32) (y[9]=y[8]∧y[8]=y[9]1∧&&(>@z(y[9], 0@z), >@z(x[9], 0@z))=TRUE∧-@z(x[8], 1@z)=x[9]1∧x[9]=x[8] ⇒ COND_EVAL4(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL4(TRUE, x[8], y[8])≥EVAL(-@z(x[8], 1@z), y[8])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (32) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(33) (>@z(y[9], 0@z)=TRUE∧>@z(x[9], 0@z)=TRUE ⇒ COND_EVAL4(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL4(TRUE, x[9], y[9])≥EVAL(-@z(x[9], 1@z), y[9])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (33) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(34) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (34) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(35) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (35) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(36) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
We simplified constraint (36) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(37) (y[9] ≥ 0∧-1 + x[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(38) (y[9] ≥ 0∧x[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]), COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]), EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(39) (y[9]=y[8]∧-@z(x[8], 1@z)=x[10]∧y[8]=y[10]∧&&(>@z(y[9], 0@z), >@z(x[9], 0@z))=TRUE∧x[9]=x[8] ⇒ COND_EVAL4(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL4(TRUE, x[8], y[8])≥EVAL(-@z(x[8], 1@z), y[8])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (39) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(40) (>@z(y[9], 0@z)=TRUE∧>@z(x[9], 0@z)=TRUE ⇒ COND_EVAL4(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL4(TRUE, x[9], y[9])≥EVAL(-@z(x[9], 1@z), y[9])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (40) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(41) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (41) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(42) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (42) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(43) (-1 + x[9] ≥ 0∧y[9] + -1 ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
We simplified constraint (43) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(44) (-1 + x[9] ≥ 0∧y[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
We simplified constraint (44) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(45) (x[9] ≥ 0∧y[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]), COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]), EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(46) (-@z(x[8], 1@z)=x[7]∧y[8]=y[7]∧y[9]=y[8]∧&&(>@z(y[9], 0@z), >@z(x[9], 0@z))=TRUE∧x[9]=x[8] ⇒ COND_EVAL4(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL4(TRUE, x[8], y[8])≥EVAL(-@z(x[8], 1@z), y[8])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (46) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(47) (>@z(y[9], 0@z)=TRUE∧>@z(x[9], 0@z)=TRUE ⇒ COND_EVAL4(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL4(TRUE, x[9], y[9])≥EVAL(-@z(x[9], 1@z), y[9])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (47) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(48) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (48) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(49) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (49) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(50) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (50) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(51) (y[9] ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (51) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(52) (y[9] ≥ 0∧x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
For Pair EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(53) (EVAL(x[9], y[9])≥NonInfC∧EVAL(x[9], y[9])≥COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥))
We simplified constraint (53) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(54) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (54) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(55) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (55) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(56) (0 ≥ 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0)
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∧0 ≥ 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 = 0)
For Pair COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)) the following chains were created:
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]), COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)), EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]) which results in the following constraint:
(58) (&&(>@z(y[7], 0@z), >=@z(0@z, x[7]))=TRUE∧-@z(y[11], 1@z)=y[7]1∧y[7]=y[11]∧x[11]=x[7]1∧x[7]=x[11] ⇒ COND_EVAL3(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL3(TRUE, x[11], y[11])≥EVAL(x[11], -@z(y[11], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (58) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(59) (>@z(y[7], 0@z)=TRUE∧>=@z(0@z, x[7])=TRUE ⇒ COND_EVAL3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL3(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (59) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(60) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-2 + (-1)Bound + y[7] ≥ 0∧0 ≥ 0)
We simplified constraint (60) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(61) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-2 + (-1)Bound + y[7] ≥ 0∧0 ≥ 0)
We simplified constraint (61) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(62) ((-1)x[7] ≥ 0∧-1 + y[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧-2 + (-1)Bound + y[7] ≥ 0)
We simplified constraint (62) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(63) (x[7] ≥ 0∧-1 + y[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧-2 + (-1)Bound + y[7] ≥ 0)
We simplified constraint (63) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(64) (x[7] ≥ 0∧y[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧-1 + (-1)Bound + y[7] ≥ 0)
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]), COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)), EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(65) (&&(>@z(y[7], 0@z), >=@z(0@z, x[7]))=TRUE∧y[7]=y[11]∧x[11]=x[9]∧-@z(y[11], 1@z)=y[9]∧x[7]=x[11] ⇒ COND_EVAL3(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL3(TRUE, x[11], y[11])≥EVAL(x[11], -@z(y[11], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (65) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(66) (>@z(y[7], 0@z)=TRUE∧>=@z(0@z, x[7])=TRUE ⇒ COND_EVAL3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL3(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (66) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(67) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-2 + (-1)Bound + y[7] ≥ 0∧0 ≥ 0)
We simplified constraint (67) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(68) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-2 + (-1)Bound + y[7] ≥ 0∧0 ≥ 0)
We simplified constraint (68) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(69) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧-2 + (-1)Bound + y[7] ≥ 0)
We simplified constraint (69) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(70) (-1 + y[7] ≥ 0∧x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧-2 + (-1)Bound + y[7] ≥ 0)
We simplified constraint (70) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(71) (y[7] ≥ 0∧x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧-1 + (-1)Bound + y[7] ≥ 0)
- We consider the chain EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7]), COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z)), EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(72) (-@z(y[11], 1@z)=y[10]∧&&(>@z(y[7], 0@z), >=@z(0@z, x[7]))=TRUE∧y[7]=y[11]∧x[11]=x[10]∧x[7]=x[11] ⇒ COND_EVAL3(TRUE, x[11], y[11])≥NonInfC∧COND_EVAL3(TRUE, x[11], y[11])≥EVAL(x[11], -@z(y[11], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (72) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(73) (>@z(y[7], 0@z)=TRUE∧>=@z(0@z, x[7])=TRUE ⇒ COND_EVAL3(TRUE, x[7], y[7])≥NonInfC∧COND_EVAL3(TRUE, x[7], y[7])≥EVAL(x[7], -@z(y[7], 1@z))∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥))
We simplified constraint (73) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(74) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-2 + (-1)Bound + y[7] ≥ 0∧0 ≥ 0)
We simplified constraint (74) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(75) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-2 + (-1)Bound + y[7] ≥ 0∧0 ≥ 0)
We simplified constraint (75) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(76) (-1 + y[7] ≥ 0∧(-1)x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-2 + (-1)Bound + y[7] ≥ 0)
We simplified constraint (76) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(77) (-1 + y[7] ≥ 0∧x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-2 + (-1)Bound + y[7] ≥ 0)
We simplified constraint (77) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(78) (y[7] ≥ 0∧x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-1 + (-1)Bound + y[7] ≥ 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
- (x[10] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- (x[10] ≥ 0 ⇒ 0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 ≥ 0∧0 = 0)
- (x[10] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 = 0)
- EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
- (0 = 0∧0 = 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 ≥ 0)
- EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
- (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧(UIncreasing(COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])), ≥)∧0 ≥ 0)
- COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
- (y[9] ≥ 0∧x[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
- (x[9] ≥ 0∧y[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
- (y[9] ≥ 0∧x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧0 ≥ 0)
- EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
- (0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 = 0)
- COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
- (x[7] ≥ 0∧y[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧-1 + (-1)Bound + y[7] ≥ 0)
- (y[7] ≥ 0∧x[7] ≥ 0 ⇒ (UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧0 ≥ 0∧-1 + (-1)Bound + y[7] ≥ 0)
- (y[7] ≥ 0∧x[7] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(x[11], -@z(y[11], 1@z))), ≥)∧-1 + (-1)Bound + y[7] ≥ 0)
The constraints for P> respective Pbound 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 Pbound.
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
POL(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(COND_EVAL4(x1, x2, x3)) = x3 + x1
POL(FALSE) = -1
POL(>@z(x1, x2)) = -1
POL(COND_EVAL3(x1, x2, x3)) = -1 + x3 + x1
POL(>=@z(x1, x2)) = -1
POL(EVAL(x1, x2)) = -1 + x2
POL(1@z) = 1
POL(undefined) = -1
The following pairs are in P>:
EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
The following pairs are in Pbound:
COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
The following pairs are in P≥:
COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
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
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ 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:
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(7): EVAL(x[7], y[7]) → COND_EVAL3(&&(>@z(y[7], 0@z), >=@z(0@z, x[7])), x[7], y[7])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(4) -> (9), if ((y[4] →* y[9])∧(-@z(x[4], 1@z) →* x[9]))
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(8) -> (7), if ((y[8] →* y[7])∧(-@z(x[8], 1@z) →* x[7]))
(8) -> (10), if ((y[8] →* y[10])∧(-@z(x[8], 1@z) →* x[10]))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(4) -> (7), if ((y[4] →* y[7])∧(-@z(x[4], 1@z) →* x[7]))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(4) -> (9), if ((y[4] →* y[9])∧(-@z(x[4], 1@z) →* x[9]))
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(8) -> (10), if ((y[8] →* y[10])∧(-@z(x[8], 1@z) →* x[10]))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]), COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(1) (-@z(x[4], 1@z)=x[9]∧x[10]=x[4]∧y[10]=y[4]∧y[4]=y[9]∧>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL1(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2) (>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL1(TRUE, x[10], y[10])≥EVAL(-@z(x[10], 1@z), y[10])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + x[10] ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + x[10] ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧-1 + (-1)Bound + x[10] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6) (-1 + x[10] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + x[10] ≥ 0∧0 = 0)
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (x[10] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧(-1)Bound + x[10] ≥ 0∧0 = 0)
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]), COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(8) (-@z(x[4], 1@z)=x[10]1∧x[10]=x[4]∧y[10]=y[4]∧y[4]=y[10]1∧>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL1(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:
(9) (>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL1(TRUE, x[10], y[10])≥EVAL(-@z(x[10], 1@z), y[10])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + x[10] ≥ 0∧0 ≥ 0)
We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + x[10] ≥ 0∧0 ≥ 0)
We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + x[10] ≥ 0)
We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧-1 + (-1)Bound + x[10] ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14) (x[10] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + x[10] ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
For Pair EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(15) (EVAL(x[10], y[10])≥NonInfC∧EVAL(x[10], y[10])≥COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥))
We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧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∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 = 0∧0 = 0∧0 = 0)
For Pair COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]), COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]), EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(20) (y[9]=y[8]∧y[8]=y[9]1∧&&(>@z(y[9], 0@z), >@z(x[9], 0@z))=TRUE∧-@z(x[8], 1@z)=x[9]1∧x[9]=x[8] ⇒ COND_EVAL4(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL4(TRUE, x[8], y[8])≥EVAL(-@z(x[8], 1@z), y[8])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (20) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(21) (>@z(y[9], 0@z)=TRUE∧>@z(x[9], 0@z)=TRUE ⇒ COND_EVAL4(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL4(TRUE, x[9], y[9])≥EVAL(-@z(x[9], 1@z), y[9])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(22) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(23) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(24) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (24) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(25) (y[9] ≥ 0∧-1 + x[9] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (25) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(26) (y[9] ≥ 0∧x[9] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]), COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]), EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(27) (y[9]=y[8]∧-@z(x[8], 1@z)=x[10]∧y[8]=y[10]∧&&(>@z(y[9], 0@z), >@z(x[9], 0@z))=TRUE∧x[9]=x[8] ⇒ COND_EVAL4(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL4(TRUE, x[8], y[8])≥EVAL(-@z(x[8], 1@z), y[8])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (27) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(28) (>@z(y[9], 0@z)=TRUE∧>@z(x[9], 0@z)=TRUE ⇒ COND_EVAL4(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL4(TRUE, x[9], y[9])≥EVAL(-@z(x[9], 1@z), y[9])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(29) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(30) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (30) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(31) (-1 + x[9] ≥ 0∧y[9] + -1 ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧1 ≥ 0)
We simplified constraint (31) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(32) (-1 + x[9] ≥ 0∧y[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧1 ≥ 0)
We simplified constraint (32) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(33) (x[9] ≥ 0∧y[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧1 ≥ 0)
For Pair EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(34) (EVAL(x[9], y[9])≥NonInfC∧EVAL(x[9], y[9])≥COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥))
We simplified constraint (34) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(35) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (35) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(36) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (36) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(37) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (37) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(38) (0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 = 0∧0 = 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
- (x[10] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧(-1)Bound + x[10] ≥ 0∧0 = 0)
- (x[10] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + x[10] ≥ 0∧0 = 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
- EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
- (0 ≥ 0∧0 = 0∧0 ≥ 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 = 0∧0 = 0∧0 = 0)
- COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
- (y[9] ≥ 0∧x[9] ≥ 0 ⇒ 1 ≥ 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
- (x[9] ≥ 0∧y[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧1 ≥ 0)
- EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
- (0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 = 0∧0 = 0)
The constraints for P> respective Pbound 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 Pbound.
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 + x2
POL(TRUE) = 0
POL(&&(x1, x2)) = 0
POL(COND_EVAL4(x1, x2, x3)) = -1 + x2 + (-1)x1
POL(EVAL(x1, x2)) = -1 + x1
POL(FALSE) = 2
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
The following pairs are in Pbound:
COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
The following pairs are in P≥:
EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
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
↳ IDP
↳ IDependencyGraphProof
↳ 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:
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(8) -> (10), if ((y[8] →* y[10])∧(-@z(x[8], 1@z) →* x[10]))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ IDP
↳ IDependencyGraphProof
↳ 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:
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]), COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]), EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(1) (y[9]=y[8]∧y[8]=y[9]1∧&&(>@z(y[9], 0@z), >@z(x[9], 0@z))=TRUE∧-@z(x[8], 1@z)=x[9]1∧x[9]=x[8] ⇒ COND_EVAL4(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL4(TRUE, x[8], y[8])≥EVAL(-@z(x[8], 1@z), y[8])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2) (>@z(y[9], 0@z)=TRUE∧>@z(x[9], 0@z)=TRUE ⇒ COND_EVAL4(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL4(TRUE, x[9], y[9])≥EVAL(-@z(x[9], 1@z), y[9])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧(-1)Bound + (2)x[9] ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧(-1)Bound + (2)x[9] ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧(-1)Bound + (2)x[9] ≥ 0)
We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6) (y[9] ≥ 0∧-1 + x[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧(-1)Bound + (2)x[9] ≥ 0)
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (y[9] ≥ 0∧x[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧2 + (-1)Bound + (2)x[9] ≥ 0)
For Pair EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(8) (EVAL(x[9], y[9])≥NonInfC∧EVAL(x[9], y[9])≥COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥))
We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11) (0 ≥ 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0)
We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12) (0 ≥ 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
- (y[9] ≥ 0∧x[9] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧2 + (-1)Bound + (2)x[9] ≥ 0)
- EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
- (0 ≥ 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
The constraints for P> respective Pbound 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 Pbound.
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(TRUE) = 0
POL(&&(x1, x2)) = 1
POL(EVAL(x1, x2)) = 1 + (2)x1
POL(COND_EVAL4(x1, x2, x3)) = (2)x2 + x1
POL(FALSE) = -1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
The following pairs are in Pbound:
COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
The following pairs are in P≥:
EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
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
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ 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:
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ IDP
↳ IDependencyGraphProof
↳ 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:
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(11): COND_EVAL3(TRUE, x[11], y[11]) → EVAL(x[11], -@z(y[11], 1@z))
(11) -> (10), if ((-@z(y[11], 1@z) →* y[10])∧(x[11] →* x[10]))
(4) -> (9), if ((y[4] →* y[9])∧(-@z(x[4], 1@z) →* x[9]))
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(8) -> (10), if ((y[8] →* y[10])∧(-@z(x[8], 1@z) →* x[10]))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(11) -> (9), if ((-@z(y[11], 1@z) →* y[9])∧(x[11] →* x[9]))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ IDP
↳ IDependencyGraphProof
↳ 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:
(4): COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(4) -> (9), if ((y[4] →* y[9])∧(-@z(x[4], 1@z) →* x[9]))
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(4) -> (10), if ((y[4] →* y[10])∧(-@z(x[4], 1@z) →* x[10]))
(8) -> (10), if ((y[8] →* y[10])∧(-@z(x[8], 1@z) →* x[10]))
(10) -> (4), if ((x[10] →* x[4])∧(y[10] →* y[4])∧(>@z(x[10], 0@z) →* TRUE))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]), COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(1) (-@z(x[4], 1@z)=x[9]∧x[10]=x[4]∧y[10]=y[4]∧y[4]=y[9]∧>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL1(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2) (>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL1(TRUE, x[10], y[10])≥EVAL(-@z(x[10], 1@z), y[10])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + x[10] ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + x[10] ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧-1 + (-1)Bound + x[10] ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6) (-1 + x[10] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + x[10] ≥ 0∧0 = 0)
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (x[10] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧(-1)Bound + x[10] ≥ 0∧0 = 0)
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]), COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4]), EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(8) (-@z(x[4], 1@z)=x[10]1∧x[10]=x[4]∧y[10]=y[4]∧y[4]=y[10]1∧>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[4], y[4])≥NonInfC∧COND_EVAL1(TRUE, x[4], y[4])≥EVAL(-@z(x[4], 1@z), y[4])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:
(9) (>@z(x[10], 0@z)=TRUE ⇒ COND_EVAL1(TRUE, x[10], y[10])≥NonInfC∧COND_EVAL1(TRUE, x[10], y[10])≥EVAL(-@z(x[10], 1@z), y[10])∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥))
We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + x[10] ≥ 0∧0 ≥ 0)
We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11) (-1 + x[10] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + x[10] ≥ 0∧0 ≥ 0)
We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧-1 + (-1)Bound + x[10] ≥ 0)
We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13) (-1 + x[10] ≥ 0 ⇒ 0 ≥ 0∧-1 + (-1)Bound + x[10] ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 = 0)
We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14) (x[10] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + x[10] ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 = 0)
For Pair EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) the following chains were created:
- We consider the chain EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(15) (EVAL(x[10], y[10])≥NonInfC∧EVAL(x[10], y[10])≥COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥))
We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18) (0 ≥ 0∧(UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0)
We simplified constraint (18) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(19) ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
For Pair COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]), COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]), EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(20) (y[9]=y[8]∧y[8]=y[9]1∧&&(>@z(y[9], 0@z), >@z(x[9], 0@z))=TRUE∧-@z(x[8], 1@z)=x[9]1∧x[9]=x[8] ⇒ COND_EVAL4(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL4(TRUE, x[8], y[8])≥EVAL(-@z(x[8], 1@z), y[8])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (20) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(21) (>@z(y[9], 0@z)=TRUE∧>@z(x[9], 0@z)=TRUE ⇒ COND_EVAL4(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL4(TRUE, x[9], y[9])≥EVAL(-@z(x[9], 1@z), y[9])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(22) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(23) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(24) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
We simplified constraint (24) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(25) (y[9] ≥ 0∧-1 + x[9] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
We simplified constraint (25) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(26) (y[9] ≥ 0∧x[9] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]), COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]), EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10]) which results in the following constraint:
(27) (y[9]=y[8]∧-@z(x[8], 1@z)=x[10]∧y[8]=y[10]∧&&(>@z(y[9], 0@z), >@z(x[9], 0@z))=TRUE∧x[9]=x[8] ⇒ COND_EVAL4(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL4(TRUE, x[8], y[8])≥EVAL(-@z(x[8], 1@z), y[8])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (27) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(28) (>@z(y[9], 0@z)=TRUE∧>@z(x[9], 0@z)=TRUE ⇒ COND_EVAL4(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL4(TRUE, x[9], y[9])≥EVAL(-@z(x[9], 1@z), y[9])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(29) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(30) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0∧1 ≥ 0)
We simplified constraint (30) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(31) (-1 + x[9] ≥ 0∧y[9] + -1 ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
We simplified constraint (31) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(32) (-1 + x[9] ≥ 0∧y[9] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
We simplified constraint (32) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(33) (x[9] ≥ 0∧y[9] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
For Pair EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(34) (EVAL(x[9], y[9])≥NonInfC∧EVAL(x[9], y[9])≥COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥))
We simplified constraint (34) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(35) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (35) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(36) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (36) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(37) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (37) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(38) (0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
- (x[10] ≥ 0 ⇒ 0 = 0∧0 ≥ 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧(-1)Bound + x[10] ≥ 0∧0 = 0)
- (x[10] ≥ 0 ⇒ 0 ≥ 0∧(-1)Bound + x[10] ≥ 0∧0 = 0∧(UIncreasing(EVAL(-@z(x[4], 1@z), y[4])), ≥)∧0 = 0)
- EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
- ((UIncreasing(COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
- COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
- (y[9] ≥ 0∧x[9] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
- (x[9] ≥ 0∧y[9] ≥ 0 ⇒ 1 ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
- EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
- (0 ≥ 0∧0 = 0∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
The constraints for P> respective Pbound 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 Pbound.
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 + x2
POL(TRUE) = 0
POL(&&(x1, x2)) = 0
POL(COND_EVAL4(x1, x2, x3)) = -1 + x2 + (-1)x1
POL(EVAL(x1, x2)) = -1 + x1
POL(FALSE) = 0
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
The following pairs are in Pbound:
COND_EVAL1(TRUE, x[4], y[4]) → EVAL(-@z(x[4], 1@z), y[4])
The following pairs are in P≥:
EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
At least the following rules have been oriented under context sensitive arithmetic replacement:
&&(FALSE, FALSE)1 ↔ FALSE1
-@z1 ↔
&&(TRUE, TRUE)1 ↔ TRUE1
FALSE1 → &&(FALSE, TRUE)1
FALSE1 → &&(TRUE, FALSE)1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(10): EVAL(x[10], y[10]) → COND_EVAL1(>@z(x[10], 0@z), x[10], y[10])
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(8) -> (10), if ((y[8] →* y[10])∧(-@z(x[8], 1@z) →* x[10]))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
I DP problem:
The following domains are used:
z
R is empty.
The integer pair graph contains the following rules and edges:
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
(8) -> (9), if ((y[8] →* y[9])∧(-@z(x[8], 1@z) →* x[9]))
(9) -> (8), if ((x[9] →* x[8])∧(y[9] →* y[8])∧(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)) →* TRUE))
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]), COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8]), EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(1) (y[9]=y[8]∧y[8]=y[9]1∧&&(>@z(y[9], 0@z), >@z(x[9], 0@z))=TRUE∧-@z(x[8], 1@z)=x[9]1∧x[9]=x[8] ⇒ COND_EVAL4(TRUE, x[8], y[8])≥NonInfC∧COND_EVAL4(TRUE, x[8], y[8])≥EVAL(-@z(x[8], 1@z), y[8])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2) (>@z(y[9], 0@z)=TRUE∧>@z(x[9], 0@z)=TRUE ⇒ COND_EVAL4(TRUE, x[9], y[9])≥NonInfC∧COND_EVAL4(TRUE, x[9], y[9])≥EVAL(-@z(x[9], 1@z), y[9])∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧(-1)Bound + x[9] ≥ 0∧0 ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (y[9] + -1 ≥ 0∧-1 + x[9] ≥ 0 ⇒ (UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧(-1)Bound + x[9] ≥ 0∧0 ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (-1 + x[9] ≥ 0∧y[9] + -1 ≥ 0 ⇒ (-1)Bound + x[9] ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6) (-1 + x[9] ≥ 0∧y[9] ≥ 0 ⇒ (-1)Bound + x[9] ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7) (x[9] ≥ 0∧y[9] ≥ 0 ⇒ 1 + (-1)Bound + x[9] ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
For Pair EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) the following chains were created:
- We consider the chain EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9]) which results in the following constraint:
(8) (EVAL(x[9], y[9])≥NonInfC∧EVAL(x[9], y[9])≥COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])∧(UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥))
We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0)
We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12) ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0)
To summarize, we get the following constraints P≥ for the following pairs.
- COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
- (x[9] ≥ 0∧y[9] ≥ 0 ⇒ 1 + (-1)Bound + x[9] ≥ 0∧(UIncreasing(EVAL(-@z(x[8], 1@z), y[8])), ≥)∧0 ≥ 0)
- EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
- ((UIncreasing(COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0)
The constraints for P> respective Pbound 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 Pbound.
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(TRUE) = -1
POL(&&(x1, x2)) = -1
POL(EVAL(x1, x2)) = 1 + x1
POL(COND_EVAL4(x1, x2, x3)) = -1 + x2 + (-1)x1
POL(FALSE) = 1
POL(1@z) = 1
POL(undefined) = -1
POL(>@z(x1, x2)) = -1
The following pairs are in P>:
EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
The following pairs are in Pbound:
COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
The following pairs are in P≥:
COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
At least the following rules have been oriented under context sensitive arithmetic replacement:
FALSE1 → &&(FALSE, FALSE)1
-@z1 ↔
&&(TRUE, TRUE)1 ↔ TRUE1
FALSE1 → &&(FALSE, TRUE)1
FALSE1 → &&(TRUE, FALSE)1
↳ ITRS
↳ ITRStoIDPProof
↳ IDP
↳ UsableRulesProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ 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:
(9): EVAL(x[9], y[9]) → COND_EVAL4(&&(>@z(y[9], 0@z), >@z(x[9], 0@z)), x[9], y[9])
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, 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
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ AND
↳ IDP
↳ IDP
↳ IDependencyGraphProof
↳ IDP
↳ IDPNonInfProof
↳ IDP
↳ IDependencyGraphProof
↳ 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:
(8): COND_EVAL4(TRUE, x[8], y[8]) → EVAL(-@z(x[8], 1@z), y[8])
The set Q consists of the following terms:
Cond_eval1(TRUE, x0, x1)
eval(x0, x1)
Cond_eval5(TRUE, x0, x1)
Cond_eval4(TRUE, x0, x1)
Cond_eval(TRUE, x0, x1)
Cond_eval2(TRUE, x0, x1)
Cond_eval3(TRUE, x0, x1)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.